Paper Title: How quantum and evolutionary algorithms can help each other: two examples.




[
    M. Amato^*
    August 5, 2024
==================

empty






§ INTRODUCTION


Quantum computing utilizes principles of quantum mechanics to perform certain computations faster and on/and larger scales than classical computers are capable of <cit.>. 
Quantum bits, or qubits, are key building blocks of quantum computers that enable the latter to be so advantageous in solving complex problems  <cit.>. 
The intersection of quantum computing and artificial intelligence (AI) is currently an area of active research that has the potential to revolutionize various fields and industry sectors. 
Quantum Machine Learning (ML) algorithms and, in particular, evolutionary algorithms, belong to one of the research directions that are actively explored now <cit.>. It is expected that the new results will help to speed up the optimization process further and improve the overall performance of computational algorithms. In particular, Evolutionary Quantum Computation (EQC) is an emerging approach that blends evolutionary algorithms and quantum computing to solve complex optimization problems <cit.>. 


The aim of this work is twofold. First, we would like to show how quantum algorithms can be used to solve specific problems related to the engineering of evolutionary algorithms. Next, in a complimentary manner, we want to illustrate how evolutionary algorithms can be used to design quantum circuits fulfilling specific requirements.
In particular, we will address the problem of designing quantum circuits that
process initial product states into
highly entangled states.

 
Entanglement, one of the key concepts of quantum physics, also plays a crucial role in quantum information processing <cit.>. The term 'entangled state' refers to a state of several quantum objects that are intrinsically correlated with each other, in a way different from all possible classical correlations <cit.>. F.e.,  when two qubits are entangled, changes in the state of one qubit affect the state of the other one. There are several measures proposed to quantify the strength of these essentially quantum correlations, and so a quantitative definition of the word 'highly,' strictly speaking, depends on the chosen measure. Highly entangled quantum systems are usually difficult to simulate on classical computers <cit.>, and this fact has significant implications for quantum information processing technologies. Metaphorically speaking, highly entangled qubit states serve as fuel to quantum information processing and without them, the latter loses its superiority with respect to the classical processing <cit.>.
To create a highly entangled state out of a non-entangled one is not an easy task. And even if we have managed to prepare a maximally entangled state, it can become less entangled just because the quantum system prepared in this state is interacting with the environment <cit.>. All real quantum systems suffer from this interaction whose action is conventionally described with the term "decoherence". Therefore, it is important to keep the time during which decoherence strikes the system as short as possible. Practically, this means to minimize the computation time. In addition, every time a quantum gate (that is a transformation of the state of a system of qubits) is implemented, an additional error occurs due to the non-ideal character of the implementation. After passing through many gates, the state
is no longer reliable in the sense that it is very different from what it has to be according to the corresponding quantum program. Thus, the more complicated the quantum circuit is (in terms of the number of gates it consists of), the stronger the joint action of decoherence and accumulated implementation errors <cit.>. As has been mentioned above, not all entangled states are equal in their 'strength'. This means that we need to quantify and measure the amount of entanglement between different qubits. This brings another challenge, because, in a generic quantum system, the quantification of entanglement is a non-trivial task  <cit.>.


Our study addresses two specific problems, which are discussed in Section <ref>. The first problem, addressed in  Subsection <ref>, involves realizing classical stochastic cellular automata using quantum circuits. This approach is innovative as it leverages quantum mechanics to enhance the modeling of complex systems, a topic of extensive interest 

in prior research <cit.>. The second problem, discussed in Subsection <ref>, focuses on designing quantum circuits that generate highly entangled multi-qubit states. This is, up to our knowledge, a new application of evolutionary algorithms: to optimize entanglement in quantum circuits. To this end we use the Meyer-Wallach entanglement measure as the fitness function. 

In Section <ref> we summarize the results and discuss further perspective. Finally, in Section <ref>, we describe the methods used and highlight the relevant concepts.











§ RESULTS



 §.§ Realizing stochastic cellular automata with quantum circuits






In this part, we use an evolutionary algorithm to realize cellular automata (CA) for specific rules, deterministic and stochastic.
The CAs are realized with quantum 
circuits and measurements. The circuits are constructed by assembling gates from a fixed set of five gates.  Our approach involves a mutation-based evolutionary algorithm, enabling the optimization of gate types and their placements.

Initially, a predetermined number of chromosomes is generated, corresponding to the number of quantum gates per circuit. Through iterative refinement, the genetic algorithm evolves the existing chromosomes into new variations, singling out the fittest chromosome within each generation to serve as the parent for the subsequent generation. The assessment of the performance of the derived quantum circuits revolves around the measurement of the first qubit,  q_0. This assessment entails comparing the measured probability of an initial state, computed from the population, with the corresponding target probability. To quantify the difference between two sets of probability distributions,  we use the Kullback-Leibler divergence (KL) as a fitness function. The KL-fitness score measures the discrepancy between the distributions of the output states and the target states. In our case, these distributions are discrete, each corresponding to the eight different initial states of the three qubits. If the distributions completely match, the fitness function value is zero. A decreasing fitness value close to zero indicates an increase in the efficiency of the circuits.




The resulting quantum circuits are evaluated through measurements, thereby demonstrating the algorithm's efficacy. For more details, see Supplementary Note I.

    







We consider three different types of quantum CAs, starting with a stochastic critical cellular automaton, followed by more general stochastic CA (random updates) and a few deterministic rules, namely, rule 90 and 110  <cit.>. The updates for each type of CA are shown in Table <ref>. To enhance the robustness of our method, we limit the number of chromosomes by  N_c=20 chromosomes and the number of generations by N_g=500. Simulations are then performed for N_ic=50 randomly chosen initial conditions. This approach builds upon our previous experiment, where we set N_c=20 chromosomes, N_g=150 generations, and N_ic=20 initial conditions were used <cit.>. 













The authors in <cit.> have evolved a stochastic cellular automata model to reach criticality, a property of dynamical systems that allows them to do robust computations.
For each triad pattern, a probability has been calculated through a genetic algorithm for the central cell to have state 1. These probabilities are shown in Table <ref>, second column. 

We start by considering the number of gates used, evolving quantum CAs with 3, 5, 10, 15, and 20 gates and 50 runs. The mutation probability is fixed at 10 percent. The results are presented in Figure <ref>. They show that the best fitness score per run improves with increasing the number of gates up to 15 gates, after which a gradual increase is observed until the limit of 20 gates. Therefore, for experiments 2 and 3, 15 gates are used. 
Furthermore, the results of experiments indicate that, while fitness scores initially improve with an increasing number of gates, the rate of progress diminishes as the number of gates continues to increase. 


Next, we explore how the fitness changes with the value of the mutation probability. The goal here is to test the impact of the mutation on the fitness function of the different generations. We fixed the number of gates to 15 and used different mutation probabilities,  5%, 10%, 20%, 30%, and 50%.

Figure <ref> illustrates the impact of different mutation rates on the KL-fitness score. The left panel of figure <ref> shows that all mutation rates result in a decrease in fitness scores over generations, indicating improvement. However, mutation rates of 5%, 10%, and 20% stabilize more consistently compared to higher rates. The right panel box plots show that while higher mutation rates (30%, 50%) lead to decreased fitness scores, they introduce greater variability and less stability. These results highlight that while higher mutation rates can still improve circuit efficiency, the optimal balance between exploration and stability is achieved at a 20% mutation rate.



We now test the above approach with generic deterministic and stochastic cellular CAs. We fixed parameters, by utilizing 15 gates and a 10% mutation probability. Specifically, we examine the application of our methodology to Rule 90 and Rule 110 (deterministic rules with fixed probabilities of 0 or 1 for the eight neighborhoods), as well as to eight randomly generated probabilities, each repeated three times, serving as target probabilities. The corresponding target probabilities for each experimental condition are detailed in   Table <ref>, while the ensuing outcomes are visually represented in Figure <ref>. This comprehensive analysis showcases the versatility and effectiveness of our approach across a spectrum of rule types and probabilistic settings.


The fitness scores obtained for the deterministic CAs, Rule 90 and Rule 110, are remarkably good, as shown in Figure <ref> (left). This highlights the efficiency of our approach to the synthesis of quantum circuits for deterministic CAs. Notably, the outcomes stemming from the stochastic CA approach, characterized by the incorporation of randomly generated probabilities over three separate iterations, exhibit some promise. The optimal fitness scores attained for Random1, Random2, and Random3 are 0.3364, 0.1681, and 0.0014, respectively. It is worth mentioning that while Random1 achieves a commendable fitness score of 0.3364, indicating its promising performance, there is potential for further refinement of the evolutionary process to reduce any existing disparities. A comprehensive evaluation of the fitness scores across different probability sets, as presented in Figure <ref> (right), to the number of generations, provides valuable insights into the performance trends for the deterministic CA models (Rule 90 and Rule 110), as well as the stochastic CA model utilizing randomly generated probabilities over three iterations (Random1, Random2, and Random3), all conducted with an initial set of 50 run conditions.

Figure <ref> (Left) illustrates the fitness scores corresponding to the diverse CA rules. The best fitness score of 0.294 was achieved by the critical CA. Notably, for two deterministic CA rules, namely Rule 90 and Rule 110, the fitness scores approximate zero, reflecting a deliberate pursuit of minimizing the absolute disparity between initial and final states. This trend aligns logically with the deterministic nature of these CA rules, where probability values exclusively assume binary states of 0 or 1. In contrast, the scenario changes when dealing with randomly generated stochastic CA rules. In this context, the fitness scores demonstrate variability across three distinct randomly assigned probabilities. Intriguingly, among these variations, Random 3 emerges as the front-runner, boasting the most favorable fitness scores. Remarkably, all three sets of randomly generated probabilities exhibit fitness values superior to that of the critical stochastic CA value. Nevertheless, to unravel the underlying mechanisms driving these observed phenomena, a comprehensive exploration beckons. This entails conducting a more extensive array of experiments, encompassing increased generations and a higher replication count. The intricate interplay of factors contributing to fitness score disparities between distinct randomly generated probabilities necessitates a more exhaustive investigation.
 
In Figure <ref> (Right), we show the circuit 
that significantly outperforms the average outcomes of runs associated with the critical stochastic cellular automaton. Impressively, this particular realization achieved a fitness score of 0.294, as evaluated by the KL fitness function outlined in Equation (<ref>). 





 §.§ Designing entangling quantum circuits



In this section, we present the results obtained with an evolutionary algorithm as a tool to create a circuit 
that entangles qubits. The strength of the entanglement is quantified with the Mayer-Wallach measure which plays the role of a fitness function. We consider systems of 3, 4, and 5 qubits, and the performance of the algorithm is explored by varying mutation rates and numbers of gates. We ran experiments for 500 generations, repeating it 50 times. The outcomes are shown as the average fitness along with the standard error, and the best fitness achieved, also with its standard error. 






  §.§.§ Three-qubit circuits

We start with a three-qubit case. 

Figure <ref> presents the evolutionary optimization of a three-qubit quantum circuit across 500 generations, highlighting both average and peak fitness scores. We use three different mutation rates, 5%, 10%, and 15%, and limit the number of gates in the circuits to  3, 5, 8, 10, and 12. The best fitness scores obtained for all mutation rates approached unity, indicating that the resulting circuits are indeed highly entangling. 

Since the best fitness scores for the quantum circuits approached unity, evolutionary algorithms indeed serve as a toolbox of methods to design highly entangling three-qubit circuits. For a three-qubit system (three lists of integer gates for each qubit), a mutation probability of 1/9 (11.11%) is needed to replace at least one individual population consisting of nine integer lists. Therefore, a 10% mutation rate, which can at least replace one individual population in each generation can be considered an optimal probability for the mutation rate. Experimentally, the optimal mutation rate was found to be 10%, which struck a balance between exploring different solutions in the search space and exploiting the best solutions as shown in Figure <ref> (Top: right). In addition, a lower mutation rate is capable of striking a balance between exploration and exploitation, leading to a higher average fitness score across all runs. However, the mutation rate of 10% is more effective in building and generating the circuit that performs best in each run due to the higher degree of exploration it allows for the entanglement.

We also varied the of gates in the quantum circuit in the range from 3 to 12 keeping the optimal mutation rate at 10%. The result is the fitness score decreased as the number of gates in the circuit increased as shown in Figure <ref> (Bottom: Left). These results suggest that as the quantum circuit becomes more complex with an increasing number of gates, the entanglement capability of the circuit is reduced, indicating that the circuit is less efficient at performing the desired task. This suggests that there may be a trade-off between circuit complexity and performance. Therefore choosing the optimal number of gates can provide the best balance between the complexity of the quantum circuit and the performance.

The best three-qubit entangling circuit with three gates is shown in Figure  <ref> (Bottom: right). It corresponds to the fitness score of 0.999. Similarly, a three-qubit quantum circuit with 12 gates also achieved a fitness value of 0.999. In Supplementary Note II, the detailed analyses of the state vector and reduced density matrices from optimal circuits in a 500-generation are presented. Table 1 in Supplementary Note shows these details for the state vector probability of the respective density matrices. The methodologies used for calculating the probabilities of quantum states and the reduced density matrices for the three-qubit circuits, along with an assessment of their entanglement using the Mayer-Wallach entanglement measure, are given in Supplementary Note III.





  §.§.§ Four and five-qubit circuits




In this section, we extended the designing of quantum circuits to advance our findings to include both four and five-qubit systems. The findings are shown in Figures <ref> and <ref> for four and five qubit quantum circuits, respectively. Notably, Figures (Top: Left) <ref> and <ref> show the mean fitness (illustrated by a green line) alongside its standard error (denoted by a shaded area), in addition to the average of the best fitness (represented by a red line) and its standard error (also shaded). A third-order polynomial fit applied to the mean fitness is shown by a blue dashed line. These figures representations demonstrate that the average fitness score approximates 0.6, while the mean of the highest fitness score approaches 0.8.

Further experiment with an optimal mutation probability of 10% and 5% for four and five qubit circuits, respectively, was conducted focusing on circuits with varying numbers of gates as shown in Figures (Top: right) <ref> and <ref>. Through an analysis of mutation rates of 3%, 5%, and 15%, we found that increasing the mutation rate beyond 5% results in a decrease in the average fitness value. This indicates that excessive randomness in the evolutionary process can have a negative impact on circuit performance. However, it was also observed that the average of the best fitness value was not affected by higher mutation rates. This highlights the importance of selecting an appropriate mutation rate for obtaining the quantum circuits of maximal entanglement.

The visualization of the quantum circuit with four and five qubits incorporating four and five gates, respectively, achieving a fitness score of 0.999, is shown in Figures (Bottom) <ref> and <ref>. Additionally, the five-qubit quantum circuit with 12 gates yielded a fitness score of 0.199, signifying a decrement in entanglement capabilities with an increase in the number of gates. Detailed outcomes for the most entangled four and five-qubit quantum circuits, including the state vector and reduced density matrices, are elaborated in Section II of the Supplementary Note. These results pertain to the optimal five-qubit circuit identified across 500 generations, encompassing circuits with both five and twelve gates, as illustrated in Tables 2 and 3 (Section II of the Supplementary Note).











Figure <ref> (Top: left) presents the results obtained. The plot shows the mean fitness (green line) and its shaded standard error, as well as the mean of the best fitness (red line) and its shaded standard error, against the number of generations. The blue dashed line represents the third-order polynomial fit to the mean fitness. From the figure it is clear that the mean fitness score is near 0.6, while the mean of the best fitness score is close to 0.8. The results indicates that circuits generated with the best fitness score have a very high degree of entanglement among the qubits in each generation of the evolutionary process.

The experiment with an optimal muttation probability of 10% for four-qubit circuits was repeated for circuits with varying numbers of gates as shown in Figure <ref> (Top: right).  It clear from the figure that with increasing number of gates in quantum circuits, the fitness score decreases. It is because increasing the number of gates in quantum circuits increases complexity of the circuit and hence drop of entanglement capability of the circuits. Figure <ref> (Bottom) presents a four-qubit circuit with four gates and a fitness score 0.9996.

The reduced density matrix (ρ_k) of a generated quantum circuit with state vector and a specific number of qubits (n-qubits) is computed by tracing out all qubits except for the k^th qubit. For four qubit system, k ranges over 0, 1, 2, 3. The resulting output is four 2 × 2 reduced density matrices. The calculated reduced density matrix and the respective state vector of four qubit quantum circuit generated bu the evolutionary algorithms are shown in Section II of the Supplementary Note, specifically to Table 2. In additional, the details of the mathematical formulation of the reduced density matrix and its role as an analytical tool in evaluating the entangled states is provided in Section III of the Supplementary Note.






  §.§.§ Five-qubit circuits




Figure <ref> (Top: Left) shows the results obtained for five-qubit circuits with five gates. The plot shows the average fitness, third-order polynomial fit to the average fitness, and the best fitness values. For five qubit system also, we explored the impact of different mutation rates (3%, 5%, and 15%) on the generated quantum circuits. The best fitness scores obtained for the optimal mutation rate of 3% were close to 0.8, with an target fitness value of one. This indicate a high degree of entanglement between the qubits for a system of five qubit quantum circuits. Also, the third-order polynomial fit to the average fitness shows a smooth curve that approximates the general trend of the average fitness scores.

Among the explored mutation rate of 3%, 5%, and 15%, we found that increasing the mutation rate beyond 5% results in a decrease in the average fitness value. This  indicates that excessive randomness in the evolutionary process can have a negative impact on circuit performance. However, it was also observed that the average of the best fitness value was not affected by higher mutation rates. This highlight the importance of selecting an appropriate mutation rate for obtaining the quantum circuits of maximal entanglement. Therefore, to determine the optimal mutation rate, we calculated the probability of replacing at least one individual population, consisting of 15 integer lists (three lists of integer gates for each of the 5 qubits). The probability of this occurring is 6.67%. So, mutation rate of 5% (close to 6.67%) would be an optimal and it would be sufficient to replace at least one individual population in each generation.

Next, we keep the mutation rate constant at 5% (optimal to replace at least one individual in each generation) and vary the number of gates. The results, as shown in Figure <ref> (Top: right), indicate that the fitness score is maximal or close to 1 for the 3-gate 5-qubit circuit. However, as the number of gates increases, the fitness scores decrease continuously resulting in a reduction of the entangling power of the circuits. This means that  a simpler quantum circuit with fewer gates can have better entanglement capability than a more complex circuit with higher number of gates.

The visualization of five-qubit quantum circuit consisting of five gates with a fitness score of 0.9996 is shown in Figure <ref> (Bottom). Also, the five-qubit quantum circuit with 12 gates yielded a fitness score of 0.199, indicating that increasing gates in a circuit decreases in entanglement capability. Section II of the Supplementary Note presents detailed results for the best entangled 5-qubit quantum circuit, including state vector and reduced density matrices. This results are from the best or optimal five qubit circuit among 500 generations, featuring circuits with both five and twelve gates (see Table 3).






  §.§.§ The von Neumann Entropy as a Fitness Function


In our experimental setup, we also incorporated von Neumann entropy within an evolutionary algorithm framework. The algorithm was configured for a three-qubit system equipped with ten gates, operating at a mutation rate of 10%. This procedure was iteratively performed across 50 cycles, spanning 500 generations, with each generation consisting of 20 chromosomes. The results of this implementation, illustrating the effectiveness of von Neumann entropy in the optimization process, are graphically represented in Figure <ref>.

The theoretical maximum entropy for such a system is 2.999. Nevertheless, when implementing von Neumann entropy entanglement as a fitness function in the evolutionary algorithm, we obtained a maximum entropy value of 1.05557. This result has significant implications for the entanglement within the system. The quantification of entanglement can be achieved through the computation of von Neumann entropy for the reduced density matrix of any given subsystem. 

Our findings indicate that the implementation of von Neumann entropy in the three-qubit system successfully detected the presence of entanglement within the evolved three-qubit quantum circuit. This is consistent with the understanding that the entropy of entanglement is the von Neumann entropy of the reduced density matrix for any of the subsystems.







§ DISCUSSION AND CONCLUSIONS




In this study, we explore the two examples of evolutionary algorithms in two distinct yet interconnected areas of quantum computing. Initially, we introduce a framework for generating quantum circuits that replicate specific cellular automata rules using genetic algorithms <cit.>. The implementation code is available at GitHub repository <https://github.com/Overskott/Quevo>. This framework effectively evolves quantum circuits for various CA rules, from deterministic to stochastic types. We improved our original framework by increasing the number of generations and runs, enhancing performance and more consistent fitness function outcomes. 

This initial investigation set the stage for more nuanced applications of evolutionary algorithms in quantum circuit design, suggesting future research paths.

To conclude,
building upon this groundwork, we have demonstrated the effectiveness of evolutionary algorithms in generating highly entangled quantum circuits, emphasizing their role in calculating reduced density matrices for three-, four-, and five-qubit systems. These circuits displayed a significant degree of entanglement, measured by the Mayer-Wallach entanglement measure. By tracing out all other qubits, the study illustrated how reduced density matrices could reveal detailed information about the circuit's entanglement. In this context, the algorithm's role was to optimize quantum gates and their connectivity, leading to the creation of quantum circuits. 

This contribution is, as far as we have checked, novel in its approach to optimizing quantum circuits for entanglement.
Moreover, the simulator provided by Qiskit enabled the evaluation of these circuits, gates, and connections, showcasing the potential of this approach in solving complex entanglement optimization problems. Furthermore, we highlighted the importance of circuit depth in determining the entanglement capability of a quantum circuit, and we found that increasing the circuit depth decreases its entanglement capability. 

Another important conclusion is that, an optimal mutation rate was identified for quantum circuits with different qubits and gates, marking a significant stride in optimizing quantum circuits for entanglement generation.
To note that the KL-fitness score measures the discrepancy between the distributions of the output states and the target states. In our case, these distributions are discrete, each corresponding to the eight different initial states of the three qubits. If the distributions completely match the fitness function, the KL-value is zero. A decreasing fitness value close to zero indicates an increase in the efficiency/entanglement of the circuits.
Our findings show that with mutation rates up to 20%, the optimization process effectively explores the solution space, leading to better circuit designs with lower fitness scores. However, mutation rates beyond 20% introduce excessive randomness, which disrupts the optimization process and results in higher fitness scores, indicating less efficient/entangled circuits.

All mutation rates result in a decrease in fitness scores over generations, indicating improvement. However, mutation rates of 5%, 10%, and 20% stabilize more consistently compared to higher rates. The right panel box plots show that while higher mutation rates (30%, 50%) lead to decreased fitness scores, they introduce greater variability and less stability. These results highlight that while higher mutation rates can still improve circuit efficiency, the optimal balance between exploration and stability during the "evolution" of the algorithm is achieved at a 20% mutation rate.

Finally, some last points for discussion. This paper opens avenues for future investigations into advanced evolutionary algorithms and alternative fitness functions, such as Von Neumann entropy <cit.> and Schmidt measures <cit.>, to maximize entanglement in more complex quantum circuits. 
Future work could explore the scalability of these methods to larger quantum circuits, composed by more than five qubits. 
The fitness function here introduced, measuring the entanglement between qubits could be extended to multi-dimensional optimization scheme, incorporating also measures of error and noise in quantum circuits. 
Moreover, it would also be of interest to adapt the method here proposed and integrate it in hybrid optimization approaches.
Future research could explore the maximization of entanglement in quantum circuits, drawing inspiration from recent advancements in benchmarking highly entangled states on a 60-atom analog quantum simulator <cit.> and leveraging non-locality as a resource in quantum computing demonstrated by loophole-free Bell inequality violations with superconducting circuits <cit.>.
All in all, applying optimized quantum circuits to real-world problems in quantum cryptography <cit.>, quantum simulation <cit.>, and quantum machine learning <cit.>, as well as testing them on actual quantum hardware, could further demonstrate their practical utility and effectiveness <cit.>.







§ METHODS



Our study implements two distinct methods for designing quantum circuits. The first method utilizes cellular automata rules combined with the KL fitness function. Here we advanced the original framework from Ref. <cit.> for generating quantum circuits that replicate quantum circuits with evolutionary algorithms. Upon this groundwork the second method focuses on designing quantum circuits for maximum entanglement, using the Mayer-Wallach measure and von Neumann entropy for entanglement evaluation. The comprehensive details regarding the implementation of evolutionary quantum algorithms for both methodologies are thoroughly delineated in Supplementary Note I.




 §.§ Evolutionary algorithms: Mutations and fitness function
 

The evolutionary algorithm initiates with a specific assigned number of chromosomes, each populated with a set number of quantum gates within the circuit. This initial population is then subject to evolution, where the fittest chromosomes are identified, iteratively refining the existing ones, and introducing the best-fit chromosome at the forefront of each new generation. This evolutionary cycle continued for a predetermined number of generations, with each generation's best chromosome becoming the parent for the next. Throughout this process, the chromosomes are subjected to mutations to enhance the optimization. The mutation process is introduced in two different forms: either by swapping existing gates in the chromosome with randomly selected ones from the gate pool, or by entirely replacing the chromosome to identify the best four parental candidates suitable for the next generation. The mutation process is controlled by a set of probabilities. The four best chromosomes are preserved as "elites" without changes, whereas the remaining chromosomes are selected for evolution based on their fitness, with selection changes directly linked to their performance.

In this context, we considered one-dimensional CAs where each of the cells can be in one of two possible states, 0 or 1 (Boolean CA), The evolution of these CA is governed by a set of rules, precisely 256 possible ones, that determine how the state of a cell updates at each step based on the states of its immediate neighbors. The rules correspond to each of the eight possible configurations of a cell and its two immediate neighbors, namely [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], and [1, 1, 1]. The rule dictates the next state of the central cell in these configurations. Table <ref> shows the matching in all cases considered in this paper. 

To evaluate the performance of the quantum circuits, we measure only the first qubit (q0) and consider
the fitness function as the Kullback-Leibler (KL) divergence, which measures the difference between two probability distributions and has been used in other works <cit.> as a fitness function:

    D_KL(P||Q)= ∑_ω∈Ω P(ω)log(P(ω)/Q(ω)).


In our case, the distributions are discrete, and each one has eight different values related to the eight different initial states of the three qubits. If the distributions are a perfect match, then the fitness function is zero. 



 §.§ Assessing entanglement in basic quantum states of qubits


Consider, for the start, the state of a system of two-qubit configurations of a quantum system. The most general form of a state for such a system can be written as: 

    |ψ⟩ = α_00|00⟩+α_01|01⟩+α_10|10⟩+α_11|11⟩   ,

where α_00, α_01, α_10, and α_11 are complex amplitudes of the corresponding basis states. The normalization of the basis state is

    |α_00|^2 + |α_01|^2+ |α_10|^2 + |α_11|^2 = ∑_X∈{0,1}^2|α_X|^2 = 1  ,

where X∈{0,1}^2 means all binary combinations of length 2.
For a two-qubit state, there is one fully entangled state given by <cit.>

    |ψ⟩ = 1/√(2) ( |00⟩+|11⟩ )   .

This state exemplifies quantum entanglement where the measurement outcome of one qubit directly determines the outcome of the other, showcasing the non-classical correlation inherent in quantum systems <cit.>.
To contrast, consider the following non-entangled state:

    |ψ⟩ = 1/2|00⟩+1/2|01⟩+1/2|10⟩+1/2|11⟩.

In this state, when measuring one qubit, there is no certainty about the state of the other qubit. Namely, there is a probability of 1/2 that the other qubit is in state 0 and probability 1/2 that it is in state 1.

In general, suppose we have two quantum systems Q_1 and Q_2. These systems are said to be entangled if the values of certain properties of system Q_1 (accessed with measurement) are correlated with the values that those properties will assume for system Q_2. The simplest forms of entangled state are known as the Bell states <cit.>, which enable tasks such as quantum cryptography<cit.>, super-dense coding<cit.>, teleportation<cit.>, and entanglement swapping <cit.>. 

The descriptors, expressibility, and entangling capability have been used to study the capabilities of the parametrized quantum circuit by quantifying its deviation from random circuits to approach the research question of how much generalization is effective enough in a quantum circuit for a given task <cit.>. The expressibility of a quantum circuit is the ability to generate pure states that are well representative of the Hilbert space <cit.>. In a single qubit, the expressibility corresponds to the circuit's ability to explore the Bloch sphere. Sim et al. <cit.> propose to quantify the ability of the quantum circuit to generate a pure state as a representative of Hilbert space by comparing the true distribution of the fidelities corresponding to the parameterized quantum circuit (PQC),  to the distribution of fidelities from the ensemble of Haar random states. 

For example, the measure of expressibility can be calculated by taking the Kullback-Leibner divergence between the estimated fidelity distribution and that of the Haar-distributed ensemble <cit.>. 
In this paper, we considered two measures, the Meyer-Wallach (MW)<cit.> and the entanglement entropy (EE)<cit.>.
The Meyer-Wallach entanglement measure is popular because of its scalability and ease of computation.
It considers a pure state of n qubits if the form <cit.>:


    Q(|ψ⟩) = 4/n∑_j=1^n D(|û^k⟩, |v̂^k⟩) ,

where |û^k⟩ and |v̂^k⟩ denote non-normalized vectors belonging to the complex vector space 𝐂^2𝐧-2, where n denoting the number of qubits in the quantum system. These vectors are obtained by projecting the pure state |ψ⟩ of the n-qubit system onto the local basis states of the k^th qubit, where k ∈1, 2, ..., n. The symbol "" above the vectors signifies their non-normalized status.
The function D(|û^k⟩, |v̂^k⟩) measures a distance between the two vectors |û^k⟩ and |v̂^k⟩, measured through the generalized cross-product: 

    D(|û^k⟩, |v̂^k⟩) = ∑_i<j|û_i^kv̂_j^k-û_j^kv̂_i^k|^2   .


Since the purity of the state of qubit k is given by Tr[ρ_k^2]=⟨x̂^k|x̂^k|^⟩2+⟨ŷ^k|ŷ^k|^⟩2, we obtain <cit.>  


    D(|x̂^k⟩, |ŷ^k⟩) = ∑_i<j|û_i^kv̂_j^k-û_j^kv̂_i^k|^2

It is because generalized cross product under logical unitaries, D(|û^k⟩, |v̂^k⟩)= D(|x̂^k⟩, |ŷ^k⟩) in relation to the norm of an anti-symmetric tensor M_k = |x̂^k⟩⟨ŷ^*k| - |ŷ^k⟩⟨x̂^*k|. Therefore, one arrives at an entanglement measure given by:

    Q(|ψ⟩)= 2  ( 
                         1-1/n∑_k=0^n-1Tr[ρ_k^2]
                          ),

which is the usual definition of the Meyer-Wallach measure. A detailed derivation and discussion of this measure, especially its implications for both pure and mixed states, are provided in Supplementary Note IV.


The entanglement entropy quantifies the quantum entanglement among two subsystems within a larger quantum system. Specifically, for a pure bipartite quantum state of the mixed system, it is feasible to derive a reduced density matrix that characterizes the state of one of the subsystems <cit.>. The entropy of entanglement is defined as the von Neumann entropy of the reduced density matrix for either of the subsystems <cit.>.

The von Neumann entropy quantifies the uncertainty or randomness in a quantum state and measures the degree of entanglement between the subsystems <cit.>. If the entropy of entanglement is non-zero, it implies that the subsystem is in a mixed state, and consequently, the two subsystems are entangled <cit.>. Therefore, the entropy of entanglement offers a useful method for quantifying the degree of entanglement between the subsystems of a composite quantum system.

The von Neumann entropy of entanglement is mathematically defined as the von Neumann entropy of the reduced density matrix for a subsystem within a composite quantum system <cit.>, namely

    S(ρ_A) = -Tr(ρ_A log_2 ρ_A)   ,

where Tr represents the trace operation, and log_2 denotes the base-2 logarithm. The von Neumann entropy of entanglement provides a measure of the uncertainty or randomness in the state of subsystem A. Thus, it serves as an indicator of the degree of entanglement between the subsystems <cit.>. When dealing with a composite quantum system that's in a mixed state, the von Neumann entropy of entanglement measures the least amount of entanglement present across all possible pure-state decompositions of the mixed state. This method allows for measuring entanglement in a mixed state by considering the representation of the composite system that has minimal entanglement.

In what concerns implementation, we have used the mathematical formulations presented in Equations (<ref>) and (<ref>) in a framework previously developed (<https://github.com/Overskott/Quevo>). It is further aligned with the methodologies outlined in Ref.<cit.>. The evolutionary algorithms (EAs) have emerged as a promising tool for quantum circuit design <cit.>. So our implementation of EAs for quantum circuit design aims to maximize entanglement using two specified methods. The framework employs an evolutionary algorithm to automatically generate quantum circuits that satisfy defined properties specified through a fitness function. The evolution properties can be controlled by tuning parameters such as initial population, number of generations, probability of mutation operator, number of gates in the circuit, and the chosen fitness function.  Details of the implementation process are available in Supplementary Note III.



§ DATA AVAILABILITY

The datasets used and/or analyzed during the current study are available in the QUEVO1 repository, <https://github.com/shailendrabhandari/QUEVO1>. 








§ ACKNOWLEDGEMENTS


We extend our gratitude to Sebastian T. Overskott for making the https://github.com/Overskott/QuevoQuevo framework available, which we have adapted for the entanglement assessment in our study. 



§ AUTHOR CONTRIBUTIONS STATEMENT


S.B. conducted the experiments and performed the simulations. All authors have analyzed the results and written the manuscript. S.N., S.D., and P.G.L. have revised the text and P.G.L. coordinated the overall project.



§ SUPPLEMENTARY NOTE I: IMPLEMENTATION OF AN EVOLUTIONARY QUANTUM ALGORITHM



Our approach was implemented in Python and resulted in a Python module called https://github.com/Overskott/QuevoQUEVO. This module consists of three classes: , , and . The  and the  classes are part of the genetic algorithm, while the  class is responsible for generating and simulating quantum circuits.





The  class is the core of the genetic algorithm used for generating quantum circuits. It handles the integer representation of the gates and their connections. The class contains a list of integers and functions to generate and mutate the list. It also handles the initialization of the population and the evolution of the population into a new one. Mutations in the  class can occur in two different ways: replacing gates from the pool of gates in the chromosome with a randomly generated new one or replacing the chromosome to generate the four best parents. The class is also responsible for checking the chromosome for gates that connect multiple qubits. If the gate has an invalid connection, meaning that it is connected to itself through the randomly generated integers, the class generates a valid configuration randomly.

The  class is another important class in the QUEVO module, responsible for managing the population of chromosomes. It is a collection of chromosomes that undergo evolution for a specified number of generations. After each evolution step, changes in the chromosomes in the generation occur by selecting a fixed number of chromosomes as elite and allowing the rest of them to evolve further. In each evolution step, the chromosomes are evaluated with the fitness function, and the fittest chromosomes become the parents for the next generation. The rest of the chromosomes are reset for the initial chromosomes. The  class stores a generation of chromosomes, the fitness associated with each chromosome, methods for running and retrieving fitness for two different fitness functions, and functions for printing. It provides methods for performing selection, crossover, and mutation to generate a new population of chromosomes. 

The  class within the QUEVO module, which utilizes IBM's Qiskit (<https://qiskit.org/>), plays a crucial role in generating and managing quantum circuits. It is designed to create Qiskit quantum circuits from string representations, simulate these circuits, perform measurements, and visualize the results. Operated on the Qiskit AER simulator, this class yields results in a dictionary format. It can also emulate an IBMQ backend through the AER simulator, allowing the simulator to use custom quantum gates, basis gates, and coupling maps specific to that backend.

Qiskit, a Python package provided by IBM, is integral to creating and simulating quantum circuits in QUEVO. It offers tools for writing quantum programs and can simulate quantum circuits locally or run them on actual quantum computers via the cloud. To ensure seamless integration and operation of Qiskit within the QUEVO framework, a dedicated Python virtual environment is recommended, preferably set up using Anaconda. 

The  method implements the Ma­yer-­Wa­llach measure of entanglement, as a fitness function of the EA. This method processes a state vector representing a quantum circuit's state, converting it into a 2^n x 2^n tensor, where n is the circuit's qubit count. The entanglement for each qubit k is determined by tracing out all other qubits, summing the squared eigenvalues of their respective reduced density matrices. The results are then normalized by 1 - 1/n, ensuring values fall between 0 and 1. More information and method usage are detailed in the QUEVO1 GitHub repository, accessible at <https://github.com/shailendrabhandari/QUEVO1>. Besides the well-known Mayer-Wallach measure, we evaluate the entanglement properties of quantum states using the von Neumann entropy as fitness functions. EA creates random density matrices from complex state vectors and calculates the von Neumann entropy based on their eigenvalues. This dual approach in measuring entanglement with Mayer-Wallach and von Neumann entropy enhances the quantum circuits' entanglement characteristics, optimized via quantum evolutionary algorithms.




§ SUPPLEMENTARY NOTE II: STATE VECTOR PROBABILITIES AND DENSITY MATRICES FOR MULTI-QUBIT CIRCUITS



In quantum systems, probabilities of states are derived from state vector coefficients within the wave function, typically complex numbers of the form a + bi. The probability P for a state with coefficient c = a + bi is given by:

    P = |c|^2 = a^2 + b^2,

where a represents the real part and b the imaginary part of the coefficient. Following the computation of reduced density matrices for all qubits, the entanglement of the circuit is assessed using the Meyer-Wallach entanglement measure.

This appendix explicitly presents maximally entangled states for three, four, and five qubits. Results are systematically organized in tables, each corresponding to distinct quantum circuit configurations. Table <ref> illustrates the outcomes for a three-qubit system, with reduced density matrices for circuits comprising 3 and 12 gates (Left), and their associated quantum state probabilities (Right). Table <ref> explores a four-qubit system, detailing both reduced density matrices and state probabilities for four-gate circuits. Finally, Table <ref> showcases results for a five-qubit system, encompassing reduced density matrices for circuits with five and twelve gates (Top), alongside the state probabilities (Bottom). These tables collectively provide a comprehensive view of quantum state entanglement across varying circuit complexities and gate counts.
















§ SUPPLEMENTARY NOTE III: REDUCED DENSITY MATRIX AS A TOOL FOR UNDERSTANDING THE PURITY OF A STATE



The density matrix is a representation of a quantum state that can describe both pure states and mixed states. While the state-vector representation is only suitable for describing pure states, the density matrix can also be used to represent mixed states, which are probabilistic mixtures of pure states. A mixed state can be expressed as a sum of outer products of pure states, each term weighted by a probability. In the density matrix formalism, a mixed state is represented by a Hermitian matrix with non-negative eigenvalues that sum up to one.
The density operator representation is an alternative way to express pure quantum states using a matrix formalism. It is defined as:

    ρ≡ | ψ⟩⟨ψ | ,








where |ψ⟩ is the state vector representing the pure quantum state. The expression |ψ⟩⟨ψ| represents the outer product of the state |ψ⟩ with itself, 

    ρ = 
    [ α_0; α_1;   ⋮; α_N ][ α_0^* α_1^*     … α_N^* ] 
    =
    
    
    
    
    
    [   |α_0|^2 α_0 α_1^*         … α_0 α_N^*; α_1 α_0^*   |α_1|^2         … α_1 α_N^*;         ⋮         ⋮         ⋱         ⋮; α_N α_0^* α_N α_1^*         …   |α_N|^2 ].

The density operator has several important properties. It is Hermitian, which means that it is equal to its conjugate transpose. It has a trace equal to one, reflecting the fact that it represents a pure state. Furthermore, it also satisfies the property, ρ^2 = ρ, indicating the state is pure. In other words, the density matrix characterizes the pure state.
To illustrate this, let us consider an example of a two-qubit, maximally-entangled pure state |ψ_AB⟩, which can be expressed as:

    |ψ_AB⟩ = 1/√(2) ( |00⟩ + |11⟩ ) = 1/√(2)[ 1; 0; 0; 1 ].



Using the density operator representation, we can express this state as:

    ρ_AB  = 
    
    
    
     ( 1/√(2)[ 1; 0; 0; 1 ] )  ( 1/√(2)[ 1 0 0 1 ] )  = 
    
    
    
    1/2[ 1 0 0 1; 0 0 0 0; 0 0 0 0; 1 0 0 1;   ]  .

This matrix satisfies the properties of a density operator, namely, it is Hermitian, positive semi-definite, and has a trace equal to one. The density operator representation provides a compact and convenient way to represent pure states in a matrix formalism, which can be useful for calculations and quantum information processing tasks.

The reduced density matrix ρ_A describes the state of subsystem A after tracing out the degrees of freedom of subsystem B. It is obtained by performing a partial trace of the density matrix ρ_AB over subsystem B <cit.>:

    ρ_A = Tr_B(ρ_AB) = ∑_i=1^d_B⟨ i|ρ_AB|i⟩ ,

where d_B is the dimension of the Hilbert space of subsystem B, and |i⟩ denotes a basis state of subsystem B.




The reduced density matrix ρ_A inherits some of the properties of the original density matrix ρ_AB. For example, it is Hermitian, positive semi-definite, and has a trace equal to one. However, it is generally not pure, even if the original state ρ_AB is pure. This is because entanglement between subsystems cannot be removed by taking a partial trace.

Tr_B in Equation (<ref>) is an operation known as the partial trace and is used to extract the state of a subsystem of a composite system from its overall density matrix. The partial trace over subsystem B of a tensor product of two operators |ξ_u⟩⟨ξ_v| and |χ_u⟩⟨χ_v| is given by:

    Tr_B  (| ξ_u ⟩⟨ξ_v | ⊗ | χ_u ⟩⟨χ_v |  ) ≡ | ξ_u ⟩⟨ξ_v |  Tr ( | χ_u ⟩⟨χ_v |  ) ,

where | ξ_u ⟩ and | ξ_v ⟩ are arbitrary states in the subspace of A, and | χ_u ⟩ and | χ_v ⟩ arbitrary states in the subspace of B. Tr is the standard trace operation, which for two arbitrary states Tr ( | χ_u ⟩⟨χ_v |  ) = ⟨χ_v |χ_u ⟩. Similarly, the reduced density matrix of subsystem B can be obtained by taking the partial trace over subsystem A of the tensor product of two operators:

    Tr_A  (| ξ_u ⟩⟨ξ_v | ⊗ | χ_u ⟩⟨χ_v |  ) ≡Tr ( | ξ_u ⟩⟨ξ_v |  ) | χ_u ⟩⟨χ_v | .




As an example, let's reconsider the pure entangled state:
    | ψ_AB⟩ = 1/√(2) ( | 0_A 0_B ⟩ + | 1_A 1_B ⟩ ).

 

This system is then composed of single-qubit subsystem A with basis vectors { |ξ_1 ⟩, |ξ_2 ⟩} = { | 0_A ⟩, | 1_A ⟩}, and single-qubit subsystem B with basis vectors { |χ_1 ⟩, |χ_2 ⟩} = { | 0_B ⟩, | 1_B ⟩}. We know that this system is not separable (i.e., | χ_AB⟩≠ |χ_A⟩⊗ |χ_B⟩); however, by using the reduced density matrix, we can find a full description for subsystems A and B as follows.

The density matrix of our state | ψ_AB⟩ can be expressed in terms of outer products of the basis vectors as
ρ_AB = 
1/2 [ | 0_A 0_B ⟩⟨ 0_A 0_B | + | 0_A 0_B ⟩⟨ 1_A 1_B |  .
. +| 1_A 1_B ⟩⟨ 0_A 0_B | + | 1_A 1_B ⟩⟨ 1_A 1_B |  ].
To calculate the reduced density matrix for, let's say, subsystem B, we have:

    ρ_B   = Tr_A(ρ_AB)
       = 1/2 [ Tr_A(| 0_A 0_B ⟩⟨ 0_A 0_B |) + Tr_A(| 0_A 0_B ⟩⟨ 1_A 1_B |) + Tr_A(| 1_A 1_B ⟩⟨ 0_A 0_B |) + Tr_A(| 1_A 1_B ⟩⟨ 1_A 1_B |)  ] 
       = 1/2 [ Tr(| 0_A ⟩⟨ 0_A |)| 0_B ⟩⟨ 0_B | + Tr(| 0_A ⟩⟨ 1_A |)| 0_B ⟩⟨ 1_B | + Tr(| 1_A ⟩⟨ 0_A |) | 1_B ⟩⟨ 0_B | + Tr(| 1_A ⟩⟨ 1_A |) | 1_B ⟩⟨ 1_B |  ] 
       = 1/2 [ ⟨ 0_A | 0_A ⟩ | 0_B ⟩⟨ 0_B | + ⟨ 1_A | 0_A ⟩ | 0_B ⟩⟨ 1_B | + ⟨ 0_A | 1_A ⟩ | 1_B ⟩⟨ 0_B | + ⟨ 1_A | 1_A ⟩ | 1_B ⟩⟨ 1_B |  ] 
       = 1/2 [ | 0_B ⟩⟨ 0_B | + | 1_B ⟩⟨ 1_B |  ].



Starting with an entangled pure state |ψ_AB⟩, the reduced density matrix ρ_B of subsystem B emerges as a mixed state. This matrix ρ_B statistically represents subsystem B by averaging out measurements from subsystem A. Tracing out A yields ρ_B, reflecting the probability distribution of B. This approach is crucial for analyzing entangled states in complex systems, allowing insight into individual subsystems within entanglements. Key concepts and details about density matrices are sourced from <cit.>, especially the section "The density matrix and mixed states" (<https://qiskit.org/textbook/ch-quantum-hardware/density-matrix.html##4.-The-Reduced-Density-Matrix–>) 







§ SUPPLEMENTARY NOTE IV: EVALUATION OF QUANTUM PURITY AND THE MEYER-WALLACH ENTANGLEMENT MEASURE



In this appendix, a comprehensive derivation and analysis of the quantum purity of a state is provided, particularly focusing on its relation to the Meyer-Wallach measure. 
If the state |ψ⟩ is pure, then its density matrix is given by ρ = |ψ⟩⟨ψ|, and its purity yields:

    Tr[ρ^2] = Tr[(|ψ⟩⟨ψ|)(|ψ⟩⟨ψ|)] 
    = Tr[|ψ⟩⟨ψ|ψ|⟨%s|⟩ψ] = Tr[|ψ⟩⟨ψ|] = 1.


The given equation is an expression for the quantum purity of a state |ψ⟩, denoted by Q(|ψ⟩). The purity is a measure of how pure or mixed a quantum state is, and it is defined as the trace of the square of the density matrix ρ that represents the state.
For a pure state, the expression 1 - 1/n∑_k=0^n-1Tr[ρ_k^2] is equal to zero.
For a mixed state, the density matrix can be written as a convex combination of pure states, ρ = ∑_i p_i |ψ_i⟩⟨ψ_i|, where p_i are probabilities and ∑_i p_i = 1. The purity of such a mixed state can not exceed 1, as one can conclude from: 

    Tr[ρ^2]    = Tr[(∑_i p_i |ψ_i⟩⟨ψ_i|)(∑_j p_j |ψ_j⟩⟨ψ_j|)] 
       = ∑_i p_i^2 Tr[|ψ_i⟩⟨ψ_i|] + ∑_i≠ j p_ip_j Tr[|ψ_i⟩⟨ψ_j||ψ_j⟩⟨ψ_i|] 
       = ∑_i p_i^2 + ∑_i≠ j p_ip_j |⟨ψ_i|ψ_j||⟩^2 
       ≤∑_i p_i^2 + ∑_i≠ j p_ip_j = (∑_i p_i)^2 = 1   .


Therefore, the purity of a mixed state is always less than or equal to one. Using this result, we can see that the expression 1 - 1/n∑_k=0^n-1Tr[ρ_k^2] is a measure of how mixed the state is. If the state is pure, then this expression is zero, and if the state is entangled/mixed, then this expression is positive.




 Paper Title: Nonlinearity-induced dynamical self-organized twisted-bilayer lattices in Bose-Einstein condensates.





























#15pt#1

    GThese authors contribute equally to this work.Ministry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, ChinaThese authors contribute equally to this work.Ministry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, ChinaMinistry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, ChinaMinistry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, ChinaMinistry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, Chinalishuai0999@stu.xjtu.edu.cnMinistry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, Chinaliubophy@gmail.comMinistry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, China§ ABSTRACT 
        Creating crystal bilayers twisted with respect to each other would lead to large periodic supercell structures, which can support a wide range of novel electron correlated phenomena, where the full understanding is still under debate.
        Here, we propose a new scheme to realize a nonlinearity-induced dynamical self-organized twisted-bilayer lattice in an atomic Bose-Einstein condensate (BEC).
        The key idea here is to utilize the nonlinear effect from the intrinsic atomic interactions to couple different layers and induce a dynamical self-organized supercell structure, dramatically distinct from the conventional wisdom to achieve the static twisted-bilayer lattices. To illustrate that, we study the dynamics of a two-component BEC and show that the nonlinear interaction effect naturally emerged in the Gross-Pitaevskii equation of interacting bosonic ultracold atoms can dynamically induce both periodic (commensurable) and aperiodic (incommensurable) moiré structures. One of the interesting moiré phenomena, i.e., the flat-band physics, is shown through investigating the dynamics of the wave packet of BEC. Our proposal can be implemented using available state-of-the-art experimental techniques and reveal a profound connection between the nonlinearity and twistronics in cold atom quantum simulators.
        Nonlinearity-induced dynamical self-organized twisted-bilayer lattices in Bose-Einstein condensates
            Bo Liu
            August 5, 2024
        ===================================================================================================
        
        
        
        § INTRODUCTION
        
        Superimposed crystal bilayers twisted with respect to each other provide a new knob to unveil a wide
        range of novel electron correlated phenomena <cit.>.
        In particular, moiré patterns associated  with small twist angles result in various novel physical properties <cit.>, including the flat band <cit.>, fractional Chern insulators <cit.>, quantum Hall effect <cit.>, moiré ferroelectricity <cit.>, exotic magnetism <cit.>, moiré excitons <cit.>, non-Abelian gauge potentials <cit.>, and unconventional superconductivity <cit.>.
        In parallel to these developments in solid state materials, examples of twisted-bilayer are also studied in other artificial quantum systems, such as in optics <cit.>, acoustics <cit.> and cold atoms  <cit.>.
        In photonic lattices, the moiré pattern and its induced phenomena, such as the localization of light and flat band physics, have been studied in the photorefractive medium <cit.> and nanostructured photonic moiré superlattices <cit.>.
        
        Ultracold atoms in optical lattices are another one of the most promising platform to
        investigate the remarkable physical properties of twisted artificial structures <cit.>.
        It thus has stimulated tremendous amount of efforts,  including theoretically predicting moiré physics in ultracold fermions <cit.> and bosons  <cit.>. In particular,  experimental breakthrough observation of the spatial moiré pattern and momentum diffraction of a Bose-Einstein condensate (BEC) in twisted bilayer optical lattices has recently been achieved <cit.>.
        However, the conventional wisdom in past studies focuses on achieving the static twisted-bilayer lattices <cit.>.
        How to realize such twisted lattice structures in a dynamical way has so far eluded efforts.
        
        
        
        In this work, we propose a new scheme to dynamically achieve twisted-bilayer lattices that builds on the concept of nonlinearity. Our scheme relies on the  nonlinear interaction effect, which naturally emerges in the Gross-Pitaevskii equations of interacting bosonic ultracold atoms. We find that such nonlinear interaction effect amazingly results in a dynamical self-organized twisted-bilayer lattice. This new scheme frees out the usual requirement of adding external single-particle type coupling between different layers and only relies on the nonlinear effect from the intrinsic atomic interactions,  dramatically distinguished from previous schemes  <cit.>. We show that both dynamical self-organized periodic (commensurable) and aperiodic (incommensurable) twisted-bilayer lattice structures can be realized through our proposed scheme. In particular, for the aperiodic case, we find the localization of the Gaussian BEC wave packet, resulting from the emergence of the flat bands in the nonlinearity-induced self-organized twisted-bilayer lattices. It indicates that our proposed nonlinear interaction effect can dynamically produce the tunable moiré lattices in BEC.
        
        
        
        § EFFECTIVE MODEL
        
        The system considered here is schematically illustrated in Fig.<ref>. A two-component BEC, for instance ^87Rb atoms, is used,  where a spin-dependent harmonic potential is applied <cit.>. Atoms in component 1 are trapped in a tight harmonic trap and the atoms in component 2 are loaded in a square optical lattice confined by a very shallow harmonic trapping potential. After the system reaches equilibrium, the tight harmonic trap for component 1 atoms is tuned to be very shallow and a square optical lattice is added. At the same time, the inter-species interaction is turned on. Such a system can be described by the following model Hamiltonian
        
            𝐇   =   ∫dxdy ∑_α=1,2ψ_α^†[ 𝐤^2/2m+V_trap,α(x,y)+V_α(x,y) ] ψ_α
               +   ∫dxdy(g_11n̂_1^2+g_22n̂_2^2+2g_12n̂_1n̂_2),
        
        where ψ_α(α=1,2) denote the Bose field operators for two-component bosons, respectively. The corresponding density operators  are defined as n̂_α=ψ^†_αψ_α. V_trap,α denotes the harmonic trap. The spin-dependent optical  lattices are  V_1(x,y)=V_1[sin^2(cosθ/2k_Lx-sinθ/2k_Ly)+sin^2(sinθ/2k_Lx+cosθ/2k_Ly)] and  V_2(x,y)=V_2[sin^2(cosθ/2k_Lx+sinθ/2k_Ly)+sin^2(-sinθ/2k_Lx+cosθ/2k_Ly)], respectively, with θ being the twist angle. k_L=2π/λ is the wavevector
        of the laser field with the corresponding lattice constant defined as a_L=π/k_L, and λ is the wavelength of the laser. V_α captures the corresponding lattice depth.
        g_11 and g_22 characterize the intra-species interaction strength for component 1 and  component 2 atoms, respectively. The inter-species interaction strength is captured by g_12.
        
        From Eq. (1), the following
        dimensionless energy functional can be constructed under the Gross-Pitaevskii (GP) mean-field theory
        
            ε   =   ∫dx'dy' ∑_α=1,2ψ_α^* [ -∇^2/2+V'_trap,α(x',y')+V'_α(x',y') ] 
               ×   ψ_α +∫dx'dy'(g'_11|ψ_1|^4+g'_22|ψ_2|^4+2g'_12|ψ_1|^2|ψ_2|^2),
        
        where dimensionless spatial coordinates are defined as x'=x/λ and y'=y/λ, respectively. V'_trap,α=V_trap,α/(ħ^2/mλ^2) represents the harmonic trapping potential. V'_α=V_α/(ħ^2/mλ^2) labels the dimensionless lattice depth. g'_11(12,22)=g_11(12,22)/(ħ^2/m) are the corresponding dimensionless interaction strength.
        The wavefunction is normalized as ∬ |ψ_α|^2dx'dy'=N_α with N_α being the particle number of the two-component atoms, respectively.
        Based on Eq. (2), the dynamics of the system can be well described by the following coupled GP equations
        
            i∂ψ_1/∂ t' =    [ -1/2∇^2+V'_trap,1(x',y')+V'_1(x',y')+g'_11|ψ_1|^2 
               +g'_12|ψ_2|^2 ] ψ_1  
            
            			i∂ψ_2/∂ t' =   [ -1/2∇^2+V'_trap,2(x',y')+V'_2(x',y')+g'_12|ψ_1|^2 
               +g'_22|ψ_2|^2 ] ψ_2,
        
        where t'=t/(mλ^2/ħ) denotes the  dimensionless evolution time.
        
        
        § MOIRÉ LOCALIZATION INDUCED BY THE  NONLINEAR INTERACTION EFFECT
        
        In the following, we will demonstrate that distinguished from previous studies <cit.>, even without
        external single-particle type interlayer coupling,  the existence of nonlinear interaction term in GP equation, i.e., w=g'_11|ψ_1|^2
        +g'_12|ψ_2|^2, can dynamically result in the twisted-bilayer lattices and  lead to one of the interesting moiré phenomena.
        To show that, we monitor the dynamics of BEC for different twist angles θ. Here the initial state for component 1 atoms is the ground state of the harmonic trap, which is chosen as a Gaussian wave packet (left part in Fig. <ref>(a)). While for component 2 atoms, the initial state is chosen as the ground state (right part in Fig.<ref>(a)) determined by the effective Hamiltonian in the right hand-side of Eq. (<ref>) with g'_12=0.
        
        Interestingly, as shown in Fig. <ref>, when tuning the twist angle θ, in the presence of nonlinear term w, even without external single-particle type coupling between two layers, one of the interesting moiré physics, i.e., both the localized and delocalized behavior of component 1 atoms, can be achieved.
        For instance, as shown in Fig. <ref>(b) and (c), when θ is chosen as a Pythagorean angle, for instance, θ= arctan3/4 (Fig. <ref>(b)), the Gaussion wave packet for component 1 atoms displays the delocalized behavior.
        While considering θ being the non-Pythagorean twisted angle, for instance, θ = π/6 (Fig. <ref>(c)), the wave packet for component 1 atoms turns out to be localized.
        Note that here we consider the case with large number imbalance between two components, i.e., N_1 ≪ N_2 and lager intra-species interaction, i.e., g'_22 > g'_12. The effect of inter-species interaction on the dynamics of component 2 atoms is highly suppressed and the density profile of component 2 atoms almost does not change during the evolution, as shown in the right column of Fig. <ref>.
        
        
        
        To quantitatively distinguish between the  delocalized and localized behavior of the wave packet of component 1 atoms, we analyze the following mean square displacement (MSD) <cit.>
            σ(t)=[∬dxdy (x^2+y^2)|ψ_1(t)|^2/∬dxdy|ψ_1(t)|^2]^1/2,
        
        which can characterize the width of the wave packet of component 1
        atoms. The different dynamical behaviors of the wave packet can be
        quantified as σ(t)∝ t^κ with distinct dynamical index κ.  As shown in Fig. <ref>,  when the twist angle θ is chosen as a Pythagorean angle, for instance, θ= arctan3/4, σ(t) tends to linearly increase with respect to the evolution time, i.e., σ(t) ∝ t.  The dynamical index is 1,  indicating
        its delocalized behavior. While θ being a non-Pythagorean angle, for instance, θ = π/6, σ(t) will reach a stable value in a fairly short time, indicating the dynamical index κ being 0. It thus characterizes the localized behavior of the wave packet.  Therefore, through tuning the twist angle θ, one of the moiré physics, i.e., both the localized and delocalized behavior of the wave packet of component 1 atoms, can be achieved.
        
        
        
        § NONLINEARITY-INDUCED DYNAMICAL SELF-ORGANIZED TWISTED-BILAYER LATTICES
        
        In the following, we will explain the reason for the appearance of the moiré physics in our proposed system. Let us focus on Eq. (<ref>) in the GP equation describing the two-component atomic BEC. In the presence of the nonlinear term w,  component 1 atoms suffer from a  dynamical self-organized effective potential V_eff≡ V'_trap,1+V'_1+w=V'_trap,1+V'_1+g'_11|ψ_1|^2+g'_12|ψ_2|^2.
        As shown in Fig. <ref>(a), when the twist angle θ is chosen as a Pythagorean
        angle, for instance, θ =arctan 3/4, the effective potential V_eff possess a spatially periodic structure. While considering θ being a non-Pythagorean twist angle, for instance,  θ=π/6, as shown in Fig. <ref>(b), V_eff behaves as a twisted-bilayer lattice with a spatial aperiodic structure. Therefore, distinguished from previous studies <cit.>, even without external single-particle type interlayer coupling, the nonlinear effect resulting from the intrinsic atomic interactions provides a new way to dynamically induce the twisted-bilayer lattices. Through tuning the twist angle θ, we find that both periodic (commensurable) and aperiodic (incommensurable) moiré structures can be created in such a dynamical twisted-bilayer lattice.
        
        
        
        
        To further understand the distinct dynamical behaviors of the wave packet of component 1 atoms, we rely on the analysis of the appearance of the flat band physics in our proposed nonlinearity-induced self-organized twisted-bilayer lattices. To do that, we calculate the band structure of V_eff for both periodic (commensurable) and aperiodic (incommensurable) moiré structures.
        In particular, when considering the aperiodic moiré lattice, the band structure can be calculated through approximating the non-Pythagorean twist angle by a Pythagorean one. For instance, here we use θ=arctan 120/209 to approximate θ= π/6.
        The band structure of V_eff can thus be obtained by the plane-wave expansion method through introducing the Bloch basis ψ_nk⃗=∑_G⃗ u_nk⃗,G⃗|k⃗+G⃗⟩ with the Bloch vector k⃗ and the reciprocal lattice vector G⃗. Here n labels the band index. Then, the energy spectra can be obtained through solving the eigen-problem via the following relation
        
            (k⃗+G⃗)^2λ^2/2u_nk⃗,G⃗ + ∑_G⃗'⃗⟨k⃗+G⃗ | V_eff| k⃗+G⃗'⃗⟩ u_nk⃗,G⃗'⃗   
            
            		= E'_nk⃗ u_nk⃗,G⃗,
        
        where E'_nk⃗=E_nk⃗/(ħ^2/mλ^2) is the dimensionless energy spectra.
        
        
        
        
        As shown in Fig. <ref>, the band structures of the corresponding
        twisted-bilayer lattices V_eff  demonstrated in Fig. <ref> are obtained. We find that when changing the twist angle from a Pythagorean angle to a non-Pythagorean angle, more dispersive lower bands become extremely flat, accompanying with the lattice structure changing from the periodic (commensurable) case to the aperiodic (incommensurable) one. In particular, the appearance of the lowest flat band will lead to the localization of the wave packet of component 1 atoms, since such a BEC wave packet will mainly concentrate at the lowest band after a long enough time evolution. Furthermore, it is known that the flat band can support quasi-nondiffracting localized modes. Therefore, the wave packet of component 1 atoms loaded in such a  nonlinearity induced self-organized aperiodic (incommensurable) twisted-bilayer lattice will tend to be localized.
        
        
        
        § CONCLUSION
        
        In summary, we have demonstrated a new approach to
        dynamically achieve twisted-bilayer lattices in an atomic BEC.
        To emphasize a remarkable difference, our scheme
        does not require external single-particle type
        interlayer coupling. Instead, we only require the nonlinear effect
        arising from the intrinsic atomic interactions.
        We further show that both periodic and aperiodic  moire lattices, i.e.,  nonlinearity-induced dynamical self-organized twisted-bilayer lattices, can be designed through tuning the twist angle. One of the interesting phenomena in twisted-bilayer lattices, i.e., the flat-band physics, is also shown through investigating the dynamics of the wave packet of BEC. Such nonlinearity-induced approach is rather generic to other quantum systems, such as in photonics and electrical circuits, than restricted to the setup considered in this work, since the nonlinear effects, such as  Kerr nonlinearity, is common in these systems. Our results reveal a profound connection between
        the nonlinearity and twistronics.
        
        
        § ACKNOWLEDGMENT
        
        We acknowledge helpful discussions with L. Chen. This work is supported by the National Key R&D Program of China (2021YFA1401700), NSFC (Grants No. 12074305, 12147137, 11774282), the Fundamental Research Funds for the Central Universities (Grant No. xtr052023002), the Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant No. 23JSZ003) and  Xiaomi Young Scholar Program. We also thank the HPC platform of Xi'an Jiaotong University, where our numerical calculations was performed.
        
        
        apsrev


 Paper Title: The impact of quantum noise on limit cycles and entrainment in an atom-cavity system.







National Institute of Physics, University of the Philippines, Diliman, Quezon City 1101, Philippines



National Institute of Physics, University of the Philippines, Diliman, Quezon City 1101, Philippines



§ ABSTRACT 
Quantum noise is inherent in any open quantum system as it affects not only the statistical properties of the initial state but also the time evolution of the system. A transversely pumped atom-cavity setup is a prototypical example of such an open system, wherein limit cycles (LCs) have been observed and identified as continuous time crystals (CTCs). Using truncated Wigner approximation (TWA), we show that the inherent quantum noise pushes the system to exhibit signatures of LCs for interaction strengths lower than the critical value predicted by mean-field theory, suggesting a noise-induced enhancement of temporal ordering. By comparing the oscillation frequencies of the LCs for the one-dimensional (1D) and two-dimensional (2D) regimes, we find that the LC frequencies have larger shot-to-shot fluctuations in 1D than in 2D. We demonstrate that this has an important consequence in the effectiveness of entrainment of LCs for a periodically driven pump intensity or light-matter coupling strength.


The impact of quantum noise on limit cycles and entrainment in an atom-cavity system
    Jayson G. Cosme
    August 5, 2024
====================================================================================





§ INTRODUCTION

Typically, noise is expected to inhibit well-defined oscillatory response in complex systems. Nevertheless, it has been suggested that noise could induce order, instead of the expected chaotic behavior <cit.>.
Indeed, phase synchronization in systems with common noise <cit.> and uncorrelated noise <cit.> has been observed.
Moreover, applied noisy drive has been shown to produce nontrivial ordered phases in interacting quantum systems <cit.> and, in other cases, can have the opposite effect of destabilizing an existing LC or CTC <cit.>. In Ref. <cit.>, fluctuations corresponding to the intrinsic quantum property and coupling to an external environment, and their role on the stability of discrete time crystals have been explored. In this work, we focus on the role of inherent quantum noise in the emergence and stability of LCs, particularly in the context of CTCs and their dynamical control <cit.>. These LCs are characterized by oscillating observables at a well-defined frequency even though the underlying equations governing the system's time evolution is manifestly time-independent.

The emergence of LCs in various many-body quantum systems is widely studied, especially in the context of time crystals <cit.>. In particular, for an atom-cavity system <cit.>, artificial noise applied to the light-matter interaction strength is found to destabilize the LC, which then begs the question if the inherent fluctuation associated with the photon dissipation in the cavity <cit.> could inhibit the appearance of LCs. This is important in understanding the quantum nature of LCs in open systems <cit.>.
Exploring the precise role of quantum noise can be experimentally challenging as the inherent noise cannot be turned off naturally in experiments. However, in a theoretical setting, an approach such as the truncated Wigner approximation (TWA) <cit.> allows for the possibility of choosing which noise to be included in the theoretical description of a given system. In atom-cavity systems, beyond mean-field approximation of the quantum dynamics using TWA has been shown to capture the correct qualitative behaviour of time crystals observed in experiments <cit.>.


 



In this paper, we investigate the effects of quantum noise on the robustness and dynamics of the LC phase in the 1D and 2D atom-cavity systems. To this end, we employ both mean-field simulations and TWA, wherein we sample the quantum noise of the initial state and the stochastic noise in time. In doing so, we can compare the dynamics for cases with and without quantum noise. This allows us to infer the effects of quantum noise to the emergence of LCs. Moreover, we investigate the entrainment of LCs <cit.> in both 1D and 2D by periodically driving the pump intensity. Consequently, we are able to observe a hallmark of entrainment, which is frequency locking, and we determine which of the atom-cavity systems will exhibit a more robust entrainment.


This paper is structured as follows. We first discuss the atom-cavity system in Sec. <ref>. Next, we present the mean-field diagrams for the 1D model in Sec. <ref>. We also show the spectrogram for different values of the pump intensity. In  Sec. <ref>, we analyze the average response frequencies together with the corresponding standard deviation to infer the qualitative and quantitative effects of quantum noise. We compare the entrainment of LCs for the 1D and 2D atom-cavity systems in Sec. <ref>. Lastly, we conclude the paper in Sec. <ref>.





§ ATOM-CAVITY MODEL

We focus on a model for an atom-cavity system of ^87Rb-atoms forming a Bose-Einstein condensate (BEC) with particle number N_a = 6.5× 10^4 trapped inside a Fabry-Pérot resonator as shown in Fig. <ref>(a). A pump beam with intensity ε = ϵ E_rec, where E_rec = ħω_rec is the recoil energy, is applied in the direction perpendicular to the cavity axis. The pump has a wavelength λ_p = 792.55 nm and the recoil frequency is ω_rec = 2π^2ħ/mλ_p, where m is the mass of a ^87Rb atom.
Photons leaking out of the cavity is characterised by the cavity photon decay rate κ= 2π× 3.6 kHz. We use realistic parameters based on experimental realizations in the good cavity regime κ∼ω_rec <cit.>.

The Hamiltonian for the atom-cavity system consists of  cavity, atom, and atom-cavity interaction terms

    Ĥ = Ĥ_C+Ĥ_A+Ĥ_AC.

Here, we neglect the contact interaction between the atoms, which corresponds to a mean-field breaking term in this model <cit.>.
The Hamiltonian for the cavity with a mode function cos(kz) is given by

    Ĥ_C = -ħδ_Cα̂^†α̂,

where δ_C is the pump-cavity detuning and α̂^† (α̂) is the creation (annihilation) operator for the cavity photon. Note that δ_C = δ_eff +N_a U_0/2, where U_0 is the lattice depth induced by a photon in the cavity. The Hamiltonian for the atoms is

    Ĥ_A = ∫Ψ̂^†(y,z) [-ħ^2/2m∇^2+εcos^2(ky)]Ψ̂(y,z)  dy dz

where Ψ̂(y, z) is the bosonic field operator associated with the BEC.
Lastly, we consider the atom-cavity interaction

    Ĥ_AC =∫Ψ̂^†(y,z)  ħ U_0  [cos^2(kz)α̂^†α̂
    
    	+√(ε/ħ|U_0|)cos(kz)cos(ky)(α̂^† +α̂)]Ψ̂(y,z)  dy dz.

In the 1D limit, the dynamics along the pump direction is frozen Ψ̂(y,z) →Ψ̂(z), such that Eq. <ref> reduces to just the kinetic energy along the cavity direction. Moreover, the atom-cavity interaction Eq. <ref> simplifies to only contain z-dependent terms, which can be obtained by taking cos(ky) → 1. The dynamics of the system is given by the Heisenberg-Langevin equations

    ∂/∂ tΨ̂   = i/ħ[Ĥ, Ψ̂] 
    ∂/∂ tα̂   = i/ħ[Ĥ, α̂]-κα̂+ξ,

where ξ is the stochastic noise due to the cavity dissipation with ⟨ξ^*(t)ξ(t')⟩ = κδ(t-t') <cit.>. 


In our numerical simulations, we represent the quantum operators as c-numbers using TWA. This enables us to approximate the dynamics using an ensemble of stochastic trajectories. The TWA includes the leading order quantum corrections to the mean-field theory, which is valid for N_a→∞ <cit.>. In open quantum systems, perturbations due to quantum noise in the initial states and stochastic noise ξ associated with the photon loss are present. In theory, we can choose which source of quantum noise we include in our TWA simulations.

To numerically solve the relevant stochastic differential equations, we utilize the predictor-corrector method, otherwise known as Heun's method. We consider a time step of 10^-8 s and record the values every 20 steps. For 1D simulations, we use a real-space basis for Ψ̂(z) <cit.>, while for 2D simulations, we employ a momentum-space basis for Ψ̂(y,z) <cit.>.







§ MEAN-FIELD RESULTS

In this section, we first focus on mean-field simulations to serve as a benchmark for TWA simulations, which will be shown in the next section.


To introduce the phases found in this configuration of an atom-cavity system, we implement the two-ramp pump protocol shown in Fig. <ref>(b). In Fig. <ref>(c), we show the dynamics of the intracavity photon number, |α|^2, used to distinguish between different phases in the system. The normal phase (NP) is identified by a zero intracavity photon number. By increasing the pump intensity, it will transition to the superradiant phase (SR) when ε≥ε_SR, where ε_SR is the critical value for the NP-SR transition.
The SR phase is characterized by a non-zero constant intracavity photon number <cit.> as highlighted in Fig. <ref>(c). Hence, its long-time trajectory as shown in Fig.<ref>(d) is a steady state in the phase space spanned by the real and imaginary parts of the cavity mode α. Increasing the pump intensity further leads to a transition into the LC phase. This dynamical phase is characterized by an oscillating |α|^2 at a well-defined limit cycle frequency ω_LC as shown in Fig. <ref>(c). Because it oscillates at a certain frequency, the long-time trajectory in phase space forms a closed loop as presented in Fig. <ref>(e).

In Fig. <ref>, we present the phase diagram in the δ_eff-ϵ space for a 1D system. We note that the existence of LCs in the 1D regime for blue and red atom-pump detunings was first predicted in Refs. <cit.> and <cit.>, respectively. 
To construct Fig. <ref>, we linearly increase the pump intensity until ϵ for t∈ [15, 20] ms. Then, we fix ϵ to a constant value for 15 ms. We consider the time frame of the last 5 ms to obtain the long-time average, maximum value, and the standard deviation of the rescaled peak heights of the cavity occupation. In classifying the dynamics, we apply the following threshold values. 
 
A cavity occupation dynamics is classified to be that of a LC if the standard deviations is lower than 0.025, and the difference between the maximum and average |α|^2 is greater than or equal to 10% of the average. The SR phase, on the other hand, is identified by having a standard deviation less than 10% of the average cavity occupation. A response with an average photon occupation less than or equal to 10^-3 is considered as a NP. Lastly, dynamics with standard deviation greater than or equal to 0.025 is determined as a chaotic response (CH). We obtain the LC frequency ω_LC from the dominant frequency in the power spectrum dynamics of the intracavity photon number |α|^2 for the same time window. Then, ω_LC is obtained as the frequency with the highest peak in the power spectrum. 
In Fig. <ref>, both the blue-detuned and red-detuned cases have LCs. 

In Fig. <ref>, we further investigate the dynamics of |α|^2 by including a gallery of spectrograms for different ε/ε_LC, MF, where ε_LC, MF is the intensity required to transition from SR to LC according to MF theory. This allows us to take note of how the dynamics change for different intensities in the absence of stochastic noise. Accordingly, we expect that the system is in the SR phase when ε/ε_LC, MF<1.0, and it is in the LC phase when ε/ε_LC, MF≥1.0. 
For the pump protocol, we linearly increase the pump intensity until the desired ε for 5 ms, then kept it constant for 395 ms. Afterwards, we obtain the spectrogram of |α|^2 using a moving time window of ≈ 3.3 ms. The intensity in units of dB corresponds to 
10log_10(P) where P is the power spectral density. Hence, the dominant oscillation frequencies of |α|^2 are characterized by positive intensity shown as brighter shades in the color bars. Thus, a dominant frequency seen as streak of positive intensity at a constant ω signifies the appearance of a LC. On the other hand, the SR phase is identified by the lack of nonzero frequency peak in the long-time limit due to the absence of photon oscillations. 

We find that in 1D, as depicted in Fig. <ref>(a), transient oscillations appear but they eventually vanish for long times as ε approaches ε_LC, MF. 
We note for the discussions in the next section that the transient oscillations in Fig. <ref> persist for t<100 ms.
To inspect these dynamics, we show in Fig. <ref>(c-f), the adjusted intracavity dynamics |α|^2_ctr = |α|^2-|α|^2_mean where |α|^2_mean is the mean value of |α|^2 for t∈[0.1, 0.4] s, and its corresponding spectrogram for ε/ε_LC, MF<1.0. We find that the amplitude of the |α|^2_ctr oscillation decreases and eventually vanishes. 
 

This is also reflected in the features of the spectrogram in Fig. <ref>(d). It is also striking that a consistent dominant frequency is present, and that it is larger than the expected ω_LC. On the other hand, the 2D system shows a different trend in Fig. <ref>(b). While we find that the transient oscillations vanish after some time similar to the 1D case, we find that the photon dynamics approach what appears to be plateaus that increase in value as portrayed in Fig. <ref>(e). Moreover, between these plateaus, we observe short-lived small amplitude oscillations that become more frequent the closer ε is to ε_LC, MF. The sharp increase in the photon number appears as streaks at constant times in the spectrogram in Figs. <ref>(b) and <ref>(f). 

Overall, the mean-field limit provides us with a baseline of the expected behavior of the system. To summarize, we found that transient oscillations are present in the absence of stochastic noise. Hence, one may mistakenly identify these transient oscillatory response as LC if the simulation time is not sufficiently long. We will thus classify such a behaviour with an oscillatory response faster than the expected LC frequency as a SR phase. 



§ TWA SIMULATIONS



 §.§ Noise and limit cycles

In the previous section, we discussed the mean-field dynamics of |α|^2 for different pump intensities. Next, we explore the effects of quantum noise in the initial state and stochastic noise associated with the cavity dissipation. We will assume that the stochastic noise in time can be freely switched off within TWA as done in the supplementary materials of Ref. <cit.>. By comparing the results with and without stochastic noise, we can pinpoint the fundamental effect of stochastic noise.


In Fig. <ref>, we present the results for the same pump protocol used in Fig. <ref> albeit now in the presence of stochastic noise. Note that we use exactly the same initial state for a fair comparison. As expected when temporal noise is included in the dynamics, we find that stochastic noise broadens the power spectrum, which results to a decrease in the intensities of the dominant frequencies. Interestingly, the transient high-frequency oscillations in the SR phase seen in Fig. <ref> have vanished in the presence of stochastic noise. This means that the inherent quantum noise due to dissipation overwhelms the transient oscillations making them unlikely to be observed in actual experiments. Moreover, it also pushes the system to oscillate at a well-defined frequency such that ω_peak∼ω_LC, MF as seen from the bright streaks close to the expected LC frequency in Fig. <ref>. This will become more obvious when we discuss Fig. <ref> later. The spectrograms for the SR phases close to the SR-LC transition point already show signs of a LC by having a faint bright streak at the expected LC frequency in Fig. <ref>(b), for example. In the remainder of this section, we will further explore this apparent noise-induced enhancement of the LC phase.






To further investigate the noise-induced temporal ordering, we now use 100 trajectories in our TWA simulations for varying pump intensities as depicted in Figs. <ref> and <ref> for the 1D and 2D systems, respectively. For the pump protocol, we linearly increased the intensity for 5 ms and then kept it constant for 15 ms. We then analyze the Fourier transform of the intracavity photon number for t∈[15, 20] ms, F{|α|^2(t)}= |α̃|^2(ω).
The dominant frequency ω_peak is determined as the frequency corresponding to the maximum value of the normalized Fourier spectrum given by the crystalline fraction defined as Ξ = max{|α̃|^2(ω)/∑_ω |α̃|^2(ω)} <cit.>. 

In Figs. <ref> and <ref>, we present two sets of results depending on which type of quantum noise is included in the TWA. In both cases, quantum noise in the initial state is included assuming a coherent state for the BEC and a vacuum state for the cavity mode. However, in one case, the stochastic noise is absent meaning the dynamics is fully deterministic, while in the other, it is present resulting to a stochastic dynamics of the system. Furthermore, Fig. <ref> depicts the results for the 1D case while Fig. <ref> corresponds to the 2D case.
In Figs. <ref>(a) and <ref>(a), the average values of ω_peak for 100 trajectories are indicated as circles and diamonds for those with and without stochastic noise, respectively. The error bars denote the standard deviation for the recorded peak frequencies. In Figs. <ref>(b)-(g) and <ref>(b)-(g), we present the histograms of the crystalline fraction and peak frequency for exemplary choices of the pump intensity. Here, the intensity of the color bar represents the number of trajectories following the binning procedure for the histograms.

The SR phase manifests as a zero-frequency oscillation in MF as shown in Fig. <ref>(b), and low-frequency oscillation due to stochastic noise in Fig. <ref>(e). 
Trajectories with significantly high frequency without stochastic noise are attributed to SR phase instead of the LC phase as discussed in Sec. <ref>. These low amplitude high-frequency oscillations emerges the closer the pump intensity is to ε_LC, MF. In some trajectories, these high frequencies have a well-defined oscillation as noted by a high value of Ξ in Fig. <ref>(b) and <ref>(c). We emphasize that these fast oscillations are transient as discussed in the previous section, which means that all trajectories in Fig. <ref>(b), for example, correspond to a SR phase. Interestingly, Figs. <ref>(c), <ref>(f), <ref>(c), and <ref>(f) exhibit trajectories with ω_peak∼ω_LC, MF even though ε<ε_LC, MF. 
More importantly, comparing Figs. <ref>(c) and <ref>(f), we find that the presence of stochastic noise in time has eliminated the presence of trajectories in the SR phase oscillating at high frequencies. This phenomenon is even more pronounced in 2D as seen in Figs. <ref>(c) and <ref>(f). Hence, the suppression of the high-frequency oscillations in the SR phase by stochastic noise reveals an early signature of the onset of LCs. This noise-induced enhancement of LC is emphasized in Fig. <ref>(f), in which most trajectories possess ω_peak∼ω_LC, MF. Nevertheless, we point out that not all trajectories can be considered as LC due to some of them having a relatively low Ξ.
Furthermore, the presence of noise in the initial state leads to a shift of the phase diagram because of the slight variations of N_aU_0 for constant U_0. Thus, trajectories would either be an SR or LC near the SR-LC transition since the quantum fluctuations in the initial state effectively changes the δ_eff. 




 

To investigate the shot-to-shot fluctuations found deep in the LC regime, we employ TWA for 500 trajectories as shown in Fig. <ref>. We use the same pump protocol as in the mean-field simulations and kept it constant for 20 ms and 35 ms for 1D and 2D, respectively. We calculate the Fourier transform of the cavity occupation dynamics for each trajectory over the last 5 ms, which means that our Fourier transform can only detect frequency changes larger than 200 Hz. This is important to note when we consider multiple trajectories within TWA as they could in principle have different ω_LC. From the Fourier spectrum, we record its real and imaginary parts at the LC frequency predicted by MF, N_P(ω_LC,MF), for fixed  δ_eff and ϵ. Furthermore, we rescale the spectrum by the maximum value recorded for all trajectories. Notice that some of the trajectories do not have the same LC frequency as the mean-field prediction, which is evident from the points scattered near the origin of Figs. <ref>(a) and <ref>(b). Specifically, we note that some of the trajectories have frequency around ω_peak = ω_LC,MF± 2π× 200 Hz. This can be explained by the smooth variation of the LC frequency in the mean-field phase diagram. Quantum fluctuations in the initial state lead to shot-to-shot variations of the total particle number N_a, which means that TWA can be considered as a sampling of MF results from slightly different choices of δ_eff in the mean-field phase diagram. This then leads to different LC frequency for each trajectory, the variation of which depends on how fast the LC frequency changes as a function of δ_eff. In comparison to Fig. <ref>(a), we find that there are less fluctuations of the LC frequency in the 2D system as shown in  Fig. <ref>(b), wherein most trajectories are found along the unit circle. 






 §.§ Entrainment

We now investigate the consequence of fluctuating LC frequency on the phenomenon of entrainment when the pump intensity is periodically modulated <cit.>. Entrainment, otherwise known as injection locking in optics, is observed by periodically driving a system with a limit cycle state <cit.>. This results to the “locking" of the limit cycle frequency by driving the system to a ratio of ω_LC <cit.>. In the atom-cavity system, this was proposed to create a more robust time crystal or limit cycle phase <cit.> and was later reported experimentally <cit.>. In both of these papers, they focused on subharmonic injection locking, wherein the pump intensity is periodically driven at a frequency ω_dr= 2ω_LC.

In the following, we first choose a set of parameters deep in the LC regime with frequency ω_LC, MF. Then, we periodically modulate the pump intensity once we have a stable LC by driving it at twice the LC frequency ϵ=ϵ_0[1+f_0cos(2ω_LC, MFt)]. A successful entrainment is marked by |α|^2 oscillating at ω_LC, MF, thereby breaking the discrete time translation symmetry imposed by the periodic drive. This constitutes a phase transition from continuous to discrete time crystals <cit.>. 

 

In Fig. <ref>, we present the response frequencies for 100 trajectories for different driving strengths f_0. The mean value of the response frequency ω_peak is indicated as a circle and its standard deviation is represented by the error bar. The dashed lines refer to the LC frequency according to MF, ω_LC, MF. We observe in Fig. <ref>(a) that deviations in the response frequencies for all f_0 is present. Hence, the entrainment in a 1D system is not as robust as the one observed in Ref. <cit.>.
The weak entrainment can be attributed to the strong shot-to-shot variations of the LC frequency in 1D as discussed in the previous subsection. While ω_LC,MF is fluctuating for each trajectory, the driving frequency is fixed to just one value for all those trajectories. That is, not all trajectories are resonantly driven, leading to deviations of the response frequency even for smaller values of f_0 in 1D. In contrast, the 2D case exhibits a more robust entrainment with zero standard deviation for f_0∈ [0.1, 0.35]. This demonstrates that, while a 2D system also experiences shot-to-shot fluctuations of the LC frequency (see Fig. <ref>(b)), albeit less likely and weaker than the 1D case, it can still result to a successful entrainment as seen in Fig. <ref>(b). This is further corroborated by a comparison of the 1D case in Fig. <ref>(a) and the zoomed-in view of the results for 2D in the inset of Fig. <ref>(b), wherein the standard deviations are considerably higher in 1D than in 2D.





§ CONCLUSIONS

In summary, we have explored the role of quantum noise in the emergence and entrainment of LCs in an atom-cavity system. Specifically, we compare the photon dynamics obtained using MF and TWA in the 1D and 2D regimes. The TWA allows us to freely switch on and off the stochastic noise in time associated with the cavity dissipation. We found that the inclusion of stochastic noise results to an earlier onset of LCs for smaller pump intensities or, equivalently, light-matter interaction strengths, compared to the corresponding mean-field prediction. In particular, stochastic noise suppresses the high-frequency oscillations in the SR phase, which in turn reveals an earlier signal for the onset of LCs through weak yet detectable oscillations at the expected LC frequency. Furthermore, we have demonstrated that the shot-to-shot fluctuations of the LC frequency are larger in the 1D limit compared to the 2D limit of the atom-cavity system. This then results to less robust entrainment in 1D as some of the trajectories oscillate at frequencies that are non-resonant with respect to the applied periodic drive of the pump intensity. 

Our work highlights the unique role that quantum noise plays in the emergence and stability of dynamical phases, such as limit cycles and time crystals. In particular, we have shown that the fluctuations corresponding to the photon dissipation, a necessary ingredient for beyond mean-field approaches in open quantum systems to ensure that the bosonic commutation relations for the photons are preserved, may reveal a temporal order, or some hints of it, that is otherwise hidden or completely absent in a standard mean-field theory. Resonance phenomena involving dynamical phases, exemplified in this work by entrainment of LCs, can be adversely affected by quantum fluctuations of the initial state depending on the rigidity of the characteristic frequency of the dynamical phase. It would be interesting to explore in future works these phenomena in the full quantum regime, which may aid in the quest for understanding the quantum origins of various behaviours in nonlinear dynamics, such as the appearance of LCs and bifurcations of stable states <cit.>.




§ ACKNOWLEDGEMENTS

R.J.L.T. and J.G.C. acknowledge support from the DOST-ASTI’s COARE high-performance computing facility. This work was funded by the UP System Balik PhD Program (OVPAA-BPhD-2021-04).




















 Paper Title: Non-classical excitation of a solid-state quantum emitter.



 
lena.maria.hansen@univie.ac.at
University of Vienna, Faculty of Physics, Vienna Center for Quantum Science and Technology (VCQ), 1090 Vienna, Austria
Christian Doppler Laboratory for Photonic Quantum Computer, Faculty of Physics, University of Vienna, Vienna, Austria

University of Vienna, Faculty of Physics, Vienna Center for Quantum Science and Technology (VCQ), 1090 Vienna, Austria
Christian Doppler Laboratory for Photonic Quantum Computer, Faculty of Physics, University of Vienna, Vienna, Austria

University of Vienna, Faculty of Physics, Vienna Center for Quantum Science and Technology (VCQ), 1090 Vienna, Austria

University of Vienna, Faculty of Physics, Vienna Center for Quantum Science and Technology (VCQ), 1090 Vienna, Austria
Christian Doppler Laboratory for Photonic Quantum Computer, Faculty of Physics, University of Vienna, Vienna, Austria

Institute of Theoretical Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland

Instituto de Física Teórica, UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1040 Vienna, Austria

philip.walther@univie.ac.at
University of Vienna, Faculty of Physics, Vienna Center for Quantum Science and Technology (VCQ), 1090 Vienna, Austria
Christian Doppler Laboratory for Photonic Quantum Computer, Faculty of Physics, University of Vienna, Vienna, Austria
 University of Vienna, Research Network for Quantum Aspects of Space Time (TURIS), 1090 Vienna, Austria
Institute for Quantum Optics and Quantum Information (IQOQI) Vienna, Austrian Academy of Sciences, Vienna, Austria

juan.loredo1@gmail.com
University of Vienna, Faculty of Physics, Vienna Center for Quantum Science and Technology (VCQ), 1090 Vienna, Austria
Christian Doppler Laboratory for Photonic Quantum Computer, Faculty of Physics, University of Vienna, Vienna, Austria

§ ABSTRACT The interaction between a single emitter and a single photon is a fundamental aspect of quantum optics. This interaction allows for the study of various quantum processes, such as emitter-mediated single-photon scattering and effective photon-photon interactions. However, empirical observations of this scenario and its dynamics are rare, and in most cases, only partial approximations to the fully quantized case have been possible. Here, we demonstrate the resonant excitation of a solid-state quantum emitter using quantized input light. For this light-matter interaction, with both entities quantized, we observe single-photon interference introduced by the emitter in a coherent scattering process, photon-number-depended optical non-linearities, and stimulated emission processes involving only two photons. We theoretically reproduce our observations using a cascaded master equation model. Our findings demonstrate that a single photon is sufficient to change the state of a solid-state quantum emitter, and efficient emitter-mediated photon-photon interactions are feasible. These results suggest future possibilities ranging from enabling quantum information transfer in a quantum network to building deterministic entangling gates for photonic quantum computing.

Non-classical excitation of a solid-state quantum emitter
    J. C. Loredo
    
=========================================================



Studying light interacting with matter at the most fundamental level offers a number of possibilities for the development of quantum technologies <cit.>. On the one hand, non-classical states of light are suitable as carriers and for processing quantum information <cit.> because they virtually do not interact with their environment. However, the lack of direct photon-photon interactions also prevents access to, e.g., deterministic entangling gates on quantized states of light—an obstacle for building larger scalable quantum photonic systems.

Light strongly interacts with resonant atomic systems. For instance, coherent scattering involves the absorption of one photon subsequent re-emission of a single photon; it has been theoretically shown that even one photon can saturate an atomic transition <cit.>. Moreover, an atomic system is inherently quantum non-linear, mediating various phenomena of effective photon-photon interactions, such as electromagnetically induced transparency <cit.>, photon-blockade <cit.>, and entangling quantum logic operations <cit.>.

	
 
However, empirically observing an atomic system interacting with a single photon has remained a formidable challenge, as one requires a sufficiently controlled and strongly coupled quantum light-matter platform. Although scarce, a few experiments have reported aspects of quantized input light interacting with an atomic system, for instance, a quantum network of neutral atoms interconnected via single photons <cit.>, single photons scattered by a single Rubidium atom <cit.>, heralded single-photon absorption in a trapped ion system <cit.>, and electronic energy transfer in a photosynthetic complex induced by a single photon <cit.>. Despite impressive demonstrations, these previous works were limited by weak signals, which strongly hindered their ability to investigate the rich photon dynamics and statistics inherent in the fundamental process.



A solid-state platform offers a promising alternative for studying coherent quantum light-matter interactions. Significant progress has been made in both cavity and waveguide quantum electrodynamics <cit.>. In particular, semiconductor quantum dots (QDs) coupled to resonant cavities and photonic crystal waveguides <cit.> constitute a leading candidate for the generation of various types of quantum light: highly-efficient sources of single and entangled photons <cit.>, quantum superpositions and entangled states of photon-numbers <cit.>, and deterministically generated squeezed states <cit.>. Moreover, these solid-state systems led to multiple demonstrations of optical non-linearities involving a small number of photons on average, with works including photon-number dependent spatial <cit.> and temporal <cit.> ordering. However, these previous works employed classical weak coherent input light, thus leaving non-classical light-matter interactions in a solid-state platform without observation.

Here, we demonstrate a quantum light-matter interface in the solid-state. A single QD coupled to an optical cavity is resonantly excited with non-classical light. When the input light consists of a single photon, the output field stems from the quantum interference between the re-emitted part of the wave packet and the incoming part of the wave packet of the single photon. We demonstrate that this process is coherent by performing first- and second-order correlation measurements; indeed, the time-evolving output signal exhibits a point of destructive interference, and consecutive output wave packets display Hong-Ou-Mandel (HOM) two-photon interference <cit.>. Moreover, we observe optical photon-number depended non-linearities occurring for quantized input light, as revealed by changes in the photon-number statistics. Further, we find signatures of stimulated emission processes between two photons, verified by analyzing the temporal mode of the generated two-photon Fock state.


 §.§ Atom-mediated single-photon interference
 
Figure <ref>a depicts the general concept of this work. Here, a quantum emitter, an InAs/GaAs QD in a cavity, is excited with a non-classical wave packet. The output signal resulting from the quantum light-matter interaction is then collected for further analysis.

	
	
In our experiment, the input non-classical light consists of resonance fluorescence from the same QD. That is, we first prepare a highly efficient single-photon source <cit.> (see Methods) and then use a setup with active path control that allows us to choose between two options: either the signal is back-reflected to the emitter or it is sent to the detection stage. With this setup, we first send the resonance fluorescence signal back toward the QD and later collect the resulting output signal; see the Supplementary Information for details on the experimental setup. 

We can tune the input light statistics by choosing the pulse area of the excitation laser that generates the resonance fluorescence signal, from mostly single photons—at π-pulse excitation—to states with one- and two-photon terms—e.g., at 2π-pulse excitation <cit.>. We first study the resulting time dynamics with an input signal generated by π-pulses. In this scenario, the quantum emitter interacts with a resonant single-photon wave packet. As the wave packet reaches the emitter, part of it is transmitted, and the excitation probability of the emitter begins to increase. During the interaction the emission probability keeps increasing, leading to quantum interference between the incoming wave packet and the reemitted part. Once the input wave packet duration ends, the emitter retains the probability of being excited and spontaneously decays without interference. Hence, the observation of destructive interference in the temporal output profile is due to coherent single-photon scattering <cit.>. 

The optimal input temporal shape for achieving atomic inversion is formed as an exponentially increasing curve <cit.>—the time-reversed profile of the spontaneous emission decay. In our case, however, we see the opposite scenario because we back-reflect resonance fluorescence: a single photon arrives at the artificial atom with a field amplitude that first rises rapidly and then exponentially decays. Figure <ref>b shows the measured input time profile with a lifetime of τ = 227 ps, and the time profile of the output field after the interaction with the emitter. We observe the predicted atom-mediated single-photon interference phenomenon and identify three main regions: (1) an earlier part governed by an undisturbed signal—i.e., the input photon is not absorbed, it enters the optical cavity, and it is released back again into the collection mode; (2) an intermediate part displaying a point of minimal intensity given by destructive interference; and (3) a final part governed by spontaneous emission of an atom subject to an exponentially-decaying input field. Our measurements are confirmed by theoretical simulations that follow a cascaded master equation approach in which two identical subsystems are considered, with the output of the first system serving as input for the second system (see Methods).


 §.§ Input-output photon statistics

We now describe the photon statistics of the input and output fields for the scenario described above. We first measure the single-photon purity of the input field by using our active path-controlled setup and choosing to directly collect the resonance fluorescence at π-pulse excitation without sending it back to the artificial atom. As shown in Fig. <ref>, a time-gated Hanbury Brown and Twiss second-order autocorrelation measurement yields the value g^(2)_in(0) = 2.540(2)%, and an HOM experiment on consecutive emissions yields an indistinguishability value <cit.> of ℐ_in = 91.90(5)%. 
These measurements demonstrate the high fidelity of the single photon used as an input in the quantum light-matter interface. We note that the multi-photon component in this state is caused by re-excitation of the QD driven by ∼25 ps long pulses—shorter pump pulses result in higher purities (see Supplementary Information). However, we deliberately choose this pulse length to study scenarios with stronger varying photon statistics along Rabi oscillations, which we describe below. We now perform the same measurements on the output field resulting from the atom-light interaction and find a higher value of g^(2)_out(0) = 6.32(4)%, as well as a decrease in the output field indistinguishability ℐ_out = 75.0(4)%.

	


The lower single-photon purity in the output field results from single-photon interference and effective photon-photon interactions, leading to a higher ratio of collected photon flux for multi-photon states <cit.>. Our results experimentally verify these effects for a non-classical input state. Our findings presented in Fig. <ref>b are evidence of the coherence of this process, showing for the first time that the output field maintains indistinguishability, an important conclusion for network applications.

The origin of multi-photon components is fundamentally different for the input and the output field. For the input field, these components are created from re-excitation. That is, with a small probability, part of the driving classical pulse area produces a first photon, leaving most of the pulse area available to re-populate the excited state. Therefore, a second photon is produced with the remaining pulse area, with this photon contributing primarily to the high input single-photon purity. Note that in this scenario the two photons occupy orthogonal time-modes along the input temporal profile. However, the multi-photon components of the output field can exhibit a different temporal structure depending on the interaction with the emitter.

This can be tested by time-gated photon statistics measurements and calculating the resulting upper bound for two-photon contribution. For simplicity, we consider multi-photon components up to two-photon terms since they are dominant compared to higher multi-photon contributions. Figure <ref>c (top) displays the upper-bound of the two-photon component of the input field for increasing gate start times. The main feature of this analysis is the rapid decrease in the amount of the two-photon component soon after gating begins. That is, by neglecting the photon statistics within the signal rise-time, the two-photon component of the remaining field is largely suppressed; thus, the remaining state is much closer to one (and only one) photon. The same gated measurement on the output field leads to a different conclusion, as shown in Fig. <ref>c (bottom). Here, we do not observe a sharp decrease in the relative two-photon component. In contrast, the amount of two-photon terms, decreases more slowly, which is consistent with the decrease in intensity of the entire wave packet. Thus, we observe two-photon components in the output field at times when they were absent in the input field. In the input field, a second photon is created only during the duration of the laser field due to re-excitation. However, in the output field, two photons can be observed due to temporal restructuring caused by the scattering dynamics from the emitter, which we will investigate further in the following section.


 §.§ Effective photon-photon interaction
 
We now perform a scan of the classical pulse area responsible for generating the input non-classical light. The resulting Rabi oscillations are shown in Fig. <ref>a, together with the intensity of the output field stemming from the respective atom-light interaction. We note an increase in the output intensity for higher multi-photon contributions along the Rabi oscillations. This is consistent with theoretical predictions <cit.>, where higher photon numbers lead to higher photon flux in the output field.


	

Moreover, second-order autocorrelation measurements along the Rabi oscillations, as shown in Fig. <ref>b, exhibit an intriguing behavior: the output g_out^(2)(0) values are higher than the input g_in^(2)(0) values, except around 2π-pulse excitation, where the two-photon component of the input non-classical field is higher <cit.>. Indeed, Fig. <ref>c displays the difference between these measurements, clearly showing the change in the sign near the excitation using 2π-pulses. This is a consistent behavior that we observe also for other values of the excitation pulse length that generate the input non-classical field (see Supplementary Information). The observed feature is a result of the non-linear interaction between the emitter and the input field. The incoming temporally separated two-photon components are partially changed to one-photon components in the detected output field, leading to lower g_out^(2)(0) values. We note that linear transformations, such as unitary evolutions or optical losses, are not able to modify these photon statistics within the light field. Therefore, these measurements demonstrate the non-linear nature of the light-matter interface presented in this work.

A particularly interesting scenario occurs with the input state generated near a 2π-pulse excitation, where we can study instances involving two photons arriving at the artificial atom. Now, the input field consists of one photon that appears first and that is temporally shorter and a second photon with a longer time-scale that follows immediately after, as shown in Fig. <ref>a. The output field displays a similar two-photon structure except that the second peak is relatively stronger in comparison to the case of the input field (see Fig. <ref>b).

Time-resolved intensity correlation maps G^(2)(t_1,t_2), as shown in Figs. <ref>c,d provide insights into the processes occurring for this case of two incident photons. For the input field, one finds two-photon correlations along the horizontal and vertical axes of the map, which originate from temporally-orthogonal events—one photon first, and a second photon after. However, the output field also presents two-photon correlations along the diagonal, which are events of detecting two photons simultaneously. This correlated two-photon wave packet shows a measured decay time of τ' = 108(6) ps, extracted from the exponential fit of the decay for t_1 = t_2. This corresponds to approximately half the spontaneous emission decay time τ = 227 ps of the artificial atom (see Fig. <ref>b), a scaling factor that we also predict theoretically (see Fig. <ref>d). This significant observation explains the underlying mechanism: within all the complex quantum non-linear processes at play, sometimes one photon arrives at the artificial atom and excites it, and a subsequent photon arriving at the excited atom leads to the generation of two photons sharing the same temporal mode. These observations indicate stimulated emission processes <cit.> between only two quanta of light. This represents a fundamental form of mediated photon-photon interaction in a quantized field.



 §.§ Discussion

In this work, we reported the first observation of quantum light interacting with an atomic system in a solid-state platform. We started by employing a QD-cavity device that produces a high-fidelity single photon with high fiber-efficiency. Indeed, device efficiency has been key in allowing the observation of subtle changes in photon statistics within such processes, a task previously out of experimental reach. We then used a setup with fast and active path control that allows to send resonance fluorescence back to the quantum emitter and later observe and analyze the results of their interaction.

We studied different aspects of this process. First, we observed an artificial atom being resonantly excited by one photon. Here, we demonstrated the coherent nature of this process by observing single-photon interference mediated by an atom. An interesting aspect of this scenario is the observation that one photon is sufficient to excite the artificial atom, followed by the re-emission of a later photon. Moreover, we performed HOM experiments and showed that the output field produced at consecutive interaction instances remains indistinguishable. We also observed non-linear aspects of the light-matter interface acting on non-classical wave packets with one and two photons. We find that photon statistics change as a result of the interaction, with the output field containing more relative two-photon terms, which represents the first observation of photon-number dependent sorting occurring on non-classical input light.

Time-resolved intensity correlation maps also revealed a prime non-linear aspect of a quantum light-matter interface: stimulated emission. We observe that the output field contains simultaneous two-photon correlations, which are not present in the input non-classical field. We argue that these events occur when one resonant photon excites the atom, and a subsequent resonant photon stimulates the emission of a photon in the same spatio-temporal mode. The correlated output mode is a result of effective photon-photon interactions introduced in the highly non-linear medium.

We believe that our observations open a novel path for exploring light-matter interactions in a fully quantized system and will encourage a number of follow-up studies. For instance, one can envision performing this protocol in a sequential manner, where the output field resulting from the process is used again as input to observe a field resulting from multiple interactions, thus exploring the limits of the light-matter interface and answer questions such as up to which point the process can remain coherent or what types of non-classical light fields can be achieved asymptotically. Moreover, a QD emitting a temporal stream of photons entangled, e.g., in their polarization <cit.> or photon-number <cit.>, could be used in a similar protocol, using either the same or a different QD, to entangle spatio-temporally multiple quantum emitters without requiring remote two-photon interference protocols or to create complex states of multi-partite entangled quantum light-matter systems.





 §.§ Methods

The InAs/GaAs QD investigated is coupled to a GaAs/AlAs distributive Bragg reflector that form a micropillar.
The device is connected to a surrounding diode structure, allowing for electrical tuning through the Stark effect. 
For the QD under study, the trion state is tuned into resonance with the cavity mode by applying 0.75 V. 
The Purcell enhancement results in a detected radiation lifetime of τ = 227 ps under π-pulse laser excitation, see Supplementary Information, with the QD transition emitting single photons at a wavelength of 922.1 nm.
All the measurements are performed with the sample inside a closed-cycle cryostat (AttoDry800, Attocube) at a temperature of 4 K. To focus the excitation onto the sample and to collimate the output mode, a single aspheric lens with a numerical aperture of 0.68 is installed inside the chamber. 

The initial excitation of the QD is performed using a pulsed Ti:Sapphire laser (Tsunami, Spectra Physics) with a pulse repetition rate of 80 MHz. 
For pump preparation, we shape the laser pulse by spectral filtering with a 4f system. 
The pulse length is set to ∼25 ps, assuming a transform-limited Gaussian pulse duration. 
A cross-polarized configuration is used for excitation and collection. In the collection path, linearly polarized laser light is suppressed using a set of wave plates and a Glan-Taylor polarizer with an extinction coefficient of more than seven orders of magnitude. The measured single-photon collection efficiency in the single-mode fiber is ∼28.7 % <cit.>, including all losses from excitation to detection. 
Note that in the case of trion states, which have a circularly polarized transition, the resonant excitation scheme inherently introduces 50 % loss since linearly polarized light is collected. 

For the preparation of non-classical excitation of a quantum emitter, the QD is excited resonantly by the pulsed laser, generating a temporal sequence of single photons that is then coupled to a single-mode fiber and sent to a broadband electro-optic modulator (EOM, QUBIG GmbH). 
At this stage, single photons, which serve as non-classical input, are deterministically sent back to the QD. 
To ensure that only one type of excitation is present at a given time, the delay of the returning single photons is adjusted accordingly such that the input and output signal are separated by a delay of ∼3 ns. 
The scattered Fock state then passes through the collection path and is actively routed by the EOM to a Hanbury Brown and Twiss (HBT) setup consisting of a balanced fiber beam splitter (BS) with two superconducting nanowire single-photon detectors (SNSPDs, Single Quantum) connected to a correlation unit (Time Tagger X, Swabian Instruments). 
The SNSPDs have a system detection efficiency of 86%, a timing jitter of 30 ps, and a dark count rate of < 10 Hz.
All devices are phase-locked to the pulsed laser excitation via a field programmable gate array (FPGA), which operates the EOM at a 50% duty cycle at a frequency of 8 MHz and serves as a reference clock for the measurement with a repetition rate of 8 MHz. 

For the data analysis, the raw time tags are recorded which allows the histogram and correlation measurements to be carried out in post-processing. 
Thus, the raw time tags can be analyzed in a time-gated manner for each signal. 
For the time-resolved evaluation of the multi-photon probability P_m(t_start) shown in Fig. <ref>, the gating is defined by a start time t_start and stop time t_stop (in reference to the 8 MHz clock signal produced by the FPGA), and all events occurring outside of the gating window Δ t_gate = t_stop - t_start are disregarded.
Correlating the gated channels then gives access to a time-gated g^(2)(0, t_start) value from which P_m(t_1) is inferred <cit.>. Increasing t_start step-wise (while t_stop is fixed) excludes the early fraction of the temporal profile and reveals a significant decrease in P^(m) when the rising edge is omitted. Notably, the drop is much sharper for Fig. <ref>c (top) than for Fig. <ref>c (bottom).

Our theoretical simulations follow a cascaded master equation approach in which two identical subsystems with lifetime τ = Γ^-1 = 227 ps are considered. The first system is driven by a Gaussian pulse with a FWHM of τ_p=25 ps, leading to the output photon flux ∝Γ⟨σ^1_+σ_-^1⟩(t), shown in Fig. 1b. Here, σ_±^1 and σ_±^2 are the usual atomic ladder operators for the first and second system, respectively. The second system's output flux ⟨ b^†_2b_2⟩(t), also shown in Fig. 1b, results from the output field operator b_2(t), which is a weighted sum of the input field operator ∝√(Γ)σ_-^1(t) and a system's response operator ∝√(Γ)σ_-^2(t). To account for experimental imperfections, two different dephasing mechanisms were considered, and to account for the detector time resolution, a Gaussian detector jitter with a FWHM of 60 ps was considered. 



 §.§ Acknowledgments

The authors thank Michael Trupke, Peter Rabl, Yuri Minoguchi and Juan José García-Ripoll for helpful discussions, and Virginia Oddi and Simone Piacentini for help in initial steps of the project. This research was funded in whole, or in part, from the European Union’s Horizon 2020 and Horizon Europe research and innovation programme under grant agreement No 899368 (EPIQUS), No. 899354 (SuperQuLAN), and No 101135288 (EPIQUE), the Marie Sklodowska-Curie grant agreement No 956071 (AppQInfo), and the QuantERA II Programme under grant agreement No 101017733 (PhoMemtor); from the Austrian Science Fund (FWF) through [10.55776/F71           ] (BeyondC), [10.55776/FG5  ] (Research Group 5), and [10.55776/COE1  ] (Quantum Science Austria); by the Polish National Science Center (NCN) “Sonatina” project CARAMEL with number 2021/40/C/ST2/00155. The financial support by the Austrian Federal Ministry for Digital and Economic Affairs, the National Foundation for Research, Technology and Development and the Christian Doppler Research Association is gratefully acknowledged. 









§ SUPPLEMENTARY INFORMATION



 §.§ Experimental setup

The experimental setup for non-classical excitation of a QD is presented in Fig.<ref>. 
The setup includes a cross-polarization excitation scheme, an active stage for sending single photons back to the emitter, and an analysis setup. 
The QD is first excited using resonant laser pulses. 
The sequence of single-photon time bins is then collected in a single-mode fiber, and the polarization state is set to horizontal polarization using fiber polarization control. 

 
During the first cycle, the EOM is in the off-state, which preserves the polarization state of the single photons such that light is transmitted through the PBS, is reflected by a mirror, and is coupled in reverse into the collection fiber. 
Thus, during the off-phase of the EOM, single-photon time bins return to the emitter. The scattered output signal after interaction with the QD is coupled to the collection fiber in the forward direction and sent to the EOM.

Following this, the EOM is set to the on-state, switching the polarization state of the light to vertical polarization. The light reflects at the PBS and is collected in the output fiber and sent to an HBT setup. 
This process allows simultaneous observation of the initial single-photon stream and the output signal, ensuring identical conditions for both excitation schemes. 
By recording the raw time-tag stream of each detector and the reference clock, both signals can be analyzed independently by time gating each signal in post processing. 
 


 §.§ Data set for different excitation parameters
  
For the measurements presented in the main text, we chose a laser pulse duration of ∼25 ps such a duration enhances the two-photon probability at the beginning of the lifetime because of reexcitation during the duration of the laser pulse <cit.>. 
An additional set of measurements for a shorter laser pump pulse is given in Fig. <ref>. 
For the additional data set, two-photon contributions are suppressed by setting the laser pulse length of ∼18 ps, that enhances single-photon purity while maintaining a high count rate.



The time histograms in Fig. <ref>a indicate that the dynamics of th e output signal, as discussed in the main text, are reproducible under similar conditions. 
The interaction of the QD with the shorter laser pulse reduces the probability of two-photon emission during pulse duration as it is shown in the inset of the correlation measurement in Fig.<ref>b. 
Here, the central peak does not exhibit the characteristic emission profile of re-excitation during the duration of the laser pulse compared to the measurement presented in the main text. 
This results in a higher single-photon purity of the first generation with a second-order auto-correlation function value of g_in^(2)(0) = 1.73(2)%. 
The correlation measurement of the second generation is plotted in Fig.<ref>c. 
The measurement has a second-order auto-correlation function value of g_out^(2)(0)=3.4(2)%, indicating that the single-photon purity of the second generation is also improved as compared to the measurement presented in the main text, while remaining a lower single-photon purity relative to the first generation.

Further, we investigate Hong–Ou–Mandel (HOM) interference of each signal. The measurement results are presented in Fig. <ref>. The raw HOM visibility of the initial single-photon signal generated by laser excitation is V_in = 90.0(2)% (see Fig. <ref>a), yielding an indistinguishability value <cit.> of ℐ_in = 93.3(2)%. The raw HOM visibility of the signal after interaction with the QD is V_out = 70.4(5)% (see Fig. <ref>b), resulting in an indistinguishability value of ℐ_out = 76.4(6)%. The reported values do not account for unbalanced splitting ratios in the linear network.



The indistinguishability value is defined as ℐ = V+g^(2)(0)/1-g^(2)(0) <cit.>. The raw HOM visibility is calculated as V = 1-A_0/A_1, where A_0 is the integrated area of the central peak and A_1 is the mean integrated area of the peaks at a time delay of 12.5 ns. The measurement uncertainties are estimated following Poissonian counting statistics. 



 §.§ Time histograms for classical and non-classical excitation

The time-resolved fluorescence measurements of the QD transition (trion) are displayed in Fig. <ref>a. The data represent the detected raw measurements with a resolution of 10 ps (binwidth) and without corrections for a detector jitter of 30 ps. Figure <ref>b shows the detected time-resolved dynamics of the interaction between a single-photon wave packet and the emitter. Note that the dip and the attributed decay of the first peak originate from single-photon interference and absorption, while the second decay indicates the decay rate of the emission after the excitation of the emitter by a single photon <cit.>.

	



 §.§ Theoretical model



  §.§.§ Cavity QED system coupled to a waveguide

We model the system as a two-level system (TLS) interacting with two orthogonally polarized cavity modes (H and V) via the usual Jaynes-Cummings interaction, whose Hamiltonian reads (ħ=1) <cit.>

    H_ JC= ω_0/2σ_z+ω_ Ha^†_ H a_ H+ ω_ Va^†_ V a_ V +i g_ H( σ_+ a_ H - σ_-a_ H^†) + g_ V( σ_+ a_ V + σ_-a_ V^†).

Here, ω_0 and σ_z,+,- are the frequency and Pauli operators characterizing the TLS, ω_ H,V and a^†_ H,V (a_ H,V) are the frequencies and creation (annihilation) operators characterizing the cavity modes, and g_ H,V are the light-matter coupling strengths. The one-sided cavity is coupled to the continuum with a total photon decay rate κ. In the “bad cavity" limit, κ≫ g_ H,V, the cavity modes can be adiabatically eliminated, and the dynamics of TLS is then governed by the Lindblad master equation

    ρ̇=ℒ_0 ρ= -iδ_L[σ_z,ρ]/2+ Γ D[σ_-]ρ + γ_d D[σ_z]ρ,

where ρ(t) is the density matrix, δ_L=ω_0-ω_L is the detuning determined by the frequency of the rotating frame ω_L, and D[X]ρ=Xρ X^† -1/2(X^† X ρ + ρ X^† X) is the Lindblad super-operator describing relaxation (σ_-) or dephasing (σ_z). On the one hand, the total decay rate of the TLS Γ=Γ_ H+Γ_ V+γ^* accounts for losses via the cavity modes (Γ_ H,V= 4g_ H,V^2/[κ (1+4δ_ H,V^2/κ^2)] with δ_ H,V=ω_0-ω_ H,V) or via other modes in the system (γ^*≪Γ). On the other hand, γ_d≪Γ accounts for a phenomenological dephasing rate. 

For the sake of simplicity, in the following we consider g_ H=g_ V, δ_ H=δ_ V=0 and γ^*=0. As a result, Γ_H=Γ_V=Γ/2. Furthermore, we assume our cavity to be one sided. Then, the following input-output relations hold

    b^ H,V_out=b_in^ H,V+√(Γ_ H,V) σ_-= b_in^ H,V+√(Γ/2) σ_-,

where b^ H,V_in and b^ H,V_out are the input and output field operators, respectively. Note that in the absence of a horizontally-polarized input field, ⟨ b_in, H⟩=0, the measured output field flux N(t)=⟨ b_out, H^†(t)b_out, H(t)⟩ is proportional to the probability that the TLS is in the excited state P_e(t)=1/2(1+⟨σ_z(t)⟩), which can be calculated via the density matrix ρ(t) as ⟨ X⟩ = Tr(ρ (t)X). 



  §.§.§ Laser driving and first-generation photon

In our system, one of the cavity modes (V) is used to drive the TLS, while the other one (H) is used for measurement. We model the driving laser as a Gaussian pulse leading to an input field <cit.>

    ⟨ b^ V_in⟩(t)=√(n_L)(4 ln(2)/πτ_p^2)^1/4exp(-2ln(2) (t-t_c)^2/τ_p^2)

where τ_p is the pulse width, t_c is the instant characterizing its central point, and n_L corresponds to the number of photons in the pulse, proportional to the laser intensity. This translates into a driving term in the effective TLS Hamiltonian, which in a frame rotating with the laser frequency ω_L reads

    H(t)=Ω (t)/2(σ_++σ_-),

where Ω(t)=4g_ V⟨ b^ V_in,+⟩(t)/√(κ) is the Rabi frequency. It is convenient to express the Rabi frequency Ω(t) in terms of the pulse area A as

    Ω(t)=A(2 ln(2)/πτ_p^2)^1/2exp(-2ln(2) (t-t_c)^2/τ_p^2),

ensuring ∫_0^∞ dt' Ω(t')=A and removing dependance on parameters κ, g_ V or n_L. Furthermore, we include a phonon-induced dephasing term as in Ref. <cit.>. Then, the Lindblad master equation for the first generation photon looks

    ρ̇=ℒ_Ωρ=-i[H(t),ρ] + Γ D[σ_-]ρ + γ_d D[σ_z]ρ + B Ω^2(t) D[σ_z]ρ,

while the measured output flux is
N_1(t)∝⟨ b_out, H^†(t)b_out, H(t)⟩=Γ P_e(t) /2.



  §.§.§ Single-photon driving and second-generation photon


In the experiment, the horizontally polarized light that comes out, b_out,1^ H, is reflected back to the cavity, and becomes the input field b_in,2^ H at a later time, i.e. b_in,2^ H=√(η_ loss) b_out,1^ H. Here, η_ loss accounts for the fraction of photons that survive after traveling through the optical setup back into the cavity. As the light entering the cavity is not coherent, we cannot represent it as a Hamiltonian term. Instead, we make use of a cascaded master equation approach <cit.>, where the output of a first system, driven by a laser pulse and with Pauli operators σ_α^(1) (with α={+,-,z}), is used as the input for a second system, with Pauli operators σ_α^(2). The full master equation then reads

    ρ̇=(ℒ^(1)_Ω +ℒ^(2)_0  )ρ - √(η_ loss)/2Γ([σ_+^(2),σ_-^(1)ρ] +[ρσ_+^(1),σ_-^(2)] ).

The horizontally polarized output field operator for the second system now reads

    b^ H_out,2=b_in,2^ H+√(Γ/2) σ_-^(2).

As in reality our cavity is not perfectly one-sided, only a fraction of the light received by the atom will come out from the out mode b^ H_out,2. We model this by assuming a percentage η_ re of the input light gets reflected right before entering the cavity. Then this fraction destructively interferes with the output light field b^ H_out,2. The modified measured field operator then reads

    b^ H_out,2=√(Γ/2) [(√(η_ loss) -√(η_ re))σ_-^(1)+ σ_-^(2)]

where η_ loss=(1-η_ re)η'_ loss. 



  §.§.§ Two-photon correlation functions


The measured two-time correlation function G^(2)(t_1,t_2)=⟨ b^†_out(t_1)b^†_out(t_2) b_out(t_2)b_out(t_1)⟩ is equivalent to G^(2)(t,t+τ), where τ=|t_2-t_1|, thus, we define g^(2)(0) as

    g^(2)(0)=2∫_0^∞ dt∫_0^∞ dτ G^(2)(t,t+τ)/|∫_0^∞ dt ⟨ b^†_out(t)b_out(t)⟩|^2,

Under the previously considered approximations, the two-time correlation function for the first generation signal is

    G^(2)(t,t+τ)= Γ^2/4 Tr(σ_+^(1)σ_-^(1) 𝒰^(τ)[σ_-^(1)𝒰^(t)[ρ_0] σ_+^(1)]),

while, for the second generation, this is

    G^(2)(t,t+τ)=Γ^2/4 Tr(S_+ S_-  𝒰^(τ)[S_-𝒰^(t)[ρ_0] S_+]),

where S_±=(√(η_ loss) -√(η_ re))σ_±^(1)+ σ_±^(2). Here, ρ_0 is the initial state ρ_0=| g⟩⟨ g|_1⊗ | g⟩⟨ g|_2, and 𝒰^(τ) is the time-evolution super-operator associated to the cascaded master evolution in Eq. (<ref>).  



  §.§.§ Numerical simulation and relevant parameters


The dynamics of our system is described by the cascaded master equation introduced in Eq. (<ref>). The free parameters are selected so as to maximise agreement with the experimental results. In particular, the value of the decay rate Γ is chosen Γ^-1=227 ps, and the dephasing rates γ_d=0.035 Γ and B=1.3× 10^-4Γ^-1. Also, the reflection parameter η_ re is chosen to be η_ re=0.05, while we find the studied observables independent of the loss parameter η'_ loss. Furthermore, the pulse parameters are chosen to be τ_p=25 ps and t_c=200 ps. Finally, all time-dependent observables, e.g. N(t), are assumed to have a finite time resolution limited by the detector's jitter, which is assume to be Gaussian with a full width at half maximum of 60 ps.


 Paper Title: Private and Robust States for Distributed Quantum Sensing.



   

1,2,3,4]Luís Bugalho

4]Majid Hassani
1,2,3]Yasser Omar
4]Damian Markham

[1]Instituto Superior Técnico, Universidade de Lisboa, Portugal
[2]Physics of Information and Quantum Technologies Group, Centro de Física eEngenharia de Materiais Avançados (CeFEMA), Portugal
[3]PQI – Portuguese Quantum Institute, Portugal
[4]Sorbonne Université, CNRS, LIP6, 4 Place Jussieu, Paris F-75005, France


Private and Robust States for Distributed Quantum Sensing
    [
    July 31, 2024
=========================================================




Distributed quantum sensing enables the estimation of multiple parameters encoded in spatially separated probes. While traditional quantum sensing is often focused on estimating a single parameter with maximum precision, distributed quantum sensing seeks to estimate some function of multiple parameters that are only locally accessible for each party involved. 
In such settings it is natural to not want to give away more information than is necessary. To address this, we use the concept of privacy with respect to a function, ensuring that only information about the target function is available to all the parties, and no other information.
We define a measure of privacy (essentially how close we are to this condition being satisfied), and show it satisfies a set of naturally desirable properties of such a measure. Using this privacy measure, we identify and construct entangled resources states that ensure privacy for a given function under different resource distributions and encoding dynamics, characterized by Hamiltonian evolution.
For separable and parallel Hamiltonians, we prove that the GHZ state is the only private state for certain linear functions, with the minimum amount of required resources, up to SLOCC. Recognizing the vulnerability of this state to particle loss, we create families of private states, that remain robust even against loss of qubits, by incorporating additional resources.
We then extend our findings to different resource distribution scenarios and Hamiltonians, resulting in a comprehensive set of private and robust states for distributed quantum estimation. These results advance the understanding of privacy and robustness in multi-parameter quantum sensing.





§ INTRODUCTION

Quantum metrology and quantum sensing have been longstanding areas of interest in quantum information research <cit.>. In particular, they are known to surpass the classical bounds for the estimation physical parameters in experiments and reach the Heisenberg scaling <cit.>.
The basic idea is that a probe state interacts with some physical process, which encodes parameters into it, it is then measured and the results are used to estimate the parameters.
In the case of single parameter estimation, one is interested in estimating one parameter of a system with the maximum precision <cit.>. However, in the multi-parameter scenario, not only the precision is important, but also what information is available, and entangled probes across sites have been seen to be useful. With the undergoing development of quantum networks, both over large distances <cit.> and local-area types of networks <cit.>, distributed quantum sensing has emerged as a promising area of research, taking the multi-parameter scenario to applications with near-term potential <cit.>. There have been several experiments demonstrating the quantum advantage for this setting <cit.>. 

Distributed quantum sensing has numerous use cases, such as clock synchronization protocols <cit.>, optical interferometry proposals <cit.>, some preliminary work regarding gravity and dark-matter experiments  <cit.> and physical implementation of sensors capable of measuring such quantities <cit.>. One can also find different approaches to solve the optimal estimation of sensing parameters, such as variational quantum sensing <cit.>, covert sensing <cit.>, error-corrected <cit.> and error-mitigated <cit.> quantum sensing, and controlled enhanced quantum sensing <cit.>.

The framework for distributed quantum sensing has been well-established as quantum sensor networks <cit.>, where each quantum node holds a set of resources (in our case, qubits), and has access to a local parameter θ_μ (encoded by some local Hamiltonian) see Fig. <ref>). We are interested in when they want to estimate not the local parameters themselves, but some function of these parameters f({θ_μ}) ≡ f(). 
More recently, security concerns have been addressed through several works <cit.>, along with the critical concept of privacy <cit.>. 
On the one hand, in a network setting there may be eavesdroppers who want to gain unauthorised access to the value of the parameters, or even simply disrupt or corrupt the estimation process <cit.>. On the other hand, some parties who are involved in the estimation iteself, may like to gain more information than they should - for example the value of other parties local parameters themselves. This is addressed by the notion of privacy, which ensures  that, only information about a target estimator can be obtained <cit.>. 
Who has access to what information about the parameters typically depends on the initial state used in the estimation scenario, prompting the natural question of identifying private states. 
In <cit.> privacy is defined for a specific function (the average value of the parameters), and the questions of general functions, the optimality of the states, and how to deal with noise were left open. 

In this work we develop a broad framework for privacy in networks of quantum sensors. We tackle these questions by finding a way to define and quantify privacy through the analysis of the quantum Fisher information (QFI) matrix, a key metric in a quantum estimation scenario <cit.>. From here we verify that the amount of resources and the control over the dynamics that encode the parameters determine the functions of parameters available at hand. We are able to find a general expression for the QFI matrix of any arbitrary quantum pure state, starting with stabilizer states, in the multi-parameter scenario. We use these results to prove which ones are the only private states, i.e. states where the information available is only the target function. Our findings relate the amount of resources and the properties of the encoding dynamics to the functions available at hand for an estimation scenario. In particular, for local and separable encoding dynamics, it will be required a minimum amount of distributed resources to even be able to estimate a target function privately. We generalize our findings then for local but not necessarily separable dynamics, by analyzing the eigenvalues of the local Hamiltonians. In creating these families of private states, we then question their robustness against several types of common Pauli noise, and for qubit-loss, which is know to hinder quantum sensing. We find that some of our private states retain information even after particle loss, while at the same time, remaining private.  

This paper is structured as follows: we start in Section <ref> by analysing the methods commonly used to tackle the problem of multiparameter sensing and define exactly our setup problem as a network problem. This encompasses defining three things beforehand: (i) what are the target functions of parameters; (ii) defining what are the resources and introducing a notation to deal with them, and finally (iii) creating a measure for privacy from a set of premises. 
In Section <ref> we address the problem of finding private states. We start by introducing important concepts and notations to deal with the upcoming problem. Then we present statements about the privacy of states both for stabilizer states and arbitrary pure states for local and separable Hamiltonians, generalizing then to arbitrary local Hamiltonians. 
We provide conditions for a given function to be private and find the unique minimum private states, give some examples and then provide relaxations of optimality to allow robustness.
Finally, over Section <ref>, we analyse the effect of different noise types over our found families of private states. Over the manuscript we try to be consistent notation wise, using greek letters for nodes, latin letters for individual qubits, boldsymbols to denote a higher dimensional object, wether it be a partition on a set, a vector of parameters or a larger operator decomposed by smaller operators.






§ ESTIMATION SCENARIOS AND PRIVACY 


There are three important inputs of an estimation scenario: (i) the initial state, (ii) the encoding dynamics and (iii) the measurement (see Fig. <ref>). 
Consider first the single parameter case. The initial state is transformed via the encoding dynamics to state ρ_θ, which is them measured giving results x,  with probability p(x|θ), from which one would like to estimate the parameter theta.
In the frequentist approach to the estimation problem, we are interested in retrieving information of the parameter over successive measurements. In this scenario, the overall precision of the parameter is bounded by the Fisher information (FI), F(θ), and the number of measurements made m <cit.>, in the assymptotic regime:

    Δθ^2    ≥1/m F(θ)
    
        F(θ)    = ∫ p( | θ) (∂_θln p( | θ))^2 d .


When the estimation scenario involves a quantum initial state and quantum dynamics, the Fisher information becomes its quantum counterpart. The probability distribution is now given by the Born rule p( | θ) =  [Π_ρ_θ], where the set {Π_}_ denotes a POVM satisfying ∑_Π_=1. The quantum version of the FI can then be written as:

    ℐ(p( | θ))    = ∫ p( | θ) (∂_θln p( | θ))^2 d
       = ∫1/p( | θ)(∂_θ  p( | θ))^2 d
       = ∫1/ [Π_ρ_θ](∂_θ [Π_ρ_θ])^2 d
       = ∫ [Π_ρ̇_θ]^2/ [Π_ρ_θ] d  ,

where ρ̇_θ=∂ρ_θ/∂θ. By maximising over all possible POVMs, the quantum Fisher information (QFI), 𝒬(θ) appears by implicitly defining an additional operator named the symmetric logarithmic derivative (SLD), and Eq. <ref> becomes:


    Δθ^2    ≥1/m 𝒬(θ)
    𝒬( θ)    = ρ_θ L_ρ_θ^2 ,   ∂_θρ_θ = ρ_θ L_θ + L_θρ_θ/2.


The QFI depends on the state of the system immediately before measurement, meaning it will depend on the initial state of the system and the dynamics of the encoding (and additional sources of noise which we leave for discussion on the robustness section of this work). Eq. <ref> has a simpler counterpart for the case of pure states, as in this case the SLD is simply given by L_θ = 2∂_θρ_θ:


    𝒬 (ρ_θ) = 4 ⟨∂_θψ_θ | ∂_θψ_θ|-⟩⟨∂_θψ_θ | ψ_θ|⟨%s|%s⟩⟩ψ_θ | ∂_θψ_θ.


In quantum sensing, and in particular in the distributed scenario, the goal is often to estimate a function of parameters that are spatially distributed. This means that the dynamics that encode the parameters are in spatially separated locations and the resources only have access to their own local dynamics. The natural way of discussing this problem is on a network, where each channel encoding one parameter is associated with one node. While the structure of the network impacts the initial state distribution, it does not impact the overall estimation scenario. Given a graph G = (V,E), in a distributed scenario we consider a set of nodes {μ∈ V(G) = 1,2,⋯,k}, each of them with access to some quantum dynamics described by a quantum channel Λ_θ_μ(ρ) (if unitary, this is described by a Hamiltonian), which encodes its own parameter θ_μ, as we outline in Fig. <ref>. 

In this case, the QFI becomes more complex as it involves multiple parameters. Given that the Crámer-Rao bound is a bound on the variance, in the multi-parameter scenario, the variance becomes a covariance matrix. Therefore, it makes sense that the QFI is also a matrix. The description follows naturally from Eq. <ref> by substituting ∂_θ→∂_θ_μ, ∂_θ_ν. This way, the SLD is now defined with respect to one parameter L_θ_μ and the Crámer-Rao bound and QFI matrix entries are given by <cit.>:


    COV()    ≽1/m𝒬()^-1
    𝒬_μν (ρ_)    =ρ_ L_θ_μ L_θ_ν + L_θ_ν L_θ_μ/2  ,   ∂_θ_μρ_θ = ρ_θ L_θ_μ + L_θ_μρ_θ/2.

Expanding for pure states where ρ_ = |ψ_⟩⟨ψ_|, just as in Eq. <ref> using L_θ_μ = 2 ∂_θ_μρ_, one can verify that the QFI simplifies to:

    𝒬_μν (ρ_)    =4 ⟨∂_θ_μψ_ | ∂_θ_νψ_|-⟩⟨∂_θ_μψ_ | ψ_|⟨%s|%s⟩⟩ψ_ | ∂_θ_νψ_.

Since each node only has access to its own parameters in distributed scenarios, it is reasonable to assume the encoding dynamics are local and unitary (see Fig. <ref>). There are multiple scenarios one can consider for this subset still (see Fig. <ref>). Let us detail the most general one given in Fig. <ref>c :


    Λ_θ_μ(ρ)    = e^-i θ_μH_μ ,   H_μ = ∑_P_j ∈𝒫_μ b_j P_j ,   𝒫_μ = { P ∈𝒫_n : P = _j∉μ1_j _j∈μσ_j },

where 𝒫_n is the set of all Pauli strings with size n, which is the total amount of qubits (see subsection detailing the notation for resources further ahead). Note that the Pauli strings of size n form an orthogonal basis for the vector space of complex 2^n×2^n matrices. Although this description can be quite general, what is usually the subject of sensing scenarios are channels which are local and separable for each qubit <cit.>(see Fig. <ref>b):


    Λ_θ_μ(ρ) = e^-i θ_μG_μ ,   G_μ = ∑_j∈μ G_j ,  G_j ≡1⊗⋯⊗ G_j ⊗⋯⊗1Λ_θ_μ(ρ) = _j∈μ e^-i θ_μ G_j ,

where we did not fixed a given generator of translation for any qubit, i.e. the operators G_j are described by x⃗·σ⃗, where x⃗∈ℝ^3 is normalized, and σ⃗ is the Pauli vector. As it is known, the Pauli group on 1 qubit exponentiates to the unitary group on 1 qubit U(2) <cit.>. This consists on the most general local and separable encoding dynamics, where each qubit acquires the phase once. One can also refer to this case as parallel, in the sense that if every qubit encoding is separable from one another, then each local encoding can be done at the same time, in parallel, and consequently so can the global dynamics. If one were to consider more general encoding dynamics, such as the case in Eq. <ref> and Fig. <ref>c, one may arrive at a problem that quickly becomes infeasible, and the definition of sampling the system starts to drift. This is due to the fact that, for separable dynamics, one might argue that each qubit goes through the parameter channel only once, or is sampled only one time <cit.>. With more complex Hamiltonians, this is not always the case. Some Hamiltonians cannot be directly rewrote into only one access to a channel for each qubit. We will discuss this case further, and provide a statement to building private states for a given arbitrary Hamiltonian.

Nonetheless, for separable encoding dynamics, even though we do not fix the generators G_j, we can reduce the problem complexity by fixing the dynamics and resorting to the following lemma:



lemmaunitaryequiv
Consider two different generators for single qubit unitary operators G and G', and a unitary operator W.

    ∀ G, G'; ∃ W : W^† G W = G'



[Interplay Dynamics ↔ Initial State]corollaryunitaryequivalence
Consider two different unitary encodings, in terms of two different generators U_θ = e^-iθ G and U_θ'= e^-iθ G', and a unitary operator W.

    ∀ U_θ, U_θ'; ∃ W : W^† U_θ W = U_θ'



We provide proofs for the above lemma and corollary over Appendix <ref>. From Corollary <ref> we get that changing the dynamics is equivalent to changing the initial state locally, and vice-versa. This in turn means that, for every separable dynamics we are always able to find a state with access to the same information, by applying a local unitary gate. This allows us to fix either one part or the other when trying to optimize the estimation. Because of this, we will often fix the local dynamics to a unitary of the type e^iθσ_j where σ_j is a Pauli operator (we will choose Z whenever we need to), and say that the initial state attaining a certain estimation outcome is equivalent up to a certain local-unitary (LU) operation.




 §.§ Resources


Unlike the single parameter scenario of quantum sensing, in distributed sensing the resources are spatially distributed over a structure resembling a network, hence the name quantum sensor networks <cit.>. To convey their distribution over the nodes (or parties), it makes sense to have a notation that attributes a corresponding node to each qubit. The ensemble of qubits constitutes the set of  resources for the estimation scenario, even though one could, equivalently, consider the amount of samples or duration of access to the encoding dynamics. 
The case where the encoding dynamics are not separable is more involved. 
We will comment further on this over Section <ref>, but for now let us define the number of resource in terms of amount of qubits available locally.

 Introducing the notation for the resources sets we will use throughout the paper: we denote by 𝒩_μ as the set of qubits (or local resources) belonging to node μ (see Fig. <ref>), which amounts to the total set of qubits given by 𝒩= ∪_μ=1^k 𝒩_μ. For this reason, we denote 𝒩 = {𝒩_μ}_μ∈ V as a partition of the set, correspondent to the distributed resources. Moreover, the number of qubits in each set is given by |𝒩_μ| = n_μ, which means the vector n⃗ = (n_1,n_2,⋯,n_k) is the vector of resources and |𝒩|=n⃗ = n is the total amount of resources. Moreover, we will assume that the states can be picked from the complete Hilbert space of all qubits, which we denote by ℋ (𝒩).

Another important aspect that will be used throughout the paper, is to be able to order different vectors, such as the vector of resources. Since we are usually working with a vector-space defined over some ordered ring, we resort to the regular product order, which has appeared in quantum information under the name of majorization as well (e.g. <cit.>), defined as follows:


Let 𝕂 be a ordered-ring and H ⊆𝕂^k. Let a⃗, b⃗ be two vectors belonging to H. Let ≤ be a total order of the field 𝕂. The product order is defined as:

    a⃗≼b⃗    iff.      a_j ≤ b_j  ∀ j ∈{1,2,⋯,k}.

Moreover, we say a⃗≺b⃗ if a⃗≼b⃗ and ∃ j: (a⃗)_j < (b⃗)_j.


Note that unlike ≤, ≼ is not a total order, i.e. there exists vectors such that neither a⃗≼b⃗ nor b⃗≼a⃗. In this case we use the notation a⃗⋠b⃗. 



    (2 , 4, 3) ≼ (2,4,3)    ,    (2 , 4, 3) ≺ (2,6,4)     ,     (1 , 3, 2) ⋠(3,2,4)





 §.§ Functions of Parameters


Before discussing the bounds on the precision and information metrics, let us first clear up the idea of what is the “good” definition for functions of parameters. The first thing to notice is that, if one is able to estimate f(), then one can also estimate α f() for any α. For this reason, let us establish an equivalence between functions: we say f() ∼ g() if f() = α g() for some α. Since we are working with linear functions of parameters, the functions can be described as an inner-product between a vector of coefficients and a vector of the parameters : f() = a⃗·, similarly to previous works <cit.>. 

As we will see in this paper, the linear coefficients available are somehow related with the eigenvalues of the Hamiltonian. For this reason, we start by finding a way to describe them, that is able to convey the distributed scenario we are working with. Taking the most general local Hamiltonians description in Eq. <ref>, one can find an orthonormal basis for the local states by choosing the eigenvectors of the local Hamiltonians. Associated to each of these eigenvectors we then have their corresponding eigenvalues, which we can arrange as follows:

    H_μ|λ_j^μ⟩ = λ_j^μ|λ_j^μ⟩,   ℬ^μ = {|λ_j^μ⟩}_i ∈μ ,   𝒪^μ = {λ_j^μ}_j ∈μ,

where ℬ^μ is a local basis for the qubits belonging to node μ, and 𝒪^μ is the list of non-identical eigenvalues for the local Hamiltonian of node μ. Note that this list does not have the same size as the basis if the Hamiltonian is degenerate, although this will not pose any problem.

Going from the local Hamiltonians to the full network states can be easily done by choosing an orthonormal basis from the tensor product of the local basis: 

    |λ_j⃗⟩ = _μ|λ_j_μ^μ⟩ ,   H_μ|λ_j⃗⟩ = λ_j_μ^μ|λ_j⃗⟩,   ℬ = _μℬ^μ ,   𝒪 = ∏_μ𝒪^μ,

where the vector j⃗ is a labeling vector such that each component (j⃗)_μ≡ j_μ corresponds to one of the local basis states of node μ. Moreover, the set 𝒪 can be seen as a discrete subset of the k-orthotope 𝒪 defined by [λ_min^1,λ_max^1] ×⋯× [λ_min^k,λ_max^k].

As we will verify further ahead, the natural functions one can estimate correspond to vectors living inside of the k-orthotope 𝒪. We will start by analysing the case of separable dynamics and later generalize the most general encoding dynamics.
For  separable dynamics we have that:

    
  * Controlled scenario: a⃗∈ℝ^k – we are able to control the local encoding dynamics by an additional parameter t, Λ = Λ(θ,t) = Λ (θ t) ∼ e^iθ t σ_j (see Fig. <ref> d);
    
  * Non-controlled scenario: a⃗∈ℤ^k – only the local number of qubits per node or party is controllable (Λ = Λ(θ) ∼ e^iθσ_j).     

Note that from the equivalence of functions introduced above, any f() = a⃗· such that a⃗∈ℚ^k is equivalent to some g() = a⃗'· such that a⃗' ∈ℤ^k. This extends the non-controlled scenario to not only integers, but also fractionals. However, there will exist an overhead in the amount of resources. This overhead can be calculated from finding the smallest number α one has to multiply a⃗ such that αa⃗ = a⃗' ∈ℤ^k, which is given by the least common multiplier of all denominators. This depends entirely on the vector a⃗.

For the controlled scenario one can establish an equivalence with the non-controlled, by taking one qubit in every node (n⃗=1⃗) and control each local parameter t_μ in order to get a⃗. One can additionally change n⃗ and t⃗ such that n_μ t_μ = a_μ, and get to the same result. This should provide that any statement made for the non-controlled scenario should also verify for the controlled scenario. On the other hand, in the non-controlled scenario, we are limited by the number of qubits in each node, which only takes values over the integers. Taking the quotient of the equivalence relation, we get that, in this case, it suffices to analyse functions such that every element of a⃗ has no greatest common divisor than 1. For this we introduce the following definition:


Let a⃗ be a integer vector of size k, a⃗∈ℤ^k, a⃗ = (a_1,a_2,⋯,a_k). 

    gcd(a⃗) = gcd(a_1,a_2,⋯,a_k)


This means that the representative of each equivalence class of functions has a⃗ such that gcd(a⃗)=1. Note the results for arbitrary a⃗ verifying gcd(a⃗)=1 work backwards for n⃗=1⃗ in the controlled scenario.

In the non-controlled scenario, let a⃗ and n⃗∈ℤ^k be the vector of the target function and the vector of resources, respectively. From here on, when we refer to the vector of the target function a⃗, we will assume that gcd(a⃗)=1. Using this ordering, let us define four different regions of amount of resources:


	
  * n⃗≺a⃗ or n⃗⋠a⃗ (No-Privacy Zone)
	
  * n⃗ = a⃗ (Minimal Privacy Zone)
	
  * a⃗≺n⃗ and (n⃗≺ 2a⃗ or  n⃗⋠2a⃗) (Minimal plus Ancilla Privacy Zone)
	
  * 2a⃗≼n⃗ (Multiple Privacy Zone)

The labelling will become obvious as we will be able to prove different statements at each zone.




 §.§ Privacy Measure


The privacy of a quantum estimation scenario, given a target linear function was introduced in <cit.>. It states that a state is private as long as, over a distributed estimation scenario, the information accessible by the various parties is given only by the target function. Not to confuse with security, which concerns a set of malicious parties cooperating strategies that allow them to get additional information. 
The definition of privacy can be stated from the following three conditions:

	
  * Let H and D be the subsets of honest and dishonest parties, respectively;
	
  * All μ∈ H can only know information about f()
	
  * All μ∈ D can only know the function f() and all θ_μ for μ∈ D

In the case everyone is honest, we regain that all honest parties only have information about the target function, which is similar to <cit.>. If there are dishonest parties, then they cannot get access to any function in the honest subspace other than f(_H) = ∑_μ a_μθ_μ, which is equivalent to the statement in the third condition.

To motivate the construction of the a privacy measure, let us first describe one property of the QFI matrix. Given that a linear function can be described as f(θ⃗) = a⃗·θ⃗, one can always construct a non-unique orthonormal basis that includes the vector a⃗/a⃗ and defines a unitary transformation. Call this matrix A, where every vector belongs to one orthonormal basis. Then:

    θ⃗'    = A  θ⃗     with θ'_1 = ∑_i A_1iθ_i = a⃗/a⃗·θ⃗∼ f(θ⃗).


Since the QFI depends on the derivatives, and the derivative changes through linear transformations, finding the new derivatives becomes straightforward:


    ∂/∂θ_i'→∑_j ∂θ_j/∂θ_i'∂/∂θ_j∇_θ⃗'⃗→ A^T ·∇_θ⃗'⃗.

This in turn implies that the QFI changes in the following way when changing variables <cit.>:

    𝒬'_ij (ρ_)    = ∫1/ [Π_ρ_θ]∂_θ'_i(  [Π_ρ_θ] ) ∂_θ'_j(  [Π_ρ_θ] ) d
       ↓
    𝒬'_ij (ρ_)    =∑_k,l∫1/ [Π_ρ_θ] A_ik^T∂_θ_k(  [Π_ρ_θ] ) A_jl^T ∂_θ_l(  [Π_ρ_θ] ) d
       ↓
    𝒬'(ρ_)   = A^T 𝒬 A.


This has an interesting consequence: by calculating the QFI for the canonical θ set of parameters, and then diagonalizing the matrix we are finding the linear transformation that retrieves the set of independent orthonormal parameters that the estimation scheme is sensitive to. This means the eigenvectors of the QFI matrix correspond to the natural set of functions available at hand given the estimation scenario, and their corresponding eigenvalues are the precision at which we can estimate them. Note however, that this does not mean that every function precision is always attainable, given the not-guaranteed commutation of the measurements which would allow one to saturate the Cramér-Rao bound <cit.>. Nonetheless, this allows to introduce a notion of privacy - having a rank-1 QFI matrix, with the only positive eigenvector spanning the space of the target function. Note that a rank-1 matrix is naturally non-invertible, making the multiparameter Crámer-Rao bound in Eq. <ref> ill-defined as all the functions belonging to the null-space of the matrix would have an associated infinite variance. This is coherent with the fact that one could not estimate these functions, as they are not accessible. One could make use of a pseudo-inverse to address this fact, or transform into a single-parameter scenario by applying a weight matrix in the vector direction <cit.>. 
It is also important and useful to be able to quantify such a property - no state will perfectly satisfy such a condition due to inevatible noise for example, it is then important to be able to say how close we are to being private. Though <cit.> quantifies privacy for the special case considered there, it is not general, not easily calculable, and its precise meaning in terms of information leaked is not clear. 

With this in mind, we list some properties that are naturally desirable for a measure if one is to quanutify how close one is to being perfectly private:
	
  * 𝒫 = 𝒫 (𝒬,a⃗), where 𝒬 is the QFI matrix with respect to parameters = {θ_1, θ_2, ... ,θ_k } and a⃗ is the vector that defines the target function f() = a⃗·.
	
  * 𝒫 (𝒬,a⃗) ∈ [0,1] as it should lie somewhere in the real valued space.
	
  * 𝒫 (𝒬,a⃗) = 1 iff. 𝒬 = αa⃗a⃗^T, meaning is completely private.
	
  * 𝒫 (B𝒬B^T,Ba⃗) = 𝒫 (𝒬,a⃗) as changing the basis should be equivalent changing the target function.
	
  * ∀ϵ > 0, A ≽ 0,  ∃δ > 0 : | 𝒫 (𝒬+ϵ A,a⃗) - 𝒫 (𝒬,a⃗)| ≤δ to ensure that continuity of the QFI implies the continuity of the privacy.


We now present our privacy measure, which satisfies these properties:

The privacy measure of a multi-parameter estimation problem, which results in a quantum Fisher information matrix 𝒬, with respect to a target linear function f()= a⃗· with ||a⃗||=1 is given by:

    𝒫(𝒬,a⃗) = a⃗^T 𝒬a⃗/𝒬≡[ 𝒬a⃗a⃗^T ]/𝒬 = [ 𝒬 W_a⃗]/𝒬.

In case ||a⃗|| ≠ 1, simply redefine a⃗ = a⃗/||a⃗||.


In Appendix <ref> we prove that privacy measure in Def. <ref> verifies all the proposed properties, and address as well the meaning behind the numerical value of 𝒫. Nonetheless, we provide some intuition below: 

	
  * 𝒫=1, means complete privacy, and in particular a state that verifies this is guaranteed to satisfy all conditions. 
	
  * For 𝒫=0, this implies there is a function of parameters, orthogonal to the target function f(), that the dishonest parties can access optimally, which by the privacy condition (iii), the dishonest parties should have no information of. On the other hand, if everybody is honest, no information about f()=a⃗· could possibly be obtained.
	
  * If 𝒫=ϵ, then one can do the same as above, but less efficiently, i.e. the best strategy can only gather less information about the same function than when 𝒫=0. In particular, the amount of information accessible decreases with the increase of ϵ, which is equivalent to needing more repetitions to get the same variance on the information available.  


We note here that in our setting the dynamics are part of the problem setting, so that privacy above depends only on the state, hence the notion of private state. 
The idea of the private sensing protocols of <cit.> is to certify that one posses such a private state, then use it for sensing, in this way ensuring privacy. 
Our definition of privacy contains several implications for a distributed secure sensing protocol. Namely, and as a consequence of the definition of privacy, if one certifies a private state, then one is able to guarantee that a private estimation protocol is also completely secure <cit.>. All the private states we find here can be certified using the techniques of <cit.>, and so can be fit into protocols, guaranteeing private sensing.



§ BUILDING PRIVATE STATES 


To build private states, we start by developing an efficient way to characterize the QFI matrix as a function of the input state and fixed separable and parallel dynamics, first for stabilizer states using the stabilizer formalism, and next taking an arbitrary state as input, by choosing an appropriate basis. Using Theorem <ref> this generalizes to any local single-qubit (or separable) dynamics, up to some local single-qubit operation. Later we will extend into the case of the most general dynamics, by finding another basis with respect to the more general encoding dynamics. Using this, and applying the privacy definition, we are able to prove which states and superpositions of states will be private (𝒫 = 1), allowing us to create families of private states. Moreover, we do this in function of the amount of resources, as one will see is key to being able to find private states.



 §.§ Initial Concepts

Before diving into building private states, let us introduce some well-known concepts and adaptations of them, that allow us to build the tools for our proofs of privacy. Starting with the Hamming-weight, a function that, given a bit string, outputs the number of ones of the string:


Let s be a string of bits with size n, s ∈𝔽_2^n. The Hamming weight of s, h(s) is given by:

    h(s) = ∑_j=1^n s_j  .



Note, one key property of the Hamming-weight is its invariance under permutations of bits within the string s:

Let X = {1,2,...,n} be the set of positions of a bit string s of size n. Let S_X be the permutation group over X.

    h(s) = h(σ(s))   ∀σ∈ S_X, ∀ s


This in fact allows one to create equivalence classes on binary strings for the Hamming-weight, as permutations generate the group that results in the invariance of the Hamming weight:


[column sep=small]
 s  ∈    𝔽_2^n [rrd, "h"] [d, "σ"']          

 [s ] ∈    𝔽_2^n/S_X [rr, "h̃"']          ℤ



Where h̃ is an isomorphism. This will mean that doing the quotient 𝔽_2^n over S_X will create the equivalent classes under the Hamming weight. We denote them [s], and they constitute the objects in 𝔽_2^n / S_X which are unequivocally mapped to a weight under the Hamming weight map. For us it is also useful to define a slight variant of the Hamming weight, which we call the symmetrized Hamming-weight:


Let s be a string of bits with size n, s ∈𝔽_2^n. The symmetrized Hamming weight of s, h^*(s) is given by:

    h^*(s) = ∑_j=1^n (-1)^s_j  .



Moreover, these two weight-functions are related via a linear transformation:

    h^* (s) = n - 2· h(s).

Where n is the integer size of the set of indices (or the length of the bit string). Because we are in a multiparameter scenario, where the parties have access to different amounts of resources, let us define a new vectorial Hamming-weight. We call it vectorial since this time, it outputs a vector of Hamming-weights. Using the notation introduced before for the partition of the set of resources:


Let s be a string of bits with size n, s ∈𝔽_2^n. Let 𝒩= 𝒩_1 ∪⋯𝒩_k be a size k partition over the set of indices of the bit string s, such that n_j = |𝒩_j|, and ∑_jn_j  = n. Define the 𝒩-Hamming weight of s, h⃗_𝒩(s) by:

    ( h⃗_𝒩(s) )_j =  ∑_i ∈𝒩_j s_i,

where j runs from 1 to k, meaning h⃗_𝒩(s) ∈ℤ^k.

In the same way one can define its symmetrized counterpart and rewrite the linear relation between them:

    h⃗^*_𝒩 (s) = n⃗ - 2·h⃗_𝒩 (s).


What before was invariance under any permutation of the bit string, now becomes only under a restricted set of permutations. Let us define this subset and prove that indeed the vectorial Hamming weight is invariant under this subset of permutations.


Let S_X be the permutations group over a set X. Moreover, let X = {X_j}_j=1,2,…,k be a partition of X. Define S_X⊆ S_X as:

    S_X = S_X_1× S_X_2×⋯× S_X_k.




Let X be a set and X= {X_j}_j=1,2,…,k be a partition of X. Let S_X be the permutation group over X and S_X the subset of it defined according to Def. <ref>.

    h⃗_X (s) = h⃗_X (σ(s))   ∀σ∈ S_X



Prop. <ref> is simply a consequence of the commutativity of the sum. The proof of Prop. <ref> is a direct consequence of Prop. <ref> into each of the subsets of the partition.  Similarly to before, one can create the equivalence classes of the vectorial Hamming weight. Let us denote them [s]_X, which are again the objects in 𝔽_2^n / S_X which are unequivocally mapped to a weight under the vectorial Hamming weight map. 


[column sep=small]
s ∈   𝔽_2^n [rrd, "h⃗_X"] [d, "σ"']          

 [s ]_X ∈    𝔽_2^n/S_X [rr, "h̃_X"']          ℤ^k




As an example of using the vectorial Hamming weight and its equivalence classes, consider the following bit strings:


Let n⃗ = { 1,3,2 } be a vector in ℕ^3 and s_1,s_2,s_3 ∈𝔽^6 be given by s_1 =1 110  10,s_2 = 1  101  10, s_3 = 1   011  01.

    h⃗_𝒩(s_1) = h⃗_𝒩(1_𝒩_1110_𝒩_210_𝒩_3)  =  { h(1),h(110),h(10) } = { 1, 2, 1 }
       h⃗_𝒩(s_1) =h⃗_𝒩(s_2) =h⃗_𝒩(s_3)  [s_1]_𝒩 = [s_2]_𝒩 = [s_3]_𝒩

This means that all bit strings s_1,s_2,s_3 belong to the same equivalence class. Moreover, as a direct consequence of this, there exists necessarily a permutation in S_𝒩 that transforms each one into each other one.


One can then show that any bit string s, indexed by a set 𝒩, with partition 𝒩 with correspondent resource vector n⃗ has a 𝒩-Hamming weight bounded by:

    0⃗≼h⃗_𝒩 (s) ≼n⃗         -n⃗≼h⃗^*_𝒩 (s) ≼n⃗.


We also have the following consequences for vectors of integers, which we prove over Appendix <ref> and that will later be crucial:
propositiongcdmultiple
∀ a⃗,b⃗∈ℤ^k such that gcd(a⃗)=1 then, if b⃗ = αa⃗ implies that α∈ℤ.

propositiongcdorder
Let m⃗ be a integer vector in ℤ_+^k such that gcd(m⃗)=1. Then:

    ∄ a⃗≠b⃗∈ℤ_+^k,  a⃗ , b⃗≺m⃗: (m⃗ -2a⃗) ± (m⃗ -2b⃗) = αm⃗   ,   α≠ 0.



propositiongcdorderm
Let m⃗ be a target vector in ℤ_+^k such that gcd(m⃗)=1. Let n⃗ be a resource vector in the same space, such that n⃗⋠m⃗. Then:

    ∄ a⃗≠b⃗∈ℤ_+^k,  a⃗ , b⃗≺n⃗: (n⃗ -2a⃗) ± (n⃗ -2b⃗) = αm⃗   ,   α≠ 0.



All of these concepts we have introduced focus on describing the functions available by employing integer resources in an estimation scenario. The manipulation of the Hamming weight into its symmetrized counterpart is a direct consequence (together with Corol. <ref>) of being able to choose any local basis for each qubit. If one is able to choose any basis for one qubit, then each basis will have two possible qubit eigenstates, which can be characterized by a bit (0 or 1), and will have eigenvalues ± 1. Then, each eigenstate of the entire state in such basis can be labeled by a bit string of the size of the number of qubits. Since we choose the basis in the same "direction" as the individual Hamiltonians, then the eigenvectors of the local Hamiltonian will be given by the eigenstates created. Moreover, and as a consequence of the ± 1 eigenvalues of each qubit Hamiltonian, each eigenstate characterized by the bit string s will have a correspondent eigenvalue given by the symmetrized Hamming weight of that bit string. We will use this to prove then how to find the private states, and build the entire set of private states.



 §.§ Privacy for Stabilizer States with Separable Hamiltonians


Stabilizer formalism is a very useful formalism to describe quantum states in terms of their symmetries <cit.>. As the name suggests, it involves the notion of a stabilizer, an operator that stabilizes a state (S |ψ⟩ = |ψ⟩). A n-qubit state is called a stabilizer state if it is stabilized by a set of n stabilizers s ∈𝒮 such that:


	
  * [s_i,s_j]=0, ∀ s_i,s_j ∈𝒮 (Commutation),
	
  * ∀ s_i,s_j, s_k ∈𝒮: s_i ∘ s_j ≠ s_k (Linear Independence),
	
  * s_i ∈{± 1,± i}𝒫^n such that s_i ≠𝕀 (Pauli strings).

We will denote the ensemble of stabilizer states in the Hilbert space ℋ by 𝖲𝗍𝖺𝖻(ℋ). Given these conditions, the group generated by 𝒮 is the complete set of stabilizers of the state. We call this group S = ⟨𝒮⟩. It is also know that:


    |ψ⟩⟨ψ| = 1/2^n∑_s ∈S s = 1/2^n∏_s_i ∈𝒮 (1 + s_i).


Moreover, there are multiple representations for the generators of a state. We present an example of GHZ state with three qubits, in different representations, namely a stabilizer table representation and the binary sympletic representation:

    c | c | cq_1    q_2    q_3 
    
    X_1     X_2     X_3 
    
    Z_1     Z_2    1
    
    Z_1    1    Z_3 
         or      (X|Z) = 
    (
    
        1     1     1     0     0     0 
    
        0     0     0     1     1     0 
    
        0     0     0     1     0     1
    ).


One can easily check that a star-graph centered on the first node is related with the GHZ state of three qubits via Hadamard gates being applied to both leafs of the star-graph, i.e. nodes 2 and 3. Doing so, is equivalent to applying Hadamard gates to q_2 and q_3:


    |GHZ_3⟩|S_3⟩
    
     c | c | cq_1    q_2    q_3 
    
    	X_1     X_2     X_3 
    
    	Z_1     Z_2    1
    
    	Z_1    1    Z_3 
    	   
    	=  c | c | cq_1    q_2    q_3 
    
    	X_1     Z_2     Z_2 
    
    	Z_1     X_2    1
    
    	Z_1    1    X_3 
    	≡(
        
            1     0     0     0     1     1 
    
            0     1     0     1     0     0 
    
            0     0     1     1     0     0
        ),

where the binary sympletic representation becomes (X|Z) = (1|Γ), with Γ being the adjacency matrix of the star graph. In general one can always do this for a graph state. 

Using the stabilizer formalism, one can then find an expression for the QFI in terms of number of stabilizers <cit.>. Under the assumptions in Eq. <ref>:


    |ψ_⟩   = U_|ψ⟩ = _μ∈ V e^-i θ_μG_μ|ψ⟩ ,   |∂_θ_μψ_⟩ = -i G_μ U_|ψ⟩, 
    𝒬_μν (ρ_)    =4 ⟨∂_θ_μψ_ | ∂_θ_νψ_|-⟩⟨∂_θ_μψ_ | ψ_|⟨%s|%s⟩⟩ψ_ | ∂_θ_νψ_
       = 4 ⟨ψ | G_μG_ν | ψ|-⟩⟨ψ | G_μ | ψ|⟨%s|%s⟩⟩ψ | G_ν | ψ
       = 4 ∑_j ∈μ∑_k ∈ν⟨ψ | G_j G_k | ψ|-⟩⟨ψ | G_j | ψ|⟨%s|%s⟩⟩ψ | G_k | ψ
       = 4 ∑_j ∈μ∑_k ∈ν𝕀_𝒮_ψ (G_j G_k) - 𝕀_𝒮_ψ (G_j)𝕀_𝒮_ψ (G_k).

From here we can use the stabilizer formalism to understand which terms come out different from zero. There are two cases one should analyze:
 
	
  * If both a G_j and b G_k, where a,b ∈{± 1, ± i}, are stabilizers of the state ψ, then so is ab G_j G_k, making its contribution to the QFI null;
	
  * The only scenario for which the QFI is different from zero is the case where ± G_j G_k is a stabilizer (note the real part in Eq. <ref>), but neither a G_j nor b G_k are stabilizers of the state.

The case where only one either a G_j or b G_k is a stabilizer and ± G_j G_k is also a stabilizer is impossible, since it would imply all of them are stabilizers.

The QFI then becomes a combinatorial problem. The generators G_j can always be made up such that G_j^p = G_j^p 2, which is the same type of structure when working with the vector space of 𝔽_2^n, where the sum is already made modulus 2. Suppose the following set of generators:

    g(l⃗) = G_1^l_1⊗ G_2^l_2⊗…⊗ G_n^l_n,

where l⃗ = (l_1, l_2, ... ,l_n) is a vector in 𝔽_2^n, made up of zeros and ones. Using this, we can define the trivial base of n vectors that generate all vectors of 𝔽_2^n by simply:

    𝔅_e
    
    	e_1    = (1, 0, 0, ...,0,  0) 
    
    	e_2    = (0, 1, 0, ..., 0, 0) 
         ⋮
    
    	e_n    = (0, 0, 0, ...,0 , 1) 
    .

From the vector space characteristics, the choice of basis is not unique. This means that taking { g(e⃗_j) }_e⃗_j ∈𝔅_e for our stabilizer generators is equivalent to taking any other basis g(ẽ⃗_j), resulting in all terms G_j and G_j G_k being part of the stabilizers of the state. Note that, from its construction using only G_j and 1, the commutation rule for the set of generators is already verified. This means that:

Any generator set of stabilizers in the form of Eq. <ref> with an associated complete basis for the binary vectors defines a state with no information retrievable.




For a generator set to correspond to a state with information, it should contain  terms of the form G_j G_k but not the terms of the form G_j nor G_k. Since there is an isomorphism between the group of stabilizers 𝒮 and the 𝔽_2^n, any linear independent set of vectors in 𝔽_2^n defines a linear independent set of generators for 𝒮. Moreover, taking whichever complete linearly independent basis for 𝔽_2^n we verify that it includes all bit strings with length n. This in turn implies that whichever basis we choose for Eq. <ref>, it will include both terms of the form G_j G_k and G_j or G_k, resulting in a state with no information.


This allows us to prove which states are able to remain private for estimating a family of functions. The following two theorems essentially tell us that the only private stabiliser state resource is locally equivalent to the GHZ state, and the distributed resources (number of qubits per network node) is fixed by the function. We start by proving there is a zone of resources where no private state can be built:


Let a⃗, n⃗∈ℤ^k, n = n⃗_1, gcd(a⃗)=1 and f() be the linear function given by f() = a⃗·. Then, there is no stabilizer state capable of estimating f privately with n⃗ distributed resources and access to local separable dynamics U_, such that n⃗≺a⃗ or n⃗⋠a⃗.

    ∄|ψ⟩∈𝖲𝗍𝖺𝖻(ℋ(n)): 𝒫 (𝒬 (U_|ψ⟩),a⃗) = 1



Note we have only considered the amount of qubits per node, and not the more general qubits distribution using the partition notation 𝒩. We follow with the proof:

Privacy, 𝒫=1, is equivalent to having a QFI matrix given by 𝒬 = λ_1 ·a⃗a⃗^T
where we can observe a rank one matrix in the direction of a⃗.
Given that the number of operators G_j for each node μ is an integer number bounded by the amount of qubits in that same node n_μ, the maximum of 𝒬_μν is given by n_μ n_ν. From the conditions stated one can say that ∃μ: n_μ < a_μ. However, by construction a_μ has no common divisors but 1 with all the other a_ν. Since Prop. <ref> only allows λ_1 to be integer, one could never achieve a⃗a⃗^T. For this reason, whichever basis ends up maximizing the privacy of parameter a⃗· has a projection onto a⃗ plus a non-null projection onto at least another linearly independent vector, making it not completely private.


This already means we require a minimum amount of resources per node to achieve privacy. Next we prove that the first private state requires at least n⃗=a⃗ amount of resources:


Let a⃗, n⃗∈ℤ^k, gcd(a⃗)=1 and f() be the linear function given by f() = a⃗·. Then, there is only one stabilizer state, up to LU operations, capable of estimating f privately with n⃗ =  a⃗ distributed resources and access to local separable dynamics U_.




Take Thm. <ref> to deal with LU and use G_j,G_j^⊥ as the non-commuting dual of generators for each qubit. Given that n⃗=a⃗, then the only possibility to be private is if every G_j G_k ∈𝒮, as one can only find integer multiples of a⃗ when gcd(a⃗)=1 (Prop. <ref>). 
Consider the table of stabilizers given by:

    𝒮 =
        (cG
    G^⊥) = (
        ccccc
        G_1    1   1   ⋯    G_n 
    1    G_2    1   ⋯    G_n 
    ⋮   ⋮      ⋱   ⋮
    1   1   1    G_n-1    G_n 
    
        G_1^⊥    G_2^⊥    G_3^⊥   ⋯    G_n^⊥
    ).

Note the choice of G is not unique, but if generated via Eq. <ref> we cannot use a complete basis (see Prop. <ref>). By enforcing condition 2, one may arrive at G, given that all G_j G_k ∈⟨G⟩ and no G_j ∈⟨G⟩. If one would add G_j ∉G, then ∃ G_j G_k ∈⟨G⟩: G_j G_k ∘ G_j = G_k  ∀ k which means a state with zero information. If one would remove a stabilizer from G, then ∃ G_p: G_p G_q ∉𝒮 ∀ q, meaning a non private state. The only option is then to add a new stabilizer that commutes with every other in G and is linearly independent from them. This can only happen by choosing a different orthogonal generator and every element of the generator must be the same, hence: G^⊥ = a G_1^⊥ G_2^⊥⋯ G_n^⊥, a ∈{± 1,± i}. Up to unitary equivalence (see Thm. <ref>), this defines one state (up to LU) which is able to remain private.


Note that this state is LU equivalent to the GHZ state. Additionally, our statements are only valid if our stabilizers and generators G_j,G^⊥_j belong to the Pauli strings, as it is required by the stabilizer definition. With the help of Corol. <ref> one can always work with G_j = Z, G_j^⊥ = X and then up to LU, not necessarily still a stabilizer state, the statements above are true.




 §.§ Privacy for Arbitrary States with Separable Hamiltonians


To build an arbitrary state, one needs an orthonormal basis of quantum states. The choice of basis is not fixed, which allows to choose whichever preferred basis. If one chooses the basis that contains a state that is LU equivalent to a GHZ state, the calculations become rather straightforward. For this, one can pick the generator set in Eq. <ref> as a representation of our state which is LU equivalent to a GHZ state. Since we are no longer working with just stabilizer states, let G_j, G^⊥_j have arbitrary definitions G_j = a⃗·σ⃗, G^⊥_j = b⃗·σ⃗ such that a⃗ = b⃗ = 1 and a⃗·b⃗ = 0. In particular, one can and should choose the generators of the encoding dynamics to make our life easier, meaning that the G_js should be local generators in Eq. <ref>. Call this state, represented by the stabilizer table in Eq. <ref>, |𝒢_1⟩.

Then, one can think of a generator set of a state as a n subspace of 𝔽_2^2n with the help of the symplectic notation introduced in Eq. <ref>. This means there still exists another subspace of dimension n, orthogonal to the first. One can use this orthogonal subspace to generate the set of orthogonal states, and the orthogonality conditions will appear naturally. In a more concrete way, take the following generator set:


    𝒮^⊥ = (
        ccccc
        G_1^⊥   1   ⋯   1   1
    1    G_2^⊥   ⋯   1   1
    ⋮   ⋮   ⋱   ⋮   ⋮
    1   1   ⋯    G_n-1^⊥   1
    1   1   ⋯   1    G_n).


This means one can describe an orthornormal basis as {|𝒢_j⟩ = s̃_j |𝒢_1⟩}_s̃_j ∈⟨𝒮^⊥⟩ with 2^n elements, as |⟨𝒮^⊥⟩| = 2^n. To prove orthonormality take:


    ⟨𝒢_i |𝒢_j |   ⟩=  ⟨𝒢_1|s̃_i  s̃_j |𝒢_1⟩ =  [ s̃_i  s̃_j |𝒢_1⟩⟨𝒢_1| ] = 1/2^N∑_s_k ∈⟨𝒮⟩ [ s̃_i  s̃_j s_k ] = δ_ij .

We now decompose an arbitrary state |ψ⟩ over our resources in this basis 

    |ψ⟩ = ∑_j α_j |𝒢_j⟩.


Using this, we can find a simple expression for the QFI matrix (see derivation over Appendix <ref>) using the notation introduced for our distributed qubits, namely using the partition 𝒩 with correspondent vector of resources given by n⃗: 

    𝒬 (ρ_)    = C 𝔔 C^T ,   𝔔   = Λ - v⃗v⃗^T,

where C is a matrix that can be made such that each line i ranges from i=0,…,2^n-1-1 is given by the corresponding vectorial Hamming-weight of the index i, h⃗^*_𝒩 (i) = n⃗-2c⃗_i, with 0⃗≼c⃗_i ≺n⃗. The matrix 𝔔 is a rank-1 correction of a diagonal matrix, namely Λ = diag(λ_0,...,λ_j,...,λ_n) and v⃗ = (v_0, ..., v_j, ..., v_n), such that λ_j ≥ |v_j| for all j. The structure of this decomposition is exactly what allows us the prove the privacy statements we will make. We will be able to establish bounds on the rank of the QFI matrix using this fact, together with a few additional theorems and propositions we detail in Appendix <ref>. 

As we have seen before, the functions at disposition are limited by the amount of resources used, so to provide the limits, we have defined privacy in terms of resources. Let us make a statement for each possible zone introduced in the Resources section (Sec. <ref>). Note that in the previous section we had limited ourselves to states described by stabilizers. In this section we perform equivalent statements on privacy, but for arbitrary pure states, not necessarily limited to stabilizers. The first result is that we still cannot find a private state in the No privacy zone:


Let a⃗, n⃗∈ℤ^k, n = n⃗_1, gcd(a⃗)=1 and f() be the linear function given by f() = a⃗·. Then, there is no pure state capable of estimating f privately with n⃗ distributed resources and access to local separable dynamics U_, such that n⃗≺a⃗ or n⃗⋠a⃗.

    ∄|ψ⟩∈ℋ(n): 𝒫 (𝒬 (U_|ψ⟩),a⃗) = 1




The privacy is only zero if 𝒬 (U_|ψ⟩) = αa⃗a⃗^T ≡𝒬. From Eq. <ref>, take the support of 𝔔, 𝔔 as the sub-matrix with the lines and columns such that λ_j≠ 0. Moreover, let C_𝔔 be the correspondent sub-matrix of C in the 𝔔, i.e. the lines multiplying by the non-zero values of 𝔔. Note that a⃗∉ C, and, in particular, a⃗∉ C_𝔔. This is a consequence of n⃗≺a⃗ or n⃗⋠a⃗. Look at the following table covering all possible combinations of outcomes:



The only case to check is that of rank 𝔔 = dim 𝔔-1 with rank C_𝔔 = 2. According to Thm. <ref>, 𝔔 = diag(λ⃗) - λ⃗λ⃗^T, up to a matrix made up of ± 1 in the diagonal, call it D. That means the vector D1⃗ spans the null space, therefore the positive eigenvectors of the 𝔔 are spanned by the orthogonal subspace of D1⃗, call it P^⊥. So, take the set of vectors W = {w⃗ | w⃗ = 1/√(2)(e⃗_i - e⃗_j), i≠ j }. Notice that the 𝗌𝗉𝖺𝗇 (D W) = 𝗌𝗉𝖺𝗇 (P^T) as Dw⃗;D1⃗=0  ∀ w∈ W. Using Prop. <ref> we can analyse then the vectors in DW, which are simpler to check:



    Dw⃗ C_𝔔   = 1/√(2)( h⃗_𝒩^* (i) ±h⃗_𝒩^* (j) ) 
        =
    	2/√(2)[ n⃗ - (h⃗_𝒩 (i) + h⃗_𝒩 (j) ) ] 
    
    	2/√(2)(h⃗_𝒩 (i) - h⃗_𝒩 (j) ) 
    	,

which by Props. <ref> and <ref> we see it is never proportional to a⃗, given gcd(a⃗)=1 and n⃗≺a⃗ or n⃗⋠a⃗. If it happens that it is zero, than no information can be obtained. This concludes the proof by noting there is no way to construct any state achieving privacy for the function presented.


In this proof, the tools and approach used provide a good insight on how to deal with the other cases. We present another statement for the Minimal privacy zone, for which a similar proof can be found in Appendix <ref>.

[Minimal privacy zone]theoremonecopy
Let a⃗, n⃗∈ℤ^k, gcd(a⃗)=1 and f() be the linear function given by f() = a⃗·. Then, there is only one family of pure states that is capable of estimating privately the function f using n⃗ =  a⃗ distributed resources and access to local separable dynamics U_.

    F_GHZ = {α|0⟩^⊗ n + β|1⟩^⊗ n: α,β∈ℂ∖{0}, |α|^2+|β|^2=1 }
    
    𝒫 (𝒬 (U_|ψ⟩),a⃗) = 1   ⟺  ∃ U∈ LU: U|ψ⟩∈ F_GHZ




Thms. <ref> and <ref> allow us to define the minimum amount of distributed resources so we can have a private state for a target linear function f() = a⃗·. This minimum is provided by the vector a⃗, under the separable Hamiltonians and integer target functions assumptions. 

Finally, one can also wonder what happens if we add more qubits than we need, but not sufficient enough, for example, to build two copies of a GHZ state with a⃗ resources. In order to do so, let us first provide an example of a simple transformation that preserves privacy:


Suppose we have a partition for the distributed qubits given by 𝒩 with corresponding vector of resources n⃗=(2,1,2) and we are trying to estimate a function a⃗= (1,1,1) ≺n⃗. One can simply take a GHZ state and add ancillas in a separable way:

    |ψ⟩   = 1/√(2)( 
    	|0_a_10_b_1^n_10_a_2^n_20_a_30_b_3^n_3⟩ + 
    	|1_a_10_b_1^n_11_a_2^n_21_a_30_b_3^n_3⟩), 
       =  1/√(2)( 
    	|0_a_1 0_a_2 0_a_3⟩ + |1_a_1 1_a_2 1_a_3⟩) |0_b_1 0_b_3⟩.

This state is still private. And more so, it is still private if one chooses any state for the ancillas over the possible bit strings for the qubits b⃗ = n⃗-a⃗, meaning (0_b_10_b_3), (0_b_11_b_3), (1_b_10_b_3), (1_b_11_b_3). Additionally, if one does any superposition of local permutations in S_𝒩 over the qubits, this action preserves the vectorial Hamming-weight, meaning it will also preserve the privacy.


Taking this example, let us formalize everything by introducing some families of pure states that emerge as a natural consequence of the equivalence of the vectorial Hamming weight under local permutations of qubits:


Let 𝒩 = {𝒩_μ}_μ = 1,…,k be a partition of the set of qubits and s ∈𝔽^n_2 be a bit string. We call a quantum state |ψ⟩ a distributed s-state if it can be represented by:

    |ψ⟩ = ∑_r ∈ [s]_𝒩α_r |r⟩ ,

where α_r ∈ℂ, such that ∑_r |α_r|^2 =1 for normalization purposes. Let s⃗ = h⃗_𝒩 (s), we denote this family of states by 𝒟(𝒩,s⃗).


Note that the bit strings that belong to the equivalence class [s]_𝒩 are exactly the ones for which the vectorial Hamming weight h_𝒩 is the same, as they all differ by a permutation in S_𝒩 from each other. In particular, if it was the case that the partitions have size 1 (k=1) and α_r is the same for every r, one would recover the regular Dicke state with |𝒩_1| = n qubits and h(s) = m excitations (|D_n^m⟩).

This allows us to define a countable number of families that achieves privacy at estimating a linear function:


Let 𝒩 = {𝒩_μ}_μ = 1,…,k correspond to the partition of a set of distributed qubits, with associated resource vector n⃗≽a⃗ necessarily. 

    ℱ(𝒩,a⃗,d⃗) = {α|ψ⟩ + β|ψ̃⟩,|ψ⟩∈𝒟(𝒩,d⃗), |ψ̃⟩∈𝒟(𝒩, a⃗+d⃗),
    	
    α,β∈ℂ∖{0}, |α|^2+|β|^2=1  } ,

where d⃗ can be such that d⃗∈ℕ_0^k, 0⃗≼d⃗≼n⃗-a⃗. Note that for a fixed a⃗ there will be multiple d⃗ available, depending on the number of qubits. Each of them defines one family of private states.

 
To provide some intuition, let us provide all the private families for a given set of resources and target function:

Let n⃗= (2,2,5) and a⃗=(1,2,3). This implies we have additionally in each node b⃗=(1,0,2) distributed qubits. However, we can atribute to the ancilla qubits different strings s such that h⃗_𝒩 (s) = b⃗_1 defining different families of private states. One way to see this is considering that b⃗_1 represents the amount of 1's we put in our ancilla qubits. We can additionally define a vector b⃗_0 as the amount of 0's in the ancilla qubits. Naturally b⃗_0 + b⃗_1 = b⃗. The total amount of 6 families are given by:

    s_1 = 0_a_1  0_b_1 00_a_2 000_a_3 00_b_3,   s̃_1 = 1_a_1  0_b_1 11_a_2 111_a_3 00_b_3,  d⃗_1 = (0, 0,0) 
    
    	 s_2 = 0_a_1  0_b_1 00_a_2 000_a_3 10_b_3,   s̃_2 = 1_a_1  0_b_1 11_a_2 111_a_3 10_b_3,  d⃗_2 = (0, 0,1)
    
    	 s_3 = 0_a_1  0_b_1 00_a_2 000_a_3 11_b_3,   s̃_3 = 1_a_1  0_b_1 11_a_2 111_a_3 11_b_3,  d⃗_3 = (0, 0,2)
    
    	 s_4 = 0_a_1  1_b_1 00_a_2 000_a_3 00_b_3,   s̃_4 = 1_a_1  1_b_1 11_a_2 111_a_3 00_b_3,  d⃗_4 = (1, 0,0) 
    
    	 s_5 = 0_a_1  1_b_1 00_a_2 000_a_3 10_b_3,   s̃_5 = 1_a_1  1_b_1 11_a_2 111_a_3 10_b_3,   d⃗_5 = (1, 0,1)
    
    	 s_6 = 0_a_1  1_b_1 00_a_2 000_a_3 11_b_3,   s̃_6 = 1_a_1  1_b_1 11_a_2 111_a_3 11_b_3,   d⃗_6 = (1, 0,2)
    ℱ(𝒩,a⃗,d⃗_i) = {α∑_r ∈ [s_i]_𝒩α_r |r⟩ + β∑_r̃∈ [s̃_i]_𝒩α_r̃|r̃⟩ , α,β∈ℂ∖{0}, |α|^2+|β|^2=1  } .

Which is equivalent to saying:

    ℱ(𝒩,a⃗,d⃗_i) = {α|ψ⟩ + β|ψ̃⟩ ,  |ψ⟩∈𝒟 ( 𝒩,s_i), |ψ̃⟩∈𝒟 ( 𝒩,s̃_i), 
    α,β∈ℂ∖{0}, |α|^2+|β|^2=1  } .



The number of existent families of states depend on the number of ancilla qubits in each node and the different ways one can prepare these ancillas. Considering that a node μ has n_μ qubits and is trying to estimate the linear function with correspondent a_μ, then one has b_μ = n_μ - a_μ ancilla qubits. This means we have b_μ+1 ways to find bitstrings with different Hamming-weights. The overall number of families is given by ∏_μ∈ V (b_μ+1). By construction, all the states that fit inside one of these families are private for the estimation of f() = a⃗·:

propositionprivatefamily
All families of pure states ℱ(𝒩,a⃗,d⃗), are capable of estimating privately the function f()=a⃗· using n⃗=a⃗+b⃗ distributed resources.

This allows us to finally get to the statement on privacy for when the amount of resources is larger than the minimum needed, but not as large so one could have multiples of the number of resources.

[Minimal Plus Ancilla Zone]theoremonecopyplusancillaarb
Let a⃗,b⃗, n⃗∈ℤ^k, gcd(a⃗)=1 and f() be the linear function given by f() = a⃗·. Then, all pure states, up to LU, capable of estimating privately the function f using n⃗ =  a⃗ + b⃗ distributed resources and access to local separable dynamics U_, belong a family of the set of a⃗-Private-Ancilla States introduced in Def. <ref>.

    𝒫 (𝒬 (U_|ψ⟩),a⃗) = 1   ⟺  ∃ U∈ LU, d⃗∈ℕ_0^k, 0⃗≼d⃗≼n⃗-a⃗ : U|ψ⟩∈ℱ(𝒩,a⃗,d⃗)



The proofs for the above two statements, Prop. <ref> and Thm. <ref> are provided over Appendix <ref>, following similar arguments as the proof for Thm. <ref>. Moreover, one can find an expression for QFI of the private state very similar to the one for the GHZ state, as detailed in the proof of Prop. <ref>.

Finally we can discuss in the case where we have multiples of the number of resources. In this case, we will not provide a complete set of private states, in the sense we will not prove that the states built in such fashion are the only private states. However, we are able to prove that every state built as so is indeed private. The way to think of this case is to consider that we have logical qubits such that the encoding dynamics by default already encode the desired function. This means exactly choosing our logical qubits to be the ones such that α|0_L⟩ + β|1_L⟩ generates a private state:


    |0_L⟩∈𝒟(𝒩,d⃗), 
       |1_L⟩∈𝒟 (𝒩,a⃗+d⃗),

for some d⃗, using the notation introduced in Defs. <ref>,<ref>. Using for each logical qubit a⃗+b⃗ resources, encompasses the case of b⃗=0⃗ which is the single GHZ state distributed among the a⃗ resources, and the states in Minimal plus ancilla region. Using these states, and our regular estimation scenario of (ρ,U_,a⃗·) it is easy to understand that now our dynamics when applied to a superposition of |0_L⟩ and |1_L⟩ will always be sensitive only to the desired function f() = a⃗·. Moreover, not all logical qubits need to be defined in the same way, as long as each of them are private. Let us define the states which are built like this:


Let 𝔑 = {𝒩_1 , 𝒩_2, ⋯ , 𝒩_d } be a set of partitions such that each 𝒩_l defines the partition set of a private state, with associated vector of resources n⃗_l = a⃗ + b⃗_l ≽a⃗ ∀ l. For each partition, one can define two logical states, such that:

    |0_L⟩_l ∈𝒟(𝒩_l,d⃗_l), 
       |1_L⟩_l ∈𝒟(𝒩_l,a⃗ + d⃗_l) .

The set of logical private states can be defined as follows:

    𝔉 = {|ψ⟩ = ∑_j=0^2^d-1α_j |j_1j_2⋯j_d⟩ : α_j ∈ℂ, |j_l⟩∈{|0_L⟩_l, |1_L⟩_l },⟨ψ | ψ|=⟩ 1 }.



As one can see, the problem is now combinatorial. One can arrange the qubits in multiple partitions, and pick multiple private logical states inside of these partitions. Let us make a statement about the privacy of states built like this, which we prove in Appendix <ref>:

[Multiple Privacy]theoremmultiplecopiesprivacy
All states which are described as a private logical state (Def. <ref>) define private states for quantum distributed sensing, up to LU operations.


This has an important consequence, namely, we can treat this case as single-parameter scenario. In particular, one could use the robustness results of the single-parameter case and construct equivalent states, but instead using the logical qubits. Moreover, we can use generally results of how the logical qubits are affected by error to predict how a state made of those logical qubits evolves under error. 

This section finishes with a set of private states for distributed quantum sensing with separable dynamics, and proof that, for limited resources, they constitute the only set of private states. If we extend our resources, we find a construction of sets of private states, by logically encoding multiple qubits into logical qubits that are only sensitive to our target function.



 §.§ Privacy for Arbitrary States with General Hamiltonians 


In a similar manner as for separable hamiltonians, the problem complexity can be reduced by choosing an appropriate basis. The basis we will use in this case will be the basis introduced in Eq. <ref>, constituted by the eigenvectors of the Hamiltonian. This already implies that finding a private state requires finding eigenstates of the Hamiltonian, which is not necessarily trivial as it was for the separable Hamiltonian scenario. By choosing this basis one can build a very similar system to Eq. <ref> in the following way:


    𝒬_μν (ρ_)    = 4∑_l⃗,m⃗α_l⃗^* α_m⃗⟨λ_l⃗ |   H_μH_ν | λ_m⃗|-⟩∑_l⃗,m⃗α_l⃗^* α_m⃗⟨λ_l⃗ |   H_μ | λ_m⃗|∑⟩_p⃗,q⃗α_p⃗^* α_q⃗⟨λ_p⃗ |   H_ν | λ_q⃗|⟩
       = 4∑_l⃗,m⃗α_l⃗^* α_m⃗λ^μ_l_μλ^ν_m_ν⟨λ_l⃗  | λ_m⃗|-⟩∑_l⃗,m⃗α_l⃗^* α_m⃗λ^μ_m_μ⟨λ_l⃗ | λ_m⃗|∑⟩_p⃗,q⃗α_p⃗^* α_q⃗λ^ν_q_ν⟨λ_p⃗ |   λ_q⃗|⟩
       =4 ∑_m⃗ | α_m⃗|^2 λ^μ_m_μλ^ν_m_ν - ∑_m⃗ |α_m⃗ |^2 λ^μ_m_μ∑_q⃗ | α_q⃗ |^2  λ^ν_q_ν
    𝒬 (ρ_)   = C𝔔 C^T,

where we used the fact that the basis states are |λ_j⃗⟩ with j⃗ being given by some labeling function in a space with dimension of the product of the local Hilbert spaces, e.g. j⃗∈{0, ⋯,  |ℋ_1| }×⋯×{0, ⋯,  |ℋ_k| }. In this case, for each of the vectors j⃗ with associated α_j⃗ in 𝔔, there will exist an associated c_j⃗ in C, given by  c_j⃗ = (λ^1_j_1,λ^2_j_2,⋯,λ^k_j_k) ∈𝒪. We provide a simple example:


Let k=3 meaning 3 nodes, each holding one qubit and having access to a Z hamiltonian (the local dynamics are given by U_θ_μ = e^-iθ_μ Z). Each local set of eigenvalues is the same, {-1,+1}, meaning 𝒪 = {-1,+1}×{-1,+1}×{-1,+1}. The set of possible eigenstates of the distributed Hamiltonian are then |j_1 j_2 j_3⟩ where |j_i⟩∈{|0⟩, |1⟩}. The possible set of functions can be mapped to c_j⃗ = ((-1)^j_1, (-1)^j_2, (-1)^j_3). They form the vertices of a cube centered in 0 with side equal to 2.


Note that the matrix 𝔔 can be written again in the same form as before, with a slight change: 𝔔 = diag(α⃗) - α⃗α⃗^T = Λ - α⃗α⃗^T, where we used that each entry of α⃗ corresponds to the probability amplitude |α_j⃗|^2 for each and every j⃗. This vector will have the length of the amount of vectors j⃗ there are, i.e. the total Hilbert space dimension. Given Thm. <ref>, we know this matrix has the rank equal to rank Λ-1.

From here we can already make some statements about finding private states, although we cannot go very far as it highly depends on the structure of 𝒪, which naturally depends on the Hamiltonian. In order to better analyze this let us introduce the set of vectors going from each point of 𝒪 to another, as it will become apparent later that these are the vectors defining linear functions for which we can find private states.


Let 𝒪 be a discrete subset inside an orthotope in k dimensions over some ordered ring 𝕂. Define the higher order discrete subset 𝒪^2_± as:

    𝒪^2_± = {z⃗∈𝕂^k : z⃗ = x⃗±y⃗ , ∀x⃗,y⃗∈𝒪}.



One can further extend the class of equivalence of linear functions to a class of equivalence of the discrete subset of the orthotopes. As before f() ∼ g() if ∃α∈ℝ: f()=α g(), one can say two sets of vectors are equivalent 𝒪_1 ∼𝒪_2 if ∀v⃗∈𝒪_1 ∃α∈ℝ, w⃗∈𝒪_2: v⃗ = αw⃗ and the correspondence is bijective. We say one subset contains another up to equivalence 𝒪_1 𝒪_2 if 𝒪_1 ∼𝒪̃_1 ⊆𝒪_2. Using this, we can perform a statement on a necessary condition in order to create private state in this scenario:

[Existence of Private States]theoremexistenceprivate
Let a distributed sensing scenario be governed by local encoding dynamics in the form of Eq. <ref>, with associated discrete subset 𝒪. For each vector a⃗ belonging to the higher order discrete subset 𝒪^2_-, one can find a private state with respect to the linear function f()=a⃗·θ⃗. Moreover, no vector outside the equivalence class of 𝒪^2_- has a private state.


Let us provide again another more complex example to help visualize what we are doing, by plotting an illustration of the functions such that a private state exists.

 (Cube in 3-D)
Let the three local hamiltonians {H_1,H_2,H_3} have possible eigenvalues {λ_j^μ}_j=1,...,|H_μ|^μ = 1, 2, 3. All of the vectors in blue with accompanying points correspond to the discrete subset 𝒪 and correspond to all the possible combinations of the eigenvalues in each local hamiltonian (λ^1_i,λ^2_j,λ^3_k). These are the private linear functions one has access to, since for each of them we prove the existence of a private state. The 3-cube in red (given that the dimension k=3) corresponds to the 3-orthotope 𝒪. The blue and black dots on the right side correspond to the set of 𝒪∩𝒪. The black dots on the right side also correspond to the vertices of the orthotope  V(𝒪), which can be seen as the intersection with the k-sphere. These correspond to the maximum of the 2-norm of the vectors inside the orthotope.



Moreover, one may wonder about the possibility of finding private states for functions within 𝒪, not only in 𝒪_-^2. This is again highly dependent on the Hamiltonian. For this reason we will give a sufficient condition for one to find these Hamiltonians, which relates with the inclusion of 𝒪 in the higher order subset 𝒪^2_-:

propositionsymmetrichamiltonians
Let {H_μ}_μ∈ V be the Hamiltonians describing some local general encoding dynamics. 

    ∀ μ∈ V  ∃ X_μ∈1⊗𝒫_μ: {X_μ, H_μ} = 0
        𝒪𝒪^2_-

Where 1⊗𝒫_μ are the Pauli strings where only the elements in μ can be non-identity. This means the functions in 𝒪 are available privately by choosing an appropriate state.


We prove Thm. <ref> and Prop. <ref> over Appendix <ref>. The latter is supposed to give some symmetry conditions for the Hamiltonian, under which the functions available are directly extractable from the original set 𝒪. Note that the form of 𝒪 is not always a cube centered in the origin, even though 𝒪^2_- is. Prop. <ref> also implies that 𝒪 will be symmetric with respect to the origin. From here, one can deduce a simple, yet important consequence that relates with the idea of concentration of information. 

[Information Concentration]corollaryconcentration
Given a distributed sensing scenario governed by local encoding dynamics in the form of Eq. <ref>, the maximum information extractable, given by functions of parameters in the vertices of 𝒪^2_-, is accessible only by a private state.


The proof for Corol. <ref> can also be found over Appendix <ref>. This has important consequences, and provides an intuition that has been in the background the whole time, namely, one can either try and estimate one function with maximum precision, or a set of functions with less precision. Moreover, one can retrieve the solutions for separable Hamiltonians from Thm. <ref>, by looking at the imposed condition on gcd(a⃗)=1. Taking into consideration that the orthotope 𝒪⊆ℤ^k, the existence of private states is a consequence of integers' properties.




§ ROBUSTNESS 


So far, we have been able to calculate the QFI for special types of quantum states. We started by the case of stabilizer states with separable Hamiltonians. We then generalized for any arbitrary pure state, still under the assumption of separable Hamiltonians. We then lifted this last assumption to general local Hamiltonians, at the expense of not being able to identify the state, and instead proving that private states still exist and for which functions they do exist. The next logical step is to consider mixed states as well. These types of states are often associated with the presence of noise in the state, hence the name of this section. Robustness should measure the ability of a quantum state to retain some of its properties upon being affected by noise <cit.>. In our case, our goal is to maximise precision, which is connected with the maximisation of the QFI. Robustness is then linked to how the QFI varies, as noise is applied onto the system. In this section, we will start by finding a way to calculate the QFI of mixed states and verify how the QFI changes under different noises types, keeping in mind the privacy as well. Moreover, we assume that the noise is applied to the state prior to any encoding dynamics, has this noise should be the one that could possibly break privacy.  



 §.§ QFI for Mixed States


Starting on the equation for the QFI in the multiparameter scenario:


    𝒬_μν (ρ_) =  ℛ_ρ_^-1 (∂_θ_μρ_) ρ_ℛ_ρ_^-1 (∂_θ_νρ_),

where the super operator ℛ_ρ_^-1 (Ô) is the inverse of the super operator ℛ_ρ_ (Ô), defined by the following set of equations:

    ℛ_ρ (O)   = ρ O + O ρ/2, 
    ℛ_ρ^-1 (O)   = ∑_j,k2/λ_j + λ_k O_jk|𝒢_j⟩⟨𝒢_k|.

Given a decomposition of the density matrix in its eigenvectors ρ = ∑_k λ_k |𝒢_k⟩⟨𝒢_k| and the matrix elements O_jk = ⟨𝒢_j|Ô|𝒢_k⟩. Developing Eq. <ref>, one can arrive at the following expression, similar to <cit.>:

    𝒬_μν (ρ_) = ∑_n ∈
     k∈ 2(λ_n - λ_k)^2/λ_n + λ_k⟨𝒢_n | G_μ | 𝒢_k|⟨%s|%s⟩⟩𝒢_k | G_ν | 𝒢_n + ∑_n ∈
     k∈  4  λ_k  ⟨𝒢_n | G_μ | 𝒢_k|⟨%s|%s⟩⟩𝒢_k | G_ν | 𝒢_n ,

where we used  and  to denote the indices k such that λ_k is either positive or zero. This can be further simplified when ρ = ∑_k λ_k |𝒢_k⟩⟨𝒢_k| is given in an orthonormal basis, i.e. such that every ⟨𝒢_j | 𝒢_k|=⟩δ_jk. Even if this is not the case, it is a well established property of quantum information that one can always find a basis for |𝒢_j⟩ that makes a certain density operator have such form. Nonetheless, one arrives at:

    𝒬_μν (ρ_) = ∑_n>k ∈
     n≠ k 4[ (λ_n - λ_k)^2/λ_n + λ_k - (λ_n + λ_k) ]  a_nk^μ a_kn^ν + ∑_n∈λ_n 𝒬_νμ (|𝒢_n⟩) ,

where a_nk^μ = ⟨𝒢_n | G_μ | 𝒢_k|$⟩. The entire derivation of these equations can be found in Appendix <ref>. In the following sections we will assume that the encoding dynamics are generated byZoperators, without loosing generality. If they are generated by other generatorG, assume that dephasing noise is given by an equivalent quantum channel that has a mixture of error in the direction ofG. For the bit-flip noise, assume the error is in the direction of someG^⊥. Again, we also assume that the noise is applied to the state, prior to evolution under the dynamics that encode the parameters.  




 §.§ Dephasing Noise


Dephasing noise is characterized by applying a quantum channel with a certain probability of aZerror. Taking a GHZ state withn⃗qubits distributed over the network, the usual encoding dynamics in theZdirection and the usual target functionf() = a⃗·, such thata⃗=n⃗,i.e.we are working with a perfectly private state. Since the GHZ state is made up of eigenvectors of theZoperator, dephasing noise acts identically on whichever qubit of the GHZ state. Namely consider the two orthogonal states one can get with onlyZoperations on whichever qubits|GHZ^±⟩ = 1/√(2)(|0⟩^⊗n ±|1⟩^⊗n) ≡|𝒢_0^±⟩, wheren=n⃗. Applying a dephasing noise channel to a|GHZ^±⟩results in introducing a mixture between these two states:


    𝒟_i (ρ, p)    = (1-p) ρ + p Z_i ρ Z_i,      Z |𝒢_0^±⟩ = |𝒢_0^∓⟩.


Applying this recursively for each qubit, we can find the following expression for an initial GHZ state :

    𝔇 (ρ, p⃗)    = 𝒟_1^p_1∘𝒟_2^p_2∘…∘𝒟_n^p_n (ρ) , 
    𝔇 (|𝒢_0^±⟩⟨𝒢_0^±|, p⃗)    =   E(p⃗ ) |𝒢_0^±⟩⟨𝒢_0^±| + O(p⃗) |𝒢_0^∓⟩⟨𝒢_0^∓|,

where the even,E(·), and ddd functions,O(·), correspond to the Kraus operators of the total channel with even and odd number ofZoperators, and were introduced in <cit.> to calculate the fidelity of a GHZ state.
Using Eq. <ref> to calculate the QFI of a mixed state where we have a mixture between|𝒢_0^±⟩resulting in the density matrixρ= λ_+ |𝒢_0^+⟩⟨𝒢_0^+| + λ_- |𝒢_0^-⟩⟨𝒢_0^-|. Moreover, using theZdynamics asU_= _μ e^iθ_μ∑_j∈μZ_j = _μ e^i θ_μZ_μ:

    𝒬_μν (ρ_)    = 4(λ_+ - λ_-)^2/λ_+ + λ_-⟨𝒢_0^+ | Z_μ | 𝒢_0^-|⟨%s|%s⟩⟩𝒢_0^- | Z_ν | 𝒢_0^+
       = 4(λ_+-λ_-)^2 a_μ a_ν ,

which preserves the privacy. This is a consequence of the dynamics being generated by the same operation as the error, namely byZoperators on each qubit. Moreover, ifp⃗is uniform, as in every qubit dephases at the same rate, we get thatλ_+ - λ_- = (1-2p)^n, wherenis the number of qubits suffering dephasing.



 §.§ Bit-flip Noise


Bit-flip noise on the other hand is characterized by applying a quantum channel with a certain probability ofXerror. Under the same estimation scenario as before we will arrive at the following density matrix:


    ρ   = λ_0 |𝒢_0^+⟩⟨𝒢_0^+| + ∑_i=1^2^n-1-1λ_i |𝒢_i^+⟩⟨𝒢_i^+|,

where|𝒢_m⟩ =1/√(2) ( |m⟩ +  |m⟩ ) and0≤m ≤2^(n-1)-1to prevent repetition of states, given the invariance ofm→m. Taking the derivation of the QFI in Eq. <ref> we get that all termsa_nk^μare going to be zero cause the dynamics are generated byZoperators, and the errors are generated byXoperators. This way, no operator containing only{1,Z}terms will ever take one|𝒢_j⟩ to another|𝒢_k⟩ , since this map is generated by some operator containing only{1,X}terms. This results in the QFI to be given by:


    𝒬 (ρ_)    = 4 ∑_j=0^2^n-1-1λ_j h⃗_𝒩^* (j) h⃗_𝒩^* (j)^T  ,
    λ_j    = g(p⃗,j) + g(p⃗,j),

where we introduce the function that maps a vector of reals and a bitstring into a real number, namely:

    g: ℝ^k ×𝔽_2^k    ⟶ℝ
    
        g(p⃗,j)    ⟼∏_i=1^k p_i^j_i (1-p_i)^1-j_i.


One can observe in Eq. <ref>, one looses complete privacy as soon as one qubit suffers a bit-flip. This is due to the fact that now the generators of the dynamics areZoperators and the errors only containXoperators. However, one can still make a statement using the fifth property of the privacy measure: if the noise can be maintained under a certain threshold, then one can bound the loss of privacy. Equivalently, the amount of error will bound the amount of information leaked.



 §.§ Depolarising Noise


Depolarising noise can be seen as a very symmetric noise, as it decoheres in every direction with identical probabilities. In particular, taking a GHZ state and applying a depolarising channel in each qubit, one arrives at a state which is very similar to a generalized Werner state <cit.>, apart from the identity not really being an identity matrix, but a diagonal matrix. Nonetheless, using the description of the depolarising channel as in <cit.>:


    𝒟_i^p(ρ)     =p ρ+1-p/3(X_i ρ X_i^†+Y_i ρ Y_i^†+ Z_i ρ Z_i^†) 
        =1+2 p/3ρ+2(1-p)/3Λ_i(Y_i ρ Y_i^†).


The density matrix can be expressed as:

    ρ   = λ_0^+ |𝒢_0^+⟩⟨𝒢_0^+| + λ_0^- |𝒢_0^-⟩⟨𝒢_0^-| + ∑_i=1^2^n-2λ_i |i⟩⟨i| ,

since⟨i|j|=⟩ ⟨i|𝒢_0^±|=⟩0fori≠0,2^n-1, and, as above, the dynamics are inZ, all the termsa_0^±j^μ= 0. Moreover, the QFI of each of the|i⟩⟨i|withi>0,is zero, as no information can be obtained from states which are just a binary string. This allows us to write the QFI matrix as:

    𝒬 (ρ_)    = 4(λ_0^+ - λ_0^-)^2/λ_0^+ + λ_0^-a⃗a⃗^T + ∑_j=1^2^n-2λ_j 𝒬(|j⟩) 
       = 4(λ_0^+ - λ_0^-)^2/λ_0^+ + λ_0^-a⃗a⃗^T ,

whereλ_0^±can be found be solving the following set of equations:

    λ_0^+ + λ_0^- = g((1+2p⃗)/3,0) + g(2(1-p⃗)/3,0) 
    λ_0^+ - λ_0^- = g((4p⃗-1)/3,0).


Importantly, this type of noise again preserves privacy. Unlike previous errors, this error contains both the same operators as in the encoding dynamics and orthogonal to the ones generating the dynamics. The privacy preservation in this case comes from the symmetry of this channel acting on the GHZ state which is also highly symmetric.



 §.§ Amplitude-Damping Noise


Amplitude-damping noise can be seen as a consequence of spontaneous emission of excited photons, in a way that the excited states (in our case|1⟩), with some probability,p_iper qubit, goes to the ground state (in our case|0⟩). Taking a GHZ state and applying a amplitude-damping channel in each qubit, one arrives at something very similar to the depolarising channel, with some slight changes:


    ρ   = λ_0^+ |𝒢_0^+⟩⟨𝒢_0^+| + λ_0^- |𝒢_0^-⟩⟨𝒢_0^-| + ∑_i=0^2^n-1λ_i |i⟩⟨i|
       = λ_0 |0⟩⟨0| + λ_0^+ |𝒢_0^+⟩⟨𝒢_0^+| + λ_0^- |𝒢_0^-⟩⟨𝒢_0^-| + λ_1 |1⟩⟨1| + ∑_i=1^2^n-2λ_i |i⟩⟨i|.

One can diagonalize the system into the following:

    ρ   = λ̃_0^+ |𝒢̃_0^+⟩⟨𝒢̃_0^+| + λ̃_0^- |𝒢̃_0^-⟩⟨𝒢̃_0^-| + ∑_i=1^2^n-2λ_i |i⟩⟨i| .

The QFI can then be obtained by:

    𝒬 (ρ_)    = 4(λ̃_0^+ - λ̃_0^-)^2/λ̃_0^+ + λ̃_0^-|⟨𝒢̃_0^+|Z| 𝒢̃_0^- |⟩|^2 a⃗a⃗^T ,

whereλ̃_0^±and𝒢̃_0^±can be found from the solution of the following eigensystem, written in the basis{ |0⟩^⊗n ≡|0⟩,|1⟩^⊗n≡|1⟩}:

    [ a b; b c ] =
    [  1 + ∏_i p_i ∏_i √(1-p_i); ∏_i √(1-p_i)  ∏_i (1-p_i) ] .


Yet again, this noise still preserves privacy. The justification for amplitude-damping channels comes again from the symmetric form of the GHZ state with respect to the Kraus operators of the state, and the simple form that Eq. <ref> takes.



 §.§ Particle Loss


To check what happens to the state under particle loss, let us start by the simple case where we haven⃗=a⃗resources. This means the only private state is given by the GHZ state, with the distribution of resources given bya⃗, in the regular setting. As it is known, the GHZ state has no resilience against particle loss, and all the information is lost upon particle loss. A GHZ state,|𝒢_0^+⟩, after loosing whichever particle is transformed into:


    ℰ_i (GHZ)    = _i |𝒢_0^+⟩⟨𝒢_0^+|
       =⟨0_i |𝒢_0^+⟩⟨𝒢_0^+| 0_i|+⟩⟨1_i |𝒢_0^+⟩⟨𝒢_0^+| 1_i|
    ⟩   = 1/2[ |0_𝒩∖ i⟩⟨0_𝒩∖ i|  + |1_𝒩∖ i⟩⟨1_𝒩∖ i|].

Using again the equation for the noisy QFI, we get the QFI becomes0for whichever parameter, as this state is invariant under the encoding dynamics.

Throughout the paper, we have defined families of private states when we have more ancilla qubits at our disposal, besides the minimum amount to create a private state. One can observe that there are states in these private families that show robustness against particle loss. To do this, suppose we have a state belonging to one of these families,ℱ(𝒩,a⃗,d⃗):

    |ψ⟩   = α|ψ_0⟩ + β|ψ_1⟩,     |ψ_0⟩∈𝒟(𝒩,d⃗), |ψ_1⟩∈𝒟(𝒩,a⃗,d⃗)  .


Let us first analyze what happens to the our states inℱ(𝒩,a⃗,d⃗)when a qubit is traced. Since we can choose whichever basis for the partial trace, let us measure the qubitjin theZbasis:

    |ψ_0⟩∈𝒟(𝒩,d⃗) ⟨0_j | ψ_0|∈⟩𝒟(𝒩∖ j,d⃗) , 
       |ψ_1⟩∈𝒟(𝒩,a⃗ + d⃗)  ⟨0_j | ψ_1|∈⟩𝒟(𝒩∖ j,a⃗+d⃗) , 
       ⇓
       α' ⟨0_j | ψ_0|+⟩β' ⟨0_j | ψ_1|∈⟩ℱ(𝒩∖ j,a⃗,d⃗).

This holds true if both⟨0_j | ψ_0|$⟩ and ⟨0_j | ψ_1|$⟩ are larger then zero. If not, it will originate a state with no information available. The same can be said if one measures the state|ψ⟩in|1⟩_j:

    |ψ_0⟩∈𝒟(𝒩,d⃗) ⟨1_j | ψ_0|∈⟩𝒟(𝒩∖ j,d⃗-e⃗_j) , 
       |ψ_1⟩∈𝒟(𝒩,a⃗ + d⃗)  ⟨1_j | ψ_1|∈⟩𝒟(𝒩∖ j,a⃗+d⃗-e⃗_j) , 
       ⇓
       α' ⟨1_j | ψ_0|+⟩β' ⟨1_j | ψ_1|∈⟩ℱ(𝒩∖ j,a⃗,d⃗ -e⃗_j),

where we usede⃗_jto denote the vector that is one for the nodeμsuch that qubitj∈μand zero everywhere else.
It is intuitive to understand that there is a generalization for this at each qubit that is lost. The resilience against any particle loss can be made larger by increasing the amount of ancillas. The privacy is guaranteed as, after loosing each qubit, we all obtain a mixture of orthogonal states, which either have information abouta⃗, as they belong to a family of private states, or they do not have information at all. To prove this consider:


Let |ψ⟩∈ℱ(𝒩 ,a⃗,d⃗), |ϕ⟩∈ℱ(𝒩 ,a⃗,d⃗').

    d⃗≠d⃗' ⟨ψ|ϕ|=⟩ 0

The proof comes straightforward from realizing that distributed s-states are orthogonal if their generating strings s have different vectorial Hamming-weights.

Let Ψ = {|ψ_1⟩, …, |ψ_n⟩} be a collection of states of a private family, |ψ_j⟩∈ℱ(𝒩 ,a⃗,d⃗)  ∀ j = 1,…,n.

    |Ψ⟩ = ∑_j=1^n α_j |ψ_j⟩∈ℱ(𝒩 ,a⃗,d⃗)  ∀ α_j ∈ℂ: ∑_j=1^n |α_j|^2

The proof comes again straightforward as |Ψ⟩ will only contain vectors belonging to 𝒟(𝒩,d⃗) and 𝒟(𝒩,a⃗+d⃗).


Using this we are now able to prove the following theorem:

All the private states in the region of minimal plus ancilla amount of resources are able to either remain completely private under e⃗-qubit loss, if the amount of ancillas are at least b⃗≽e⃗, or provide no information at all.

Any state belonging to the family of private states on ℱ(𝒩,a⃗,d⃗), with associated vector of resources n⃗=a⃗+b⃗ already verifies privacy, by construction (Prop. <ref>). The evolution of these states under loss can be seen from Eqs. <ref>,<ref>. Note that after the partial trace of each qubit we will have a mixture of states that belong to private families. We can group them according to the families they belong to, ensuring the states in the mixture are orthogonal and then using Prop. <ref> and Eq. <ref> to verify privacy. In the case they belong to the same family, diagonalizing them makes, by Prop. <ref>, new orthogonal states belonging to the same family. Nonetheless in this last case by analyzing Eq. <ref> we also verify that this remains private. Applying the same procedure as in these equations to all qubits j∈e⃗, we get that the final eigenstates of the density matrix all belong to either private families, or are states with zero information at hand. 


 §.§ Generalizing Results


When addressing the most general case, we use the private logical states introduced in Def. <ref>. For this, there is a condition to be able to use noise resiliency strategies already reproduced for single-parameter quantum metrology, and achieve the same outcome for multiparameter private metrology. The property is the following:


We say a logical qubit basis {|0_L⟩,|1_L⟩} is private against an error generated by a quantum channel ℰ(·) if:

    ∀α, β∈𝒞: |ψ⟩ = α|0_L⟩ + β|1_L⟩, ℰ(|ψ⟩⟨ψ|) = ∑_i λ_i |ψ_i⟩⟨ψ_i|,

such that ⟨ψ_i | ψ_j|=⟩δ_ij and ∃γ : ⟨ψ_i | G_μ | ψ_j|=⟩γ a_μ, for all i,j. 


This allows us to say that, if a logical qubit basis is private against an error, then any state generated in that basis will also be private, under that same error. The consequence follows trivially from Eq. <ref>. This is already the case for the states in the minimal privacy zone under dephasing, depolarising and amplitude-damping noises and for minimal plus ancillas private states under particle loss.




§ DISCUSSION OF THE RESULTS


We have condensed the results of this work for the existence of private families of states with the corresponding assumptions and theorems in Table <ref>.



Throughout this paper we presented several families of states, which hold an important property for distributed sensing scenarios. Privacy comes as a natural consequence of choosing a state which makes available information about one single function of parameters. In all the results for pure states, the decomposition of the QFI in Eqs. <ref> and  <ref> was crucial, and simplified the problem. Given its generality, one could potentially use this as means to efficiently calculate the QFI matrix, by following a similar procedure, and grouping states according to their vectorial Hamming weight.

The first no-go result is that, if one is not able to control the encoding dynamics, there is a minimum amount of resources such that one can estimate privately the target linear function, and this naturally depends on the Hamiltonian of the encoding dynamics. In particular, for separable Hamiltonians, with integer resources, one is only able to measure a linear function that is proportional to an integer combination of the parameters. The second result still for this class of Hamiltonians is, if we have exactly the amount of required resources, there is only one family of states that is able to estimate the target function privately, and this family is SLOCC equivalent to a GHZ state. Moreover, the GHZ state is the one for which the QFI is maximized, and therefore, the precision is also maximized. This result also affects the controllable encoding dynamics case, implying the privacy of a GHZ state with arbitrary number of qubit in each node. If one hasn_μon modeμ, and control over a linear parameter,t_μ, locally, then as long as one can maken_μt_μ= a_μfor allμ, then one finds a private state.

Another important result is that, by adding extra qubits and superpositions of states that differ by local permutations, we can gain some redundancy, allowing us to create private states that maintain privacy and information even after qubit loss. This has important consequences for practical applications in near-term experiments. 

Finally, we provide methods to analyze when multiples of the vector of resources are available. In this scenario, the scaling of the precision with the amount of resources is a pertinent question. In most cases, this is identical to dealing with the single parameter scenario by choosing logical qubits that are constituted by the basis of private states. 

Moving to more general Hamiltonians, not necessarily separable, but still local in the sense that each node can only encode their own parameter, we found a structure on the space of available private functions and a way to generate the private states. However, in this scenario, it requires knowledge about the eigenstates of the Hamiltonian which is not a trivial problem, although there are some strategies to find them <cit.>. Most importantly, this approach for general Hamiltonians provides an interesting insight to the structure of finding states for sensing. In particular it relates the maximum precision at which one can estimate a function employing a quantum sensing scheme and the privacy of such estimation, as they walk hand in hand. This was expected as the best local strategy is often to use locally entangled states, where the amount of information is the same, but scattered across each local parameter. When analyzing the best strategy for a given target function, privacy can also be perceived as an intuitive idea of concentrating all the information along a target function. 

Overall, these results obtained clarify exactly which states one should build in order to perform secure and private from construction protocols for distributed quantum sensing. The continuity the privacy measure with respect to the QFI information, allows one to infer and create protocols that provide bounds on the QFI to establish bounds on the privacy of a state. These bounds in the privacy can then be converted into security concerns by an appropriate protocol, as we have discussed, that uses state certification rounds intercalated with sensing rounds.



This work opens up a different way to look at sensing protocols, relating resources and hamiltonian dynamics to information directly. While the framework has been consistently that of quantum sensing, where measurements are repeated and from them estimators are created, one could potentially apply our results to distributed quantum computation. In particular, we dealt with only linear functions, which seems to be a consequence of the Hamiltonian dynamics not being able to encode arbitrary functions of parameters. A possible direction of future work could be coming up with protocols to access non-linear functions.

Another possible direction would be to find all of the private states when multiples of the vector of resources are available. Since technically from the definition of a Private-Ancilla state, these are not limited ton⃗ ≼2a⃗, it is possible that finding all states involves constructing a hierarchy of private states by the logical encoding formalism taking the logical qubits inside each of the(ma⃗)-Private-Ancilla states,m=1,2,…, and verify possible repetitions of families by playing with different encodings.

Finally, the structure of the private orthotope, and the fact that it is an orthotope, might hint at some more fundamental results. It would be interesting to understand exactly are the implications of this, outside of the privacy in the sensing scenario, and into an information perspective by using the intuition of concentration of information.



§ ACKNOWLEDGEMENTS


The authors acknowledge the support from the EU Quantum Flagship project QIA (101102140) and France 2030 under the French National Research Agency projects HQI ANR-22-PNCQ-0002 and the PEPR integrated project EPiQ ANR-22-PETQ-0007. L.B. and Y.O. thank the support from Fundação para a Ciência e a Tecnologia (FCT, Portugal), namely through project UIDB/04540/2020. L.B. acknowledges the support of FCT through scholarship BD/05268/2021. 


63 urlstyle[Giovannetti et al.(2006)Giovannetti, Lloyd, and
  Maccone]Giovannetti2006
Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone.
 Quantum metrology.
Physical Review Letters, 960 (1):0 13–16,
  2006.
 ISSN 10797114.
10.1103/PhysRevLett.96.010401.

[Sidhu and Kok(2019)]Sidhu2019
Jasminder S. Sidhu and Pieter Kok.
 A Geometric Perspective on Quantum Parameter Estimation.
AVS Quantum Science, 20 (1):0 014701, July
  2019.
 ISSN 2639-0213.
10.1116/1.5119961.

[Degen et al.(2017)Degen, Reinhard, and Cappellaro]Degen2017
C. L. Degen, F. Reinhard, and P. Cappellaro.
 Quantum sensing.
Reviews of Modern Physics, 890 (3):0 035002,
  July 2017.
10.1103/RevModPhys.89.035002.

[Hou et al.(2021)Hou, Jin, Chen, Tang, Huang, Yuan, Xiang, Li, and
  Guo]Hou2021
Zhibo Hou, Yan Jin, Hongzhen Chen, Jun-Feng Tang, Chang-Jiang Huang, Haidong
  Yuan, Guo-Yong Xiang, Chuan-Feng Li, and Guang-Can Guo.
 "Super-Heisenberg" and Heisenberg Scalings Achieved
  Simultaneously in the Estimation of a Rotating Field.
Physical Review Letters, 1260 (7):0 70503,
  2021.
 ISSN 1079-7114.
10.1103/PhysRevLett.126.070503.

[Liu et al.(2021)Liu, Zhang, Li, Zhang, Yin, Fei, Li, Liu, Xu, Chen,
  and Pan]Liu2021e
Li-Zheng Liu, Yu-Zhe Zhang, Zheng-Da Li, Rui Zhang, Xu-Fei Yin, Yue-Yang Fei,
  Li Li, Nai-Le Liu, Feihu Xu, Yu-Ao Chen, and Jian-Wei Pan.
 Distributed quantum phase estimation with entangled photons.
Nature Photonics, 150 (2):0 137–142, February
  2021.
 ISSN 1749-4885, 1749-4893.
10.1038/s41566-020-00718-2.

[Barbieri(2022)]Barbieri2022
Marco Barbieri.
 Optical Quantum Metrology.
PRX Quantum, 30 (1):0 010202, January 2022.
 ISSN 2691-3399.
10.1103/PRXQuantum.3.010202.

[Li et al.(2022)Li, Chen, and Cappellaro]Li2022
Changhao Li, Mo Chen, and Paola Cappellaro.
 A geometric perspective: Experimental evaluation of the quantum
  Cramer-Rao bound, April 2022, arXiv:2204.13777.
10.48550/arXiv.2204.13777.

[Paris(2009)]Paris2009
Matteo G.A. Paris.
 Quantum estimation for quantum technology.
International Journal of Quantum Information, 70
  (SUPPL.):0 125–137, 2009.
 ISSN 02197499.
10.1142/S0219749909004839.

[Wehner et al.(2018)Wehner, Elkouss, and Hanson]Wehner2018
Stephanie Wehner, David Elkouss, and Ronald Hanson.
 Quantum internet: A vision for the road ahead.
Science, 3620 (6412):0 eaam9288, October 2018.
 ISSN 10959203.
10.1126/science.aam9288.

[Proctor et al.(2017)Proctor, Knott, and Dunningham]Proctor2017
T. J. Proctor, P. A. Knott, and J. A. Dunningham.
 Networked quantum sensing, February 2017, arXiv:1702.04271.
10.48550/arXiv.1702.04271.

[Yehia et al.(2022)Yehia, Neves, Diamanti, and Kerenidis]Yehia2022
Raja Yehia, Simon Neves, Eleni Diamanti, and Iordanis Kerenidis.
 Quantum City: Simulation of a practical near-term metropolitan
  quantum network, November 2022, arXiv:2211.01190.
10.48550/arXiv.2211.01190.

[Sekatski et al.(2019)Sekatski, Wölk, and Dür]Sekatski2019
Pavel Sekatski, Sabine Wölk, and Wolfgang Dür.
 Optimal distributed sensing in noisy environments.
Physical Review Research, 20 (2):0 1–8, May
  2019.
10.1103/PhysRevResearch.2.023052.

[Knott et al.(2016)Knott, Proctor, Hayes, Ralph, Kok, and
  Dunningham]Knott2016
P. A. Knott, T. J. Proctor, A. J. Hayes, J. F. Ralph, P. Kok, and J. A.
  Dunningham.
 Local versus global strategies in multiparameter estimation.
Physical Review A, 940 (6):0 062312, December
  2016.
10.1103/PhysRevA.94.062312.

[Eldredge et al.(2018)Eldredge, Foss-Feig, Gross, Rolston, and
  Gorshkov]Eldredge2018
Zachary Eldredge, Michael Foss-Feig, Jonathan A. Gross, Steven L. Rolston,
  and Alexey V. Gorshkov.
 Optimal and secure measurement protocols for quantum sensor networks.
Physical Review A, 970 (4):0 042337, April
  2018.
 ISSN 2469-9926.
10.1103/PhysRevA.97.042337.

[Proctor et al.(2018)Proctor, Knott, and Dunningham]Proctor2018d
Timothy J. Proctor, Paul A. Knott, and Jacob A. Dunningham.
 Multiparameter Estimation in Networked Quantum Sensors.
Physical Review Letters, 1200 (8):0 1–11,
  2018.
 ISSN 10797114.
10.1103/PhysRevLett.120.080501.

[Qian et al.(2019)Qian, Eldredge, Ge, Pagano, Monroe, Porto, and
  Gorshkov]Qian2019
Kevin Qian, Zachary Eldredge, Wenchao Ge, Guido Pagano, Christopher Monroe,
  J. V. Porto, and Alexey V. Gorshkov.
 Heisenberg-scaling measurement protocol for analytic functions with
  quantum sensor networks.
Physical Review A, 1000 (4):0 042304, October
  2019.
 ISSN 2469-9926.
10.1103/PhysRevA.100.042304.

[Zhang and Zhuang(2020)]Zhang2020
Zheshen Zhang and Quntao Zhuang.
 Distributed Quantum Sensing, October 2020, arXiv:2010.14744.
10.48550/arXiv.2010.14744.

[Qian et al.(2021)Qian, Bringewatt, Boettcher, Bienias, and
  Gorshkov]Qian2020
Timothy Qian, Jacob Bringewatt, Igor Boettcher, Przemyslaw Bienias, and
  Alexey V. Gorshkov.
 Optimal measurement of field properties with quantum sensor networks.
Physical Review A, 1030 (3):0 L030601, March
  2021.
 ISSN 2469-9926.
10.1103/PhysRevA.103.L030601.

[Cao et al.(2022)Cao, Zhang, Huang, Liu, Li, Walther, and Guo]Cao2022
Huan Cao, Chao Zhang, Yun-Feng Huang, Bi-Heng Liu, Chuan-Feng Li, Philip
  Walther, and Guang-Can Guo.
 Experimental beating the standard quantum limit under non-markovian
  dephasing environment, August 2022, arXiv:2208.02543.
10.48550/arXiv.2208.02543.

[Kómár et al.(2013)Kómár, Kessler, Bishof, Jiang,
  Sørensen, Ye, and Lukin]Komar2014
Peter Kómár, Eric M. Kessler, Michael Bishof, Liang Jiang, Anders S.
  Sørensen, Jun Ye, and Mikhail D. Lukin.
 A quantum network of clocks.
Nature Physics, 100 (8):0 582–587, October
  2013.
 ISSN 1745-2473.
10.1038/nphys3000.

[Nichol et al.(2022)Nichol, Srinivas, Nadlinger, Drmota, Main, Araneda,
  Ballance, and Lucas]Nichol2022
B. C. Nichol, R. Srinivas, D. P. Nadlinger, P. Drmota, D. Main, G. Araneda,
  C. J. Ballance, and D. M. Lucas.
 An elementary quantum network of entangled optical atomic clocks.
Nature, 6090 (7928):0 689–694, September
  2022.
 ISSN 1476-4687.
10.1038/s41586-022-05088-z.

[Khabiboulline et al.(2018)Khabiboulline, Borregaard, De Greve, and
  Lukin]Khabiboulline2018
Emil T. Khabiboulline, Johannes Borregaard, Kristiaan De Greve, and Mikhail D.
  Lukin.
 Optical Interferometry with Quantum Networks.
Physical Review Letters, 1230 (7):0 070504,
  September 2018.
 ISSN 0031-9007.
10.1103/PhysRevLett.123.070504.

[Lévèque et al.(2021)Lévèque, Fallet, Mandea, Biancale,
  Lemoine, Tardivel, Delavault, Piquereau, Bourgogne, Pereira Dos Santos,
  Battelier, and Bouyer]Leveque2021b
T. Lévèque, C. Fallet, M. Mandea, R. Biancale, J. M. Lemoine,
  S. Tardivel, S. Delavault, A. Piquereau, S. Bourgogne, F. Pereira Dos Santos,
  B. Battelier, and Ph Bouyer.
 Gravity field mapping using laser-coupled quantum accelerometers in
  space.
Journal of Geodesy, 950 (1):0 1–19, 2021.
 ISSN 14321394.
10.1007/s00190-020-01462-9.

[Conlon et al.(2022)Conlon, Michel, Guccione, McKenzie, Assad, and
  Lam]Conlon2022
Lorcán O. Conlon, Thibault Michel, Giovanni Guccione, Kirk McKenzie,
  Syed M. Assad, and Ping Koy Lam.
 Enhancing the precision limits of interferometric satellite geodesy
  missions.
npj Microgravity, 80 (1):0 1–10, June 2022.
 ISSN 2373-8065.
10.1038/s41526-022-00204-9.

[Brady et al.(2022)Brady, Chen, Xiao, Xia, Liu, Harnik, Wilson, Zhang,
  and Zhuang]Brady2022
Anthony J. Brady, Xin Chen, Kewen Xiao, Yi Xia, Zhen Liu, Roni Harnik,
  Dalziel J. Wilson, Zheshen Zhang, and Quntao Zhuang.
 Entanglement-enhanced optomechanical sensor array for dark matter
  searches, October 2022, arXiv:2210.07291.

[Alonso et al.(2022)]Alonso2022
Ivan Alonso et al.
 Cold Atoms in Space: Community Workshop Summary and
  Proposed Road-Map.
EPJ Quantum Technology, 90 (1):0 30, December
  2022.
 ISSN 2662-4400, 2196-0763.
<https://doi.org/10.1140/epjqt/s40507-022-00147-w>.


[Fang et al.(2016)Fang, Dutta, Gillot, Savoie, Lautier, Cheng,
  Garrido Alzar, Geiger, Merlet, Pereira Dos Santos, and Landragin]Fang2016
B Fang, I Dutta, P Gillot, D Savoie, J Lautier, B Cheng, C L Garrido Alzar,
  R Geiger, S Merlet, F Pereira Dos Santos, and A Landragin.
 Metrology with Atom Interferometry: Inertial Sensors from
  Laboratory to Field Applications.
Journal of Physics: Conference Series, 723:0 012049,
  June 2016.
 ISSN 1742-6588, 1742-6596.
10.1088/1742-6596/723/1/012049.

[Greve et al.(2022)Greve, Luo, Wu, and Thompson]Greve2022
Graham P. Greve, Chengyi Luo, Baochen Wu, and James K. Thompson.
 Entanglement-enhanced matter-wave interferometry in a high-finesse
  cavity.
Nature, 6100 (7932):0 472–477, October 2022.
 ISSN 1476-4687.
10.1038/s41586-022-05197-9.

[Koczor et al.(2020)Koczor, Endo, Jones, Matsuzaki, and
  Benjamin]Koczor2020a
Bálint Koczor, Suguru Endo, Tyson Jones, Yuichiro Matsuzaki, and Simon C.
  Benjamin.
 Variational-state quantum metrology.
New Journal of Physics, 220 (8):0 1–14,
  August 2020.
 ISSN 13672630.
10.1088/1367-2630/ab965e.

[Meyer et al.(2021)Meyer, Borregaard, and Eisert]Meyer2021
Johannes Jakob Meyer, Johannes Borregaard, and Jens Eisert.
 A variational toolbox for quantum multi-parameter estimation.
npj Quantum Information, 70 (1):0 89, December
  2021.
 ISSN 20566387.
10.1038/s41534-021-00425-y.

[Kaubruegger et al.(2023)Kaubruegger, Shankar, Vasilyev, and
  Zoller]Kaubruegger2023
Raphael Kaubruegger, Athreya Shankar, Denis V. Vasilyev, and Peter Zoller.
 Optimal and Variational Multi-Parameter Quantum Metrology and
  Vector Field Sensing, February 2023, arXiv:2302.07785.

[Hao et al.(2022)Hao, Shi, Gagatsos, Mishra, Bash, Djordjevic, Guha,
  Zhuang, and Zhang]Hao2022
Shuhong Hao, Haowei Shi, Christos N. Gagatsos, Mayank Mishra, Boulat Bash, Ivan
  Djordjevic, Saikat Guha, Quntao Zhuang, and Zheshen Zhang.
 Demonstration of Entanglement-Enhanced Covert Sensing.
Physical Review Letters, 1290 (1):0 010501,
  June 2022.
10.1103/PhysRevLett.129.010501.

[Kasai et al.(2022)Kasai, Takeuchi, Hakoshima, Matsuzaki, and
  Tokura]Kasai2022
Hiroto Kasai, Yuki Takeuchi, Hideaki Hakoshima, Yuichiro Matsuzaki, and
  Yasuhiro Tokura.
 Anonymous Quantum Sensing.
Journal of the Physical Society of Japan, 910
  (7):0 074005, July 2022.
 ISSN 0031-9015.
10.7566/JPSJ.91.074005.

[Shettell et al.(2021)Shettell, Munro, Markham, and
  Nemoto]Shettell2021
Nathan Shettell, William J. Munro, Damian Markham, and Kae Nemoto.
 Practical limits of error correction for quantum metrology.
New Journal of Physics, 230 (4):0 043038,
  April 2021.
 ISSN 1367-2630.
10.1088/1367-2630/abf533.

[Yamamoto et al.(2022)Yamamoto, Endo, Hakoshima, Matsuzaki, and
  Tokunaga]Yamamoto2022
Kaoru Yamamoto, Suguru Endo, Hideaki Hakoshima, Yuichiro Matsuzaki, and Yuuki
  Tokunaga.
 Error-Mitigated Quantum Metrology via Virtual Purification.
Physical Review Letters, 1290 (25):0 250503,
  December 2022.
10.1103/PhysRevLett.129.250503.

[Liu and Yuan(2017)]Liu2017a
Jing Liu and Haidong Yuan.
 Control-enhanced multiparameter quantum estimation.
Physical Review A, 960 (4), 2017.
 ISSN 24699934.
10.1103/PhysRevA.96.042114.

[Rubio and Dunningham(2020)]Rubio2020
Jesús Rubio and Jacob Dunningham.
 Bayesian multiparameter quantum metrology with limited data.
Physical Review A, 1010 (3), 2020.
 ISSN 24699934.
10.1103/PhysRevA.101.032114.

[Rubio et al.(2020)Rubio, Knott, Proctor, and Dunningham]Rubio2020a
Jesús Rubio, Paul A. Knott, Timothy J. Proctor, and Jacob A. Dunningham.
 Quantum sensing networks for the estimation of linear functions.
Journal of Physics A: Mathematical and Theoretical,
  530 (34):0 344001, March 2020.
 ISSN 1751-8113.
10.1088/1751-8121/ab9d46.

[Bringewatt et al.(2021)Bringewatt, Boettcher, Niroula, Bienias, and
  Gorshkov]Bringewatt2021a
Jacob Bringewatt, Igor Boettcher, Pradeep Niroula, Przemyslaw Bienias, and
  Alexey V. Gorshkov.
 Protocols for estimating multiple functions with quantum sensor
  networks: Geometry and performance.
Physical Review Research, 30 (3):0 033011,
  July 2021.
10.1103/PhysRevResearch.3.033011.

[Shettell et al.(2022a)Shettell, Kashefi, and
  Markham]Shettell2022b
Nathan Shettell, Elham Kashefi, and Damian Markham.
 Cryptographic approach to quantum metrology.
Physical Review A, 1050 (1):0 L010401, January
  2022a.
10.1103/PhysRevA.105.L010401.

[Moore and Dunningham(2023)]Moore2023
Sean William Moore and Jacob Andrew Dunningham.
 Secure Quantum Remote Sensing Without Entanglement, February
  2023, arXiv:2302.03617.

[Shettell et al.(2022b)Shettell, Hassani, and
  Markham]Shettell2022a
Nathan Shettell, Majid Hassani, and Damian Markham.
 Private network parameter estimation with quantum sensors, July
  2022b, arXiv:2207.14450.
10.48550/arXiv.2207.14450.

[Shettell and Markham(2020)]Shettell2020c
Nathan Shettell and Damian Markham.
 Graph States as a Resource for Quantum Metrology.
Physical Review Letters, 1240 (11):0 110502,
  March 2020.
 ISSN 0031-9007.
10.1103/PhysRevLett.124.110502.

[Liu et al.(2019)Liu, Yuan, Lu, and Wang]Liu2019
Jing Liu, Haidong Yuan, Xiao-Ming Lu, and Xiaoguang Wang.
 Quantum Fisher information matrix and multiparameter estimation.
Journal of Physics A: Mathematical and Theoretical,
  530 (2):0 023001, December 2019.
 ISSN 1751-8121.
10.1088/1751-8121/ab5d4d.

[Goldberg et al.(2021)Goldberg, Sánchez-Soto, and
  Ferretti]Goldberg2021
Aaron Z. Goldberg, Luis L. Sánchez-Soto, and Hugo Ferretti.
 Intrinsic Sensitivity Limits for Multiparameter Quantum
  Metrology.
Physical Review Letters, 1270 (11):0 110501,
  September 2021.
10.1103/PhysRevLett.127.110501.

[Reilly et al.(2023)Reilly, Wilson, Jäger, Wilson, and
  Holland]Reilly2023
Jarrod T. Reilly, John Drew Wilson, Simon B. Jäger, Christopher Wilson, and
  Murray J. Holland.
 Optimal Generators for Quantum Sensing, May 2023,
  arXiv:2305.15556.

[Zwierz et al.(2010)Zwierz, Pérez-Delgado, and Kok]Zwierz2010
Marcin Zwierz, Carlos A. Pérez-Delgado, and Pieter Kok.
 General optimality of the Heisenberg limit for quantum metrology.
Physical Review Letters, 1050 (18):0 1–4,
  2010.
 ISSN 00319007.
10.1103/PhysRevLett.105.180402.

[Nielsen and Chuang(2011)]Nielsen
M.A. Nielsen and I.L. Chuang.
Quantum Computation and Quantum Information,
  volume 52.
 2011.
 ISBN 978-1-107-00217-3.
10.1080/00107514.2011.587535.

[Ragy et al.(2016)Ragy, Jarzyna, and
  Demkowicz-Dobrzański]Ragy2016
Sammy Ragy, Marcin Jarzyna, and RafałDemkowicz-Dobrzański.
 Compatibility in multiparameter quantum metrology.
Physical Review A, 940 (5):0 052108, November
  2016.
 ISSN 2469-9926, 2469-9934.
10.1103/PhysRevA.94.052108.

[Unnikrishnan and Markham(2022)]Unnikrishnan2022
Anupama Unnikrishnan and Damian Markham.
 Verification of graph states in an untrusted network.
Physical Review A, 1050 (5):0 052420, May
  2022.
10.1103/PhysRevA.105.052420.

[Hein et al.(2004)Hein, Eisert, and Briegel]Hein2004b
M. Hein, J. Eisert, and H. J. Briegel.
 Multiparty entanglement in graph states.
Physical Review A - Atomic, Molecular, and Optical Physics,
  690 (6):0 1–22, 2004.
 ISSN 10502947.
10.1103/PhysRevA.69.062311.

[Hein et al.(2006)Hein, Dür, Eisert, Raussendorf, den Nest, and
  Briegel]Hein2006
M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Van den Nest, and H. J.
  Briegel.
 Entanglement in Graph States and its Applications.
Proceedings of the International School of Physics "Enrico
  Fermi", 162:0 115–218, February 2006.
 ISSN 0074784X.
10.3254/978-1-61499-018-5-115.

[Aaronson and Gottesman(2004)]Aaronson2004
Scott Aaronson and Daniel Gottesman.
 Improved simulation of stabilizer circuits.
Physical Review A - Atomic, Molecular, and Optical Physics,
  700 (5 A):0 1–14, 2004.
 ISSN 10502947.
10.1103/PhysRevA.70.052328.

[Shettell(2022)]Shettell2022
Nathan Shettell.
Quantum Information Techniques for Quantum Metrology.
 PhD thesis, 2022.

[Tao(2023)]Tao2023
Hong Tao.
 Quantum multiparameter estimation with graph states, June 2023,
  arXiv:2306.02518.

[Demkowicz-Dobrzański et al.(2017)Demkowicz-Dobrzański,
  Czajkowski, and Sekatski]Demkowicz-Dobrzanski2017
RafałDemkowicz-Dobrzański, Jan Czajkowski, and Pavel Sekatski.
 Adaptive Quantum Metrology under General Markovian Noise.
Physical Review X, 70 (4):0 041009, October
  2017.
 ISSN 2160-3308.
10.1103/PhysRevX.7.041009.

[Ouyang et al.(2022)Ouyang, Shettell, and Markham]Ouyang2022a
Yingkai Ouyang, Nathan Shettell, and Damian Markham.
 Robust quantum metrology with explicit symmetric states.
IEEE Transactions on Information Theory, 680
  (3):0 1809–1821, March 2022.
 ISSN 0018-9448, 1557-9654.
10.1109/TIT.2021.3132634.

[Hayashi et al.(2022)Hayashi, Liu, and Yuan]Hayashi2022a
Masahito Hayashi, Zi-Wen Liu, and Haidong Yuan.
 Global Heisenberg scaling in noisy and practical phase
  estimation.
Quantum Science and Technology, 70 (2):0
  025030, April 2022.
 ISSN 2058-9565.
10.1088/2058-9565/ac5d7e.

[Rezakhani et al.(2019)Rezakhani, Hassani, and Alipour]Rezakhani2019
A. T. Rezakhani, M. Hassani, and S. Alipour.
 Continuity of the quantum Fisher information.
Physical Review A, 1000 (3):0 32317, 2019.
 ISSN 24699934.
10.1103/PhysRevA.100.032317.

[Bugalho et al.(2023)Bugalho, Coutinho, Monteiro, and
  Omar]Bugalho2023
Luís Bugalho, Bruno C. Coutinho, Francisco A. Monteiro, and Yasser Omar.
 Distributing Multipartite Entanglement over Noisy Quantum
  Networks.
Quantum, 7:0 920, February 2023.
10.22331/q-2023-02-09-920.

[Werner(1989)]Werner1989
Reinhard F. Werner.
 Quantum states with Einstein-Podolsky-Rosen correlations
  admitting a hidden-variable model.
Physical Review A, 400 (8):0 4277–4281, 1989.
 ISSN 10502947.
10.1103/PhysRevA.40.4277.

[Santagati et al.(2018)Santagati, Wang, Gentile, Paesani, Wiebe,
  McClean, Morley-Short, Shadbolt, Bonneau, Silverstone, Tew, Zhou, O'Brien,
  and Thompson]Santagati2018
Raffaele Santagati, Jianwei Wang, Antonio A. Gentile, Stefano Paesani, Nathan
  Wiebe, Jarrod R. McClean, Sam Morley-Short, Peter J. Shadbolt, Damien
  Bonneau, Joshua W. Silverstone, David P. Tew, Xiaoqi Zhou, Jeremy L. O'Brien,
  and Mark G. Thompson.
 Witnessing eigenstates for quantum simulation of Hamiltonian
  spectra.
Science Advances, 40 (1):0 eaap9646, January
  2018.
10.1126/sciadv.aap9646.

[Meister and Benjamin(2022)]Meister2022
Richard Meister and Simon C. Benjamin.
 Resource-frugal Hamiltonian eigenstate preparation via repeated
  quantum phase estimation measurements, December 2022, arXiv:2212.00846.
10.48550/arXiv.2212.00846.



§ UNITARY EQUIVALENCE OF ENCODING DYNAMICS 


In this appendix we go over the proofs of the interplay between the dynamics and the initial state. Starting on the first theorem:

*



Since G and G' are generators for single qubit unitary operators, then they can be decomposed into a linear combination of Pauli operators. Let them be G = a⃗·σ⃗ and G' = b⃗·σ⃗, such that |a⃗|=|b⃗|=1. Using the expression for the adjoint action on the Pauli vector

    R_n⃗ (-α)  σ⃗  R_n⃗ (α) = σ⃗cos(α) + n⃗×σ⃗sin(α) + n⃗ n⃗·σ⃗ (1-cos(α))
 
Taking the inner product with a⃗ we get that:

    R_n⃗ (-α)  a⃗·σ⃗  R_n⃗ (α)    = a⃗·σ⃗cos(α) + a⃗· (n⃗×σ⃗) sin(α)  + a⃗·n⃗ n⃗·σ⃗ (1-cos(α)) 
       = [ a⃗cos(α) + a⃗×n⃗sin(α) + n⃗ a⃗·n⃗ (1-cos(α))] ·σ⃗
       = [ 1cos(α) + n⃗n⃗^T (1-cos(α)) + [n⃗]_×sin(α) ] a⃗·σ⃗
       = (R_n⃗^xyz (α) a⃗) ·σ⃗
       = b⃗·σ⃗,

where R_n⃗^xyz (α) is the rotation in ℝ^3 around the axis n⃗, by an angle of α. This in turn means that we can always find values for n⃗ and α such that one can transform the vector a⃗ into vector b⃗. Setting W = R_n⃗ (α) we can always find a unitary transformation that transforms one generator into another.


Which consequently leads to:

*


Follows trivially from previous theorem:

    W^† U_θ W    =  W^† e^-iθ G W 
       = W^†[ 1 + -iθ/1! G + (-iθ)^2/2! G^2 +  ... ] W 
       =W^† W + -iθ/1! W^† G W + (-iθ)^2/2! W^† G^2 W  + ... 
       = 1 + -iθ/1! W^† G W + (-iθ)^2/2! W^† G W W^† G  W  + ... 
       = 1 + -iθ/1! G' + (-iθ)^2/2! G'^2  + ... 
       = e^-iθ G'.


§ PRIVACY MEASURE 
 
In this appendix we look at the measure of privacy introduced in Def. <ref>, and prove the properties it should follow, introduced in the main text. Looking at the second and third properties, they should come by construction, as we verify below. For the last two, it is not obvious, but we still get them:


  *  From the fact that 𝒬 is a positive semi-definite matrix, we get that 𝒫 (𝒬,a⃗) ≤ 1. The minimum value follows from:
		
    a⃗^T 𝒬a⃗/𝒬    =   a⃗^T ∑_i β_i b⃗_i b⃗_i^T a⃗/𝒬
       =  ∑_i β_i ( a⃗·b⃗_i )^2 /[ ∑_i β_i b⃗_i b⃗_i^T ] 
       =  ∑_i β_i ( a⃗·b⃗_i )^2 /∑_i β_i  
       ≤ 1,

		as a⃗·b⃗_i ≤ 1 always, choosing normalized vectors b⃗_i.
	
  *  The fact that only a QFI of the type 𝒬 = αa⃗a⃗^T saturates the bounds comes trivially from before, as a⃗·b⃗_i = 1 iff b⃗_i = a⃗. Moreover, even if the QFI does not come in a diagonal form, we can still show that:
		
    a⃗^T 𝒬a⃗/𝒬    =   a⃗^T ∑_ijβ_ijb⃗_i b⃗_j^T a⃗/𝒬
       =  a⃗^T ∑_iβ̃_ib⃗̃⃗_i b⃗̃⃗_i^T a⃗/𝒬
       =  ∑_i β̃_i ( a⃗·b⃗̃⃗_i )^2 /∑_i β̃_i  
       ≤ 1,

		as the matrix of the QFI 𝒬 is always real and symmetric, so there is always a diagonalization for it. Moreover, since it is positive matrix all β̃_i are positive. The only way the bound is saturated is if ∃ i: b⃗̃⃗_i = a⃗ and β̃_j = 0 ∀ j≠ i.
	
  * 
    𝒫 (B𝒬B^T,Ba⃗)    =   B𝒬B^T - a⃗^T B^T B𝒬B^T B a⃗/ B𝒬B^T
       =𝒬B^TB - a⃗^T 𝒬a⃗/𝒬B^TB
       =𝒬 - a⃗^T 𝒬a⃗/𝒬
       = 𝒫 (𝒬,a⃗) ,

		where we made the assumption that B is a orthonormal change of basis matrix, i.e.BB^T=1. In case it is not, the two measures are not equivalent, but they still present a valid skewed measure.
	
  * 
    𝒫 (𝒬+ϵ A,a⃗) - 𝒫 (𝒬,a⃗)    =  [𝒬+ϵ A] - a⃗^T (𝒬+ϵ A) a⃗/[𝒬+ϵ A]  -𝒬 - a⃗^T 𝒬a⃗/𝒬
       = 1-a⃗(𝒬+ϵ A)a⃗^T/𝒬+ϵ A - (1-a⃗𝒬a⃗^T/𝒬) 
       =  a⃗𝒬a⃗^T/𝒬 -  a⃗𝒬a⃗^T/𝒬+ϵ A -ϵa⃗Aa⃗^T/𝒬+ϵ A
    
    			|𝒫 (𝒬+ϵ A,a⃗) - 𝒫 (𝒬,a⃗) |    ≤|  a⃗𝒬a⃗^T/𝒬+ϵ A  + ϵa⃗Aa⃗^T/𝒬+ϵ A - a⃗𝒬a⃗^T/𝒬| 
       ≤|  a⃗𝒬a⃗^T/𝒬  + ϵa⃗Aa⃗^T/𝒬 - a⃗𝒬a⃗^T/𝒬| 
       = 	 | ϵa⃗Aa⃗^T/𝒬| ≡δ		,

	providing the continuity of the privacy upon change of the initial state.


Using this measure of private one can try and understand what the different values of𝒫mean. The complete private case𝒫=1can be verified to naturally satisfy all conditions for the privacy. If the QFI matrix is of the form𝒬= αa⃗a⃗^Tthen is only possible to find an estimator for the functionf() = a⃗·. This also means that whatever estimators for other functions would be ill-defined and in not accessible (infinite variance). Consequently all parties can not estimate but the target function, and the dishonest parties having access to all the values in the dishonest subset, could at most gather information about the honest parties functionf(_H)but not their local values.

For the case of𝒫=ϵlet us provide two examples to understand howϵwould affect the information that might be leaked in the case of dishonest parties:


Suppose the QFI matrix is given by:

    𝒬 = ϵa⃗a⃗^T + (1-ϵ)b⃗b⃗^T,

such that a⃗_H ·b⃗_H = 0, a⃗·b⃗ = 0, denoting by x⃗_H the vector restricted to the honest parties. First verify that 𝒫=ϵ for the estimation of the function f()=a⃗·. Then, since the dishonest parties can share their own values and therefore have access to all θ_μ∈ D, they also have access to b⃗_H · which is a different from f(_H), which is the private function. This is not private, since it violates the third condition of privacy. If ϵ=0, then their information about this function is the maximum possible for this estimation (𝒬).

Suppose we have k parties and d≤ k/2 dishonest parties among them. Le the QFI matrix be given by:

    𝒬 = ϵa⃗a⃗^T + 1-ϵ/d∑_j=1^d b⃗_jb⃗_j^T,

Such that a⃗·b⃗_j = 0 for all vectors b⃗_j. In particular, one can choose these vectors b⃗_j such that b⃗_j_H, which is the correspondent to slicing the vectors b⃗_j for the indices correspondent to honest parties, form a d-dimensional orthogonal subspace. Again, we first verify that 𝒫 = ϵ as every b⃗_j is orthogonal to a⃗. Second, we verify that the set of information available by considering every dishonest party knows the dishonest subspace, is given by an orthogonal basis of d vectors in the honest subspace. Namely they have information about all b⃗_j_H· after using the dishonest information. In particular, if d=k/2, then they have d linearly independent vectors to estimate the honest subspace with k-d = d functions. This means they also have access to each individual parameter of the honest parties. Again, if ϵ→ 0 then they are more efficient, in the sense the variance at which they can do this the same amount of resources is smaller.


Note however that in both of these cases, for the parties to implement a non-private strategy and acquire information about the honest parties, they would have to distribute a non-private state. If one utilizes a state certification protocol in conjunction with this <cit.>, then one can guarantee privacy and security.



§ INTEGER VECTORS PROPERTIES 
*

    If α∈ℤ, then a⃗∈ℤ^k b⃗∈ℤ^k. To prove it is the only option choose a b_i such that b_i = α a_i, where α∉ℤ = c/d. For b_j to be integer for every other j, then gcd(a⃗,d)>1, as in, every a_i. However, gcd is associative, meaning gcd(a⃗,d)=gcd(gcd(a⃗),d)=1. This means α has to be integer.
*

Since a⃗,b⃗≺m⃗, 0⃗≺a⃗ + b⃗≺ 2m⃗, -m⃗≺a⃗ - b⃗≺m⃗. gcd(m⃗)=1 if αm⃗∝m⃗ then α∈ℤ. 

    ⇔
    	2 [ m⃗ - (a⃗+ b⃗) ]  ∝m⃗
    
    	2 (a⃗- b⃗) ∝m⃗⇔[ m⃗ - (a⃗+ b⃗) ]  ∝m⃗
    (a⃗- b⃗) ∝m⃗ but 
    	-m⃗≺m⃗ - (a⃗+ b⃗) ≺m⃗
    
    	-m⃗≺a⃗ - b⃗≺m⃗ .
*


The proof follows similarly. Since a⃗,b⃗≺n⃗, 0⃗≺a⃗ + b⃗≺ 2n⃗, -n⃗≺a⃗ - b⃗≺n⃗. gcd(m⃗)=1 if αm⃗∝m⃗ then α∈ℤ. 

    ⇔
    	2 [ n⃗ - (a⃗+ b⃗) ]  ∝m⃗
    
    	2 (a⃗- b⃗) ∝m⃗⇔[ n⃗ - (a⃗+ b⃗) ]  ∝m⃗
    (a⃗- b⃗) ∝m⃗ but 
    	-n⃗≺n⃗ - (a⃗+ b⃗) ≺n⃗
    
    	-n⃗≺a⃗ - b⃗≺n⃗ .

From here we verify that as n⃗⋠m⃗, ∃ j: n_j < m_j and gcd(m⃗)=1 and therefore no proportionality can be found.


§ QFI MATRIX FOR ARBITRARY STATES 


Starting from the definition of an arbitrary pure state of Eq. <ref> can write the elements of the QFI matrix (see Eq. <ref>) as:

    𝒬_μν (ρ_)    = 4∑_l,m=1^2^nα_l^* α_m ⟨𝒢_l |   G_μG_ν | 𝒢_m |-⟩∑_l,m=1^2^nα_l^* α_m  ⟨𝒢_l |G_μ |  𝒢_m |∑⟩_p,q=1^2^nα_p^* α_q⟨𝒢_p | G_ν |  𝒢_q |⟩,

withG_μ= ∑_j∈μ G_j. To calculate the different elements above consider the following labeling for the basis states: lets̃_l^±= (G_1^⊥)^l_1 ⊗(G_2^⊥)^l_2 ⊗⋯⊗(G_n-1^⊥)^l_n-1 ⊗(G_n)^±, whereG_n^+ = 1andG_n^- = G_n. Letl ∈{0,1,⋯,2^n-1-1}andl_1l_2⋯l_n-1be the binary string associated tol. From our definition ofG,G^⊥, we also have that:

    G_j ∘ G_j^⊥∘ G_j    = -G_j^⊥ ,      G_j ∘ G_j ∘ G_j =  G_j ,
    
        G_j^⊥∘ G_j ∘ G_j^⊥   = -G_j ,      G_j^⊥∘ G_j^⊥∘ G_j^⊥ =  G_j^⊥.

Using this when applyings̃_l^±to the stabilizer table in Eq. <ref>, we get a minus sign for the linesjsuch thatl_j=1. This yields:


    ⟨𝒢_l^± | ∑_j ∈μ∑_k ∈ν G_j G_k | 𝒢_m^±|   ⟩= h^*_𝒩_μ (l) h^*_𝒩_ν (m) ⟨𝒢_l^± | 𝒢_m^±|
    ⟩⟨𝒢_l^± | ∑_j ∈μ G_j |  𝒢_m^±|   ⟩= h^*_𝒩_μ (l)   ⟨𝒢_l^± | 𝒢_m^∓|,⟩

whereh^*_𝒩_μis theμelement of the vector-Hamming weight in Def. <ref>. Note that states differing on±have the same Hamming-weight, as expected sinceG_ncommutes with allG_jG_kterms. We can rewrite then Eq. <ref> the QFI in the following way:


    𝒬_μν (ρ_)    = ∑_l=0^2^n-1-1[ |α_l^+|^2 + |α_l^-|^2 ] h^*_𝒩_μ (l) h^*_𝒩_ν (l) - 
       - [ ∑_l=0^2^n-1-1 2 α_l^+^*α_l^-  h^*_𝒩_μ (l) ] [ ∑_q=0^2^n-1-1 2α_q^+^*α_q^- h^*_𝒩_ν (q)  ] 
       = ∑_l λ_l h^*_𝒩_μ (l) h^*_𝒩_ν (l) - [ ∑_l v_l h^*_𝒩_μ (l) ] [ ∑_q v_q h^*_𝒩_ν (q)  ] .


Let𝒩 = { 𝒩_μ}_μ∈Vbe the partition of the set of qubits into each of the nodes. Assembling the last equation for all entriesμνwe get exactly:


    𝒬 (ρ_)    = ∑_m λ_m h⃗^*_𝒩 (m) h⃗^*^T_𝒩 (m)  - ∑_m v_m h⃗^*_𝒩 (m)  ∑_q v_q h⃗^*^T_𝒩 (q) 
       = ∑_m,q𝔔_mqh⃗^*_𝒩  (m) h⃗^*_𝒩  (q)^T 
       = C 𝔔 C^T,

whereCis a matrix where each line is given by a vectorn⃗-2c⃗_i, where0⃗ ≼c⃗_i ≺n⃗.
This is equivalent, under the choice ofG = Z, G^⊥= Xto the GHZ state onnqubits, and the basis formed by|𝒢_m^±⟩ ∝|m⟩ ±|m⟩wheremranges between 0 and2^n-1-1≡N, is represented by the binary string associated to the numbermandmis the binary negation ofm(e.g.m=00101, m=11010). Furthermore, we also get two additional important properties: from Eq. <ref> we can see that the QFI matrix can be decomposed in𝔔 = Λ- v⃗v⃗^T, whereΛ= diag(λ_0,...,λ_m,...,λ_N)andv⃗ = (v_0, ..., v_m, ..., v_N). On top of this, we can verify that:


    |α_j^+ ±α_j^-|^2 ≥ 0  |α_j^+|^2 + |α_j^-|^2 ≥± 2α_j^+  α_j^-^*λ_j ≥ |v_j|.


§ MATRIX RANK INEQUALITIES 


Let us introduce some additional theorems regarding symmetric real matrices, which are the ones resultant from our problem. We start by the following theorem, which allows us to pick one and only one subspace in the set of states we can choose to construct a private state:

theoremrankk
Let A ∈ℝ^n× n be a symmetric matrix given by A = Λ - v⃗v⃗^T, where Λ is a diagonal matrix with all entries real and non-negative summing to 1 and v⃗ is a vector in ℝ^n such that v⃗≤ 1 and in particular ∀ i ∈{1,…,n}: λ_i ≥ |v_i |. A is positive semi-definite and the rank of A is at least

    Λ≥ A ≥ Λ -1,

and A =  Λ -1 iff. λ_j = |v_j|, ∀ j.
 
We start by reducing the problem into considering only the non-zero entries of Λ and v⃗, as all the zero entries of Λ do not contribute for the rank and the zero entries of v⃗ have corresponding values in Λ which are already eigenvalues of A. Also, we can never have a non-zero entry in v⃗ associated to a zero value in Λ, from the conditions stated. Denote the vector λ⃗ as the vector of all the positive diagonal values of Λ. The condition imposed on v⃗ means that ∀ i ∈{1,…,n}: λ_i ≥ |v_i |, which allows us to say:

    ∀y⃗≠0⃗ , ∃ x⃗≠0⃗: y⃗^T( Λ - v⃗v⃗^T) y⃗   ≥x⃗^T ( Λ - λ⃗λ⃗^T ) x⃗
    y⃗^T A y⃗   ≥x⃗^T B x⃗.

In particular, one could consider the vector x⃗ where (x⃗)_j = |(y⃗)_j|. By doing this, we can find bound the eigenvalues of A.

    x⃗^T ( diag (λ⃗) - λ⃗λ⃗^T ) x⃗   = ∑_i x_i^2 λ_i - (∑_i x_i λ_i)^2 
       = ∑_i x_i^2 λ_i - (∑_i x_i λ_i)(∑_j x_j λ_j) 
       =  ∑_i x_i^2 λ_i (1-λ_i) - ∑_i,  j≠ i x_i λ_i x_j λ_j 
       =  ∑_i x_i^2 λ_i ∑_j≠ iλ_j - 2 ∑_i,  j> i x_i λ_i x_j λ_j 
       =  ∑_i ∑_j> iλ_i λ_j (x_i^2 - 2 x_i x_j +x_j^2 ) 
       =  ∑_i ∑_j> iλ_i λ_j (x_i - x_j )^2 .

Since each λ_i is by construction positive and non-zero, this means this expression is always positive, and is only zero when (x_i-x_j)^2 = 0  ∀ i,j, which only happens for the null vector 0⃗ and the 1⃗/√(n). 

Suppose ∃ j: λ_j > |v_j |. In this case, Eq. <ref> is never saturated, unless (y⃗)_j is zero, which implies x⃗^T B x⃗ > 0. This means A is definite positive, and consequently that it has the rank of Λ. The other possible case is if ∀  j: λ_j = |v_j |. Consider the matrix D = diag((v⃗)_j/|(v⃗)_j|) which transforms v⃗→λ⃗, Dv⃗=λ⃗. Notice this matrix is a change of basis as DD^T=1 and moreover that DAD^T = B in this case. This allows us to map the eigenvectors of B into the eigenvectors of A, proving the lower bound on the rank, as 1⃗/√(n) is the only non-zero vector than spans the null space of B. This means the rank of A is bounded between the rank of Λ and rank Λ -1, where the latest equality only happens iff.  ∀  j: λ_j = |v_j |.

Let A ∈ℝ^n× n be symmetric and B ∈ℝ^k× n. If A has full rank, then BAB^T has rank equal to the rank of B.

    A = n       BAB^T =  B

Since A is symmetric, consider its diagonalization A = Q Λ Q^T.

    BAB^T = B Q Λ Q^T B^T = B Q √(Λ) Q^T Q √(Λ) Q^T B^T = B̃B̃^T  B̃B̃^T = B̃ =  B,

where used properties of the rank, namely, B B^T =  B^T B =  B =  B^T and if A is a l× m matrix with rank m and B is a m× n matrix, AB =  B.

Let A ∈ℝ^n× n be symmetric and B ∈ℝ^k× n. If A has rank n-1, then:

    A = n-1      B ≥ BAB^T ≥ B -1 .

Consider the same as before, but in the last step apply the Sylvester inequality for the rank of √(A)B:

    B ≥min{ √(A) ,  B }≥ B̃≥ B +  √(A) - n =  B -1.

Let V,W be two vector spaces. Let V_1 and V_2 be two m× n matrices with vectors in V. If 𝗌𝗉𝖺𝗇 (V_1) = 𝗌𝗉𝖺𝗇(V_2) then 𝗌𝗉𝖺𝗇(V_1 W_1) = 𝗌𝗉𝖺𝗇 (V_2 W_1) for any n× p matrix W_1 with vectors in W.

If 𝗌𝗉𝖺𝗇 (V_1) = 𝗌𝗉𝖺𝗇(V_2) then ∃ Q: V_2 = Q V_1 and is invertible. The proof concludes by realising that 𝗌𝗉𝖺𝗇 (QA) = 𝗌𝗉𝖺𝗇(A) for any invertible Q, given the definition of a span of a set of vectors.


§ PRIVATE STATES - PROOFS 
*


The proof in the indirect way comes trivially from calculating the privacy for the family of states F_GHZ, together with Corol. <ref>, we get that difference up to a LU operation.
Doing the same as before, now we have a different 𝔔 and C matrices, as n⃗ = a⃗. When looking at the matrix 𝔔 note that the first coefficient λ_0 and v_0 are associated with the span of the family of states F_GHZ, as by varying them, we get all the states in F_GHZ. Doing the same analysis as before, we get that now a⃗∈ C and dim_a⃗ (C) = 1, i.e. it only appears once in the lines of C. Constructing the same table to analyze privacy:



Checking again the case of rank 𝔔 = dim 𝔔-1 with rank C_𝔔 = 2. Using the same reasoning we find a vector Dw⃗ in the orthogonal subspace of D1⃗, 𝗌𝗉𝖺𝗇 P^⊥, acting on two different lines of C_𝔔:

    Dw⃗ C_𝔔   = 1/√(2)( h⃗_𝒩^* (i) ±h⃗_𝒩^* (j) ) 
       =
    	2/√(2)[ n⃗ - (h⃗_𝒩 (i) + h⃗_𝒩 (j) ) ] 
    
    	2/√(2)(h⃗_𝒩 (i) - h⃗_𝒩 (j) ) 
    	.

From here there are two options:

  * i or j correspond to the GHZ state, and in that case we have that h⃗_𝒩 (i) ±h⃗_𝒩 (j) = 0⃗≡h⃗_𝒩 (j) = 0⃗, meaning that i=j and contradicting rank-2 of C_𝔔;
    
  *  neither i nor j correspond to the GHZ state, and we recover that C_𝔔 only has vectors c⃗_i ≺a⃗, which is the case for the non-existence of a private state.

This means also this case is never private. Consequently, the only private state is the one spanned by a matrix with a seul non-zero entry correspondent to λ_0-v_0^2. Moreover, this is maximized for v_0^2 = 0, meaning a perfect GHZ state, up to a LU operation.

*


This is a direct consequence of the invariance of the Hamming-weight introduced in Props. <ref>,<ref>. Looking at the terms that appear in the QFI, fix the encoding dynamics basis at Z, as we have chosen our distributed s-states (Def. <ref>) in the computational basis. Notice this does not affect the generality of our proof, as again we are always an LU operation from any other basis (Thm. <ref>). Using this fact, and the fact that the vectorial Hamming-weight is invariant under the distributed permutations, we get that:

    ∑_j∈μ Z_j |ψ⟩   = h_𝒩_μ(|s(b_1_μ,a_μ+b_0_μ)⟩) |ψ⟩ = (a_μ+b_0_μ - b_1_μ) |ψ⟩ ,
    ∑_j∈μ Z_j |ψ̃⟩   = h_𝒩_μ(|s(a_μ+b_1_μ,b_0_μ)⟩) |ψ̃⟩ = (-a_μ +b_0_μ - b_1_μ) |ψ̃⟩,

Where we used that b⃗ = b⃗_0 + b⃗_1 which correspond to the amount of 0's and 1's respectively. Note that d⃗=b⃗_1 as well. After assembling in the QFI matrix we get:

    𝒬   = a⃗a⃗^T [ |α^2| + |β|^2 - (|α|^2-|β|^2)^2 ],

which means private. Moreover, it is maximised for |α|=|β|.
*


The inverse direction is a direct consequence from Thm. <ref>. The direct proof comes again from the QFI. Construct the same decomposition into C𝔔C^T as before, where the basis of 𝔔 are the families correspondent to F_GHZ. 



Now, for the first time we have that {c⃗_j ∈ C: c⃗_j= a⃗}≡𝔔_a⃗ is either 0 or larger than 1. It is larger than 1 if b⃗ = n⃗-a⃗ has all entries divisible by two, and one can find the family of states such that b⃗_0 = b⃗_1. This is a consequence of the symmetric Hamming weight that remains the same when adding the same amount of 0s and 1s to a bitstring.

Looking at the case of rank 𝔔 = dim 𝔔-1 with rank C_𝔔 = 2, we can find a way to construct all the others families. Using the same reasoning as before, let D be the matrix  such that D1⃗ spans the null space of 𝒬. Denote the orthogonal space of this vector by P^⊥. Let W be the set of vectors given by W^± = {w| w= e⃗_i ±e⃗_j , sign (D_ii) = ∓sign(D_jj)}, and Prop. <ref> allows us to use this vectors to continue our proof, as (P^⊥)=𝗌𝗉𝖺𝗇(W^+∪ W^-). Looking first at W^-,the options let us check which vectors c⃗_i - c⃗_j are proportional to a⃗:

    h⃗_𝒩^* (i) - h⃗_𝒩^* (j)    = δa⃗,     δ = {± 1, ± 2 }
    n⃗ - 2h⃗_𝒩 (i) - ( n⃗ - 2 h⃗_𝒩 (j) )    =  δa⃗,     δ = {± 1, ± 2 }
    h⃗_𝒩 (i) - h⃗_𝒩 (j)     =  δ̃a⃗  ,     δ̃ = {± 1}
    h⃗_𝒩 (i) = a⃗ + b⃗_j    ∧h⃗_𝒩 (j) = b⃗_j ,

where we used the fact that h⃗_𝒩 (·), is an positive integer vector and gcd(a⃗)=1. Moreover, we introduce the notation b⃗_i as a vector between 0⃗≼b⃗_i ≺b⃗. Substituting for h⃗_𝒩^* (i) and h⃗_𝒩^* (j) and using b⃗_i =b⃗-b⃗_j we observe: 

    h⃗_𝒩^* (i)    = n⃗ - 2( a⃗ + b⃗_j) =  a⃗ + b⃗ - 2( a⃗ + b⃗_j) = - a⃗ + b⃗ - 2 b⃗_j = - (a⃗ + b⃗ - 2 b⃗_i ) = -(n⃗-2b⃗_i)  
    h⃗_𝒩^* (j)    = n⃗ -2 b⃗_j  =  a⃗ + b⃗ - 2 b⃗_j.

The equations for the subspace W^+ are:

    h⃗_𝒩^* (i) + h⃗_𝒩^* (j)    = δa⃗,     δ = { 0, ± 1, ± 2 }
    n⃗ - 2h⃗_𝒩 (i) + ( n⃗ - 2 h⃗_𝒩 (j) )    =  δa⃗,     δ = { 0, ± 1, ± 2 }
    n⃗ - (h⃗_𝒩 (i) + h⃗_𝒩 (j) )    =  δ̃a⃗  ,     δ̃ = { 0, ± 1}
    ( h⃗_𝒩 (i) + h⃗_𝒩 (j)  = a⃗ + b⃗)    ∨( h⃗_𝒩 (i)  + h⃗_𝒩 (j) = b⃗) ∨(  h⃗_𝒩 (i) + h⃗_𝒩 (j) = 2a⃗ +b⃗).

This means the overall solution are the vectors inside C_𝔔, such that:

    c⃗_i = n⃗ - 2 (a⃗ + b⃗_i ) ∨c⃗_i = n⃗ - 2  b⃗_i, 
    c⃗_j = n⃗ - 2 (a⃗ + b⃗_j ) ∨c⃗_j = n⃗ - 2  b⃗_j,
    	,     b⃗_i + b⃗_j = b⃗.

Substituting we actually see that n⃗ - 2 (a⃗ + b⃗_i ) = -( n⃗ - 2  b⃗_j ), meaning they are in fact spanned by a state in the same family (see Prop. <ref>). This means choosing only the combination c⃗_i = n⃗ - 2  b⃗_i ∧c⃗_j = n⃗ - 2  b⃗_j is sufficient. The set of states with the same vectorial Hamming-weight, are exactly the states belonging to the equivalence class, by definition. From the construction of matrix 𝔔, we get that each line is associated with the following family of states:

    F_GHZ_m = {α|m⟩ + β|m⟩, α,β∈ℂ, |α|^2+|β|^2=1 }
       λ_i = |α_i|^2+|α_i+1|^2 ,	v_i = 2α_i α_i+1^*, α = α_i + α_i+1/√(2), β = α_i - α_i+1/√(2)
       λ_i = |v_i|  α = 1 , β = 0 , v_i/|v_i| = 1 
    α = 0 , β = 1 , v_i/|v_i| = -1 
    	.

This means we can only take states i such that, up to permutation of qubits (see Prop. <ref>), that have h⃗_𝒩(i) = b⃗_i, together with states j that h⃗_𝒩(j) = a⃗ + b⃗_j. Given that we can only choose b⃗_i, b⃗_j such that b⃗_i + b⃗_j = b⃗ (Eq. <ref>) and h⃗_𝒩(j)  = n⃗ - h⃗_𝒩(j) =  n⃗ -(a⃗ + b⃗_j) = b⃗-b⃗_j, then this are exactly the family of states given by ℱ(𝒩,a⃗,b⃗_i). By doing the same for each possible set of b⃗_i,b⃗_j, one generates each of the possible families of private states. Note however that one cannot create a superposition of states of different families, as that means the rank of C_𝔔 would be larger than 2, which would always mean a non-private state (check table). Moreover, one recovers the fact that if b⃗ is divisible by 2, then for b⃗_i = b⃗_j = b⃗/2 is the case of a rank-1 C_𝔔. Given that all other possibilities create either not rank-1 QFI matrices, or QFI matrices which are not proportional to a⃗a⃗^T, this are the only sets of private states. 


Below we provide two additional propositions that allowed some simplifications on the theorems above.


Let σ∈ S_𝒩. Let |GHZ_m⟩ = α|m⟩ + β|m⟩, where m is a binary string with length n⃗_1 and (·) is the binary conjugation of (·).

    h_𝒩 ( m ) = h_𝒩( σ(m) ) ⟺𝒬 (|GHZ_m⟩) = 𝒬(σ(|GHZ_m⟩))

Let |GHZ_m⟩ = α|m⟩ + β|m⟩ and |GHZ_m⟩ = α|m⟩ + β|m⟩, where m is a binary string with length n⃗_1 and (·) is the binary conjugation of (·).

    𝒬 (|GHZ_m⟩) = 𝒬(|GHZ_m⟩)


Finally, the statement for when we have multiples of the vector of resources:

*


From Thm. <ref> we get that we can fix dynamics as Z dynamics, without loosing any generality, and work with the regular computational basis for the qubits. This comprises the statement up to LU operations. Then, starting from:

    ∑_j∈ l Z_j |0_L⟩_l ≡Z_μ^l |0_L⟩_l    = (a_μ+b_0_μ^l - b_1_μ^l ) |0_L⟩  = (a_μ+Δ b_μ^l ) |0_L⟩ ,
    ∑_j∈ l Z_j |1_L⟩_l ≡Z_μ^l |1_L⟩_l     = (-a_μ +b_0_μ^l - b_1_μ^l ) |1_L⟩ = (-a_μ+Δ b_μ^l )|1_L⟩,

define |j⟩ = | j_1  j_2 ⋯  j_d⟩, such that Z_μ^l |j_L⟩_l = ( (-1)^j_l a_μ + Δ b_μ^l )  |j_L⟩_l and note that:

    Z_μ|j⟩ = ∑_l = 1^d Z_μ^l |j⟩   = ( h^*(j) a_μ + ∑_l=1^d Δ b_μ^l )  |j⟩.

This means that, for |ψ⟩∈𝔉, the QFI will have the following form:

    𝒬_μν   ∝⟨ψ|Z_μZ_ν|ψ⟩ - ⟨ψ|Z_μ|ψ⟩⟨ψ|Z_ν|ψ⟩
       = ∑_j,m=0^2^d-1α_j^* α_m ⟨j|Z_μZ_ν|m⟩ -∑_j,m=0^2^d-1α_j^* α_m ⟨j|Z_μ|m⟩∑_p,q=0^2^d-1α_p^* α_q ⟨p|Z_ν|k⟩
       = ∑_j=0^2^d-1 |α_j|^2 ( h^*(j) a_μ + ∑_l=1^d Δ b_μ^l ) ( h^*(j) a_ν + ∑_l=1^d Δ b_ν^l ) 
                - ∑_j,p=0^2^d-1 |α_j|^2 |α_p|^2 ( h^*(j) a_μ + ∑_l=1^d Δ b_μ^l ) ( h^*(p) a_ν + ∑_l=1^d Δ b_ν^l )  
       = [ ∑_j=0^2^d-1 |α_j|^2 h^*(j)^2 - ∑_j,p=0^2^d-1 |α_j|^2 |α_p|^2 h^*(j) h^*(p) ] a_μ a_ν ,

which makes the QFI matrix always proportional to a⃗a⃗^T, resulting always in a private state.



§ PRIVATE STATES EXISTENCE FOR GENERAL HAMILTONIANS 


Let us start by repeating and proving Thm. <ref>:

*



Given the rewriting of the QFI matrix in Eq. <ref>, we already now from Thm. <ref> that the rank of this matrix is given by the rank  𝔔-1, similarly to what we have done before. By choosing the orthogonal subspace to the null space of 𝔔 we realize that the span of this space is generated by the vectors in W = {w| w= e⃗_i⃗ - e⃗_j⃗}. Note that W C_𝔔 generates a space of vectors 𝒞 = {c⃗| c⃗= c⃗_i⃗ - c⃗_j⃗ , c⃗_i⃗, c⃗_j⃗∈𝒪}. This is exactly the higher order discrete subspace generated by 𝒪, i.e.𝒞𝒪^2_-. This already proves the first part of the theorem about the existence of private states, by choosing states with α_i⃗,α_j⃗ to be the only non-zero terms, associated to vectors c⃗_i⃗-c⃗_j⃗ = a⃗∈𝒪^2_-.

The fact that only vectors in the equivalence class of 𝒪^2_- have private states is a consequence of 𝗌𝗉𝖺𝗇(𝒞) = {αa⃗, α∈ℝ} in order for privacy to be verified. If αa⃗∉𝒪^2_- than one cannot build 𝗌𝗉𝖺𝗇(𝒞), as every c⃗_i⃗-c⃗_j⃗ has to be proportional to a⃗ and c⃗_i⃗,c⃗_j⃗ can only be chosen from 𝒪.


To prove Prop. <ref> let us first introduce and prove some additional propositions that will allows us to prove it:


Let X ∈𝒫_n be a Pauli string and ℋ⊆𝒫_n a subset of Pauli strings. Define an Hamiltonian as H = ∑_P ∈ℋ c_P P, where all c_P ≠ 0. Then:

    [X,H]_± = 0 ⇔ [X,P]_± = 0 ,  ∀ P ∈ℋ.

The inverse implication comes trivially from the linearity of the (anti)commutator [·,·]_±. The direct implication comes as follows: divide ℋ into two disjoint sets ℋ = ℋ_+ ∪ℋ_- such that [X,P]_± = 0,  ∀ P ∈ℋ_±. This is always possible given that two Pauli strings either commute or anticommute. Then:

    [X,H]_±   = ∑_P ∈ℋ c_P [X,P]_±
       = ∑_P ∈ℋ_± c_P [X,P]_± + ∑_P ∈ℋ_∓ c_P [X,P]_±
       = 0 +  ∑_P ∈ℋ_∓ c_P [X,P]_±
       = 2∑_P ∈ℋ_∓ c_P PX 
       = 2∑_P̃∈ℋ_∓ X c_P̃P̃ ,

which can only be zero if all c_P̃ are 0, as the set of Pauli strings generates a basis for complex matrices, leading to a contradiction.

Let ℋ⊆𝒫_n be a subset of Pauli strings. Define an Hamiltonian as H = ∑_P ∈ℋ c_P P, where all c_P > 0. Then:

    ∃ X ∈𝒫_n : {X, H } = 0 ⇔{λ_H } = { + λ_H }∪{ -λ_H}

Meaning for each eigenvalue of the Hamiltonian H, λ_H, its symmetric counterpart, -λ_H, is also an eigenvalue of H.

The direct proof follows very simply. X ∈𝒫_n  X^2=1.

    (H-λ1)    =  (H-λ1) (X^2) 
       = (X)(H-λ1) (X)
       =(XHX-λ X1X)
       = (-H-λ1),

which means the eigenvalues admit solutions for (H-λ1) and (H+λ1) and therefore if λ_H ∈ ev, -λ_H ∈ ev. The inverse proof comes as a consequence of the Pauli strings forming a basis for the space of complex matrices ℂ^2^n×ℂ^2^n. Take U to be a unitary operation that switches from positive eigenstates to the negative ones: U|λ_H^±⟩ = |λ_H^∓⟩. Then:

    U H |λ_H^±⟩   = ±λ_H U |λ_H^±⟩ = ±λ_H |λ_H^∓⟩
    
         H U |λ_H^±⟩   =  H |λ_H^∓⟩ =  ∓λ_H |λ_H^∓⟩
        (UH + HU)|λ_H^±⟩ = 0,   ∀ |λ_H^±⟩,

which means there is a unitary satisfying the anti-commutation relation. Given that U is a complex matrix, one can find a decomposition in terms of Pauli strings. Using Prop. <ref> and similar arguments for the proof of Prop. <ref>, then one has to choose the decomposition among the set of anti-commuting Pauli strings only. This consequently means there is at least one Pauli string that anti-commutes with the Hamiltonian.


This allows us to prove Prop. <ref>:
*

Follows directly from Prop. <ref>. The stated conditions for each local Hamiltonian imply that the local eigenvalues λ_μ belong to a set such that if +λ exists so does -λ. This means that if the vector c⃗_i belongs to the discrete subset 𝒪, so does the vector -c⃗_i. This consequently means that 𝒪^2_- where the vectors are built from c⃗_i - c⃗_j will, in particular, include the vectors c⃗_i-(-c⃗_i) = 2 c⃗_i. So 𝒪∼ 2𝒪⊆𝒪^2_- ≡𝒪𝒪^2_-. 


Finally, we are able to prove Corol. <ref>:

*

This is a consequence of the decomposition of 𝒬=C𝔔C^T of Eq. <ref> and the fact that a⃗_2 is maximum for a⃗∈ V(𝒪^2_-). Among only the private states, a⃗∈𝒪^2_-, one can calculate the maximal QFI as:


    max𝒬(a⃗) = maxa⃗^T𝒬a⃗ =a⃗_2^2 = c⃗_i - c⃗_j_2^2 .


The privacy measure already incapsulates the fact that any private state will maximise information about the target function. This means the maximum information one can get about the maximum function is necessarily private.


§ QFI FOR MIXED STATES - DERIVATIONS 


As said in the main text, to find the expression for the QFI in the mixed state case, we start by the most elementar expression for the QFI:


    𝒬_μν (ρ_) =  ℛ_ρ_^-1 (∂_θ_μρ_) ρ_ℛ_ρ_^-1 (∂_θ_νρ_).


Under the assumptions of the main text regarding the superoperatorℛ_ρ_, which corresponds to the symmetric logarithmic derivative, we can derive it in terms of the density matrix eigenvectors:

    ℛ_ρ^-1 (∂_θ_μρ_)   = ∑_j,k2/λ_j + λ_k⟨𝒢_j|∂_θ_μρ_|𝒢_k⟩|𝒢_j⟩⟨𝒢_k| .


Using the notation|𝒢_k'⟩ = U_|𝒢_k⟩:


    ∂_θ_μρ_   = ∂_θ_μ∑_k∈λ_k |𝒢_k'⟩⟨𝒢_k'|
       = ∑_k∈λ_k [ |∂_θ_μ𝒢_k'⟩⟨𝒢_k'| + |𝒢_k'⟩⟨∂_θ_μ𝒢_k'|],

where we assumed that the encoding is unitary and done over the density matrix.


    D_ij^μ≡⟨𝒢_i'|∂_θ_μρ_|𝒢_j'⟩   =⟨𝒢_i'|∑_k λ_k [ |∂_θ_μ𝒢_k'⟩⟨𝒢_k'| + |𝒢_k'⟩⟨∂_θ_μ𝒢_k'|] |𝒢_j'⟩
       = ∑_k λ_k [ ⟨𝒢_i' | ∂_θ_μ𝒢_k'|δ⟩_kj + δ_ik⟨∂_θ_μ𝒢_k' | 𝒢_j'|⟩] 
       =𝕀_ (j) λ_j ⟨𝒢_i' | ∂_θ_μ𝒢_j'|+⟩𝕀_ (i) λ_i ⟨∂_θ_μ𝒢_i' | 𝒢_j'|
    ⟩   =[𝕀_ (i) λ_i - 𝕀_ (j)λ_j ] ⟨∂_θ_μ𝒢_i' | 𝒢_j'|,⟩

where the last step was taking into consideration that:

    ∂_θ_μ (⟨𝒢_i' | 𝒢_j'|)⟩ = ∂_θ_μ (⟨𝒢_i | 𝒢_j|)⟩   = ∂_θ_μ (δ_ij) = 0 
       =  ⟨∂_θ_μ𝒢_i' | 𝒢_j'|+⟩⟨𝒢_i' | ∂_θ_μ𝒢_j'|.⟩


Plugging it in the QFI we get that:

    𝒬_μν (ρ_)    =  ℛ_ρ_^-1 (∂_θ_μρ_) ρ_ℛ_ρ_^-1 (∂_θ_νρ_)
       = [ ∑_i,j2/λ_i + λ_j D_ij^μ|𝒢_i'⟩⟨𝒢_j'|·∑_k∈λ_k |𝒢_k'⟩⟨𝒢_k'|·∑_m,n2/λ_m + λ_n D_mn^ν|𝒢_m'⟩⟨𝒢_n'|] 
       = [ ∑_i,j,m,n2D_ij^μ/λ_i + λ_j·2 D_mn^ν/λ_m + λ_n∑_k∈λ_k |𝒢_i'⟩⟨𝒢_j' | 𝒢_k'|⟨%s|%s⟩⟩𝒢_k' | 𝒢_m'⟨𝒢_n'|] 
       = [ ∑_i,n 
     k ∈2D_ik^μ/λ_i + λ_k·2 D_kn^ν/λ_k + λ_nλ_k |𝒢_i'⟩⟨𝒢_n'|] 
       = ∑_n∈
     k∈2D_nk^μ/λ_n + λ_k·2 D_kn^ν/λ_k + λ_nλ_k +  ∑_n∈
     k∈2D_nk^μ/λ_n + λ_k·2 D_kn^ν/λ_k + λ_nλ_k 
       =∑_n∈
     k∈2D_nk^μ D_kn^ν/λ_n + λ_k·( λ_k/λ_k + λ_n + λ_n/λ_n + λ_k) + ∑_n∈
     k∈2D_nk^μ/λ_k·2 D_kn^ν/λ_kλ_k 
       = ∑_n ∈
     k∈2D_nk^μ  D_kn^ν/λ_n + λ_k +  ∑_n ∈
     k∈4D_nk^μ  D_kn^ν/λ_k
       =∑_n ∈
     k∈  2(λ_n - λ_k)^2/λ_n + λ_k⟨∂_θ_μ𝒢_n' | 𝒢_k'|⟨%s|%s⟩⟩𝒢_k' | ∂_θ_ν𝒢_n' + ∑_n ∈
     k∈  4  λ_k ⟨𝒢_n' | ∂_θ_μ𝒢_k'|⟨%s|%s⟩⟩∂_θ_ν𝒢_k' | 𝒢_n'
       = ∑_n ∈
     k∈ 2(λ_n - λ_k)^2/λ_n + λ_k⟨𝒢_n | G_μ | 𝒢_k|⟨%s|%s⟩⟩𝒢_k | G_ν | 𝒢_n+ ∑_n ∈
     k∈  4  λ_k ⟨𝒢_k | G_μ | 𝒢_n|⟨%s|%s⟩⟩𝒢_n | G_ν | 𝒢_k.


Let us retrieve the pure state equation from this. Assume the quantum state density matrix can be described by that of a pure stateρ= |ψ⟩⟨ψ|. This meansλ_1 = 1and every otherλ_j = 0. Moreover, since independent of the|ψ⟩choice, one can always find an orthonormal basis such that|ψ⟩is part of. This means in particular that∃ℬ = { |ψ_j⟩ }_j=1,⋯,2^n, such that|ψ_1⟩ = |ψ⟩and∑_j=1^2^n |ψ_j⟩⟨ψ_j| = 1. Pluggingρinto Eq. <ref> we get:

    𝒬_μν (ρ_)    = 4 ⟨ψ_1 |G_μ(  ∑_ k=2^2^n|ψ_k⟩⟨ψ_k|)  G_ν|ψ_1⟩
       =   4  ⟨ψ_1 |G_μ(  1 - |ψ_1⟩⟨ψ_1|) G_ν|ψ_1⟩
       =   4 ⟨ψ_1 |G_μG_ν|ψ_1⟩ -  ⟨ψ_1 |G_μ|ψ_1⟩⟨ψ_1|G_ν|ψ_1⟩ .


This allows us to get one additional way to calculate the QFI of a mixed state. Suppose the density matrix is a sum of orthogonal states (even if it is not, there is always a decomposition into orthogonal states by changing basis, consequence of a diagonalization of the matrix). Let:

    ρ = ∑_j ∈λ_j |𝒢_j⟩⟨𝒢_j|,  such that ⟨𝒢_j | 𝒢_k |=⟩δ_jk.

Then:

    𝒬_μν (ρ_)    = ∑_n ∈
     k∈ 2(λ_n - λ_k)^2/λ_n + λ_k⟨𝒢_n | G_μ | 𝒢_k|⟨%s|%s⟩⟩𝒢_k | G_ν | 𝒢_n+ ∑_n ∈
     k∈  4  λ_k ⟨𝒢_k | G_μ | 𝒢_n|⟨%s|%s⟩⟩𝒢_n | G_ν | 𝒢_k
       = ∑_n ∈
     k∈ 2(λ_n - λ_k)^2/λ_n + λ_k a_kn^μ a_nk^ν + ∑_k∈  4  λ_k  ⟨𝒢_k |G_ν[ ∑_n∈|𝒢_n⟩⟨𝒢_n|]  G_μ|𝒢_k⟩
       = n ∈
     k∈ 2(λ_n - λ_k)^2/λ_n + λ_k a_kn^μ a_nk^ν + k∈  4  λ_k ⟨𝒢_k |G_ν[ 1 - n∈|𝒢_n⟩⟨𝒢_n|]  G_μ|𝒢_k⟩
       = n ∈
     k∈ 2(λ_n - λ_k)^2/λ_n + λ_k a_kn^μ a_nk^ν + k∈  4  λ_k ⟨𝒢_k |G_ν[ 1 - |𝒢_k⟩⟨𝒢_k| -n∈
     n≠ k|𝒢_n⟩⟨𝒢_n|]  G_μ|𝒢_k⟩
       = n ∈
     k∈ 2(λ_n - λ_k)^2/λ_n + λ_k a_kn^μ a_nk^ν + k∈λ_k 𝒬_μν (|𝒢_k⟩) - n,k ∈
     k≠ n  4  λ_k  a_kn^μ a_nk^ν
       = ∑_n,k ∈
     n≠ k[ 2(λ_n - λ_k)^2/λ_n + λ_k - 4 λ_k ]  a_kn^μ a_nk^ν + ∑_k∈λ_k 𝒬_μν (|𝒢_k⟩) 
       = ∑_n>k ∈ 4[ (λ_n - λ_k)^2/λ_n + λ_k - (λ_n + λ_k) ]  a_nk^ν a_kn^μ + ∑_k∈λ_k 𝒬_μν (|𝒢_k⟩) .


In the case the state density matrix is described by a sum of non-orthogonal vectors:

    ρ = ∑_j ∈λ_j |ϕ_j⟩⟨ϕ_j|,  such that ⟨ϕ_j | ϕ_k |=⟩α_jk≤ 1,

such thatG_μ|ϕ_j⟩is equal for every|ϕ_j⟩, then diagonalizingρinto vectors that are orthogonal between themselves, call it|ϕ̃_j⟩still holds the fact thatG_μ|ϕ̃_j⟩is equal for every|ϕ̃_j⟩. If, more than that, a_nk^νa_kn^μ ∝𝒬_νμ (|𝒢_n⟩) , thenλ_nshould simply be scaled into aλ̃_n, but the information is still structurally the same,i.e.it is the same function of parameters. 

This gives some intuition about how to analyze Eq. <ref> and when to use it over Eq. <ref>. In Eq. <ref> the first term corresponds to a distinguishability between the states under the encoding dynamics. If the generators of the dynamics are orthogonal to the noise, then this goes to zero. The second part of Eq. <ref> can be seen as the local informations of each states in a perfect scenario weighted by the corresponding weights. Note that the expression:

    4[ (λ_n - λ_k)^2/λ_n + λ_k - (λ_n + λ_k) ] ,

is always negative, meaning that if they are completely distinguishable, then we recover the notion of convexity of the QFI. If they are not completely distinguishable, then the amount of information is decreased by a factor proportional to  a_nk^νa_kn^μ . On the other hand, when the noise is only in the same direction of the generators of the encoding, then Eq. <ref> becomes simpler, as the second part goes to zero, and we are left simply with an expression that weights the distinguishability of the states, under the encoding dynamics.

 Paper Title: Enhancing the Harrow-Hassidim-Lloyd (HHL) algorithm in systems with large condition numbers.

 





 ./figures/ 

















 
















custompmatrix
  ([ )


 




















theoremTheorem
lemma[theorem]Lemma
claimClaim
corollary[theorem]Corollary
definitionDefinition
lemma1Lemma
exampleExample
conditionCondition


APS/123-QED 
]Enhancing the Harrow-Hassidim-Lloyd (HHL) algorithm in systems with large condition numbers

Centre for Quantum Engineering, Research and Education, TCG CREST, Sector V, Salt Lake, Kolkata 700091, India

Centre for Quantum Engineering, Research and Education, TCG CREST, Sector V, Salt Lake, Kolkata 700091, India

Centre for Quantum Engineering, Research and Education, TCG CREST, Sector V, Salt Lake, Kolkata 700091, India
Graduate School of Science and Technology, Keio University, 7-1 Shinkawasaki, Saiwai-ku, Kawasaki, Kanagawa 212-0032, Japan
Quantum Computing Center, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan

Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada

Institute for Advancing Intelligence, TCG CREST, Sector V, Salt Lake, Kolkata 700091, India
Academy of Scientific and Innovative Research (AcSIR), Ghaziabad- 201002, India



srinivasaprasannaa@gmail.com
Centre for Quantum Engineering, Research and Education, TCG CREST, Sector V, Salt Lake, Kolkata 700091, India
Academy of Scientific and Innovative Research (AcSIR), Ghaziabad- 201002, India


§ ABSTRACT 
Although the Harrow-Hassidim-Lloyd (HHL) algorithm offers an exponential speedup in system size for treating linear equations of the form Ax⃗=b⃗ on quantum computers when compared to their traditional counterparts, it faces a challenge related to the condition number (κ) scaling of the A matrix. In this work, we address the issue by introducing the post-selection-improved HHL (Psi-HHL) approach that involves a simple yet effective modification of the HHL algorithm to extract a feature of |x⟩, and which leads to achieving optimal behaviour in κ (linear scaling) for large condition number situations. This has  the important practical implication of having to use substantially fewer shots relative to the traditional HHL algorithm. We carry out two sets of simulations, where we go up to 26-qubit calculations, to demonstrate the ability of Psi-HHL to handle situations involving large κ matrices via: (a) a set of toy matrices, for which we go up to size 64 × 64 and κ values of up to ≈ 1 million, and (b) a deep-dive into quantum chemistry, where we consider matrices up to size 256 × 256 that reach κ of about 466. The molecular systems that we consider are Li_2, RbH, and CsH. Although the feature of |x⟩ considered in our examples is an overlap between the input and output states of the HHL algorithm, our approach is general and can be applied in principle to any transition matrix element involving |x⟩. 


[
    V. S. Prasannaa
    August 5, 2024
===================







§ INTRODUCTION


The Harrow-Hassidim-Lloyd (HHL) algorithm was proposed to solve equations of the form Ax⃗=b⃗, where A is a known and sparse 2^N × 2^N matrix and b⃗ is 
a 2^N × 1 vector, on quantum computers with a time complexity that is exponentially lower in system size than those of the best known classical methods <cit.>. Further, via amplitude encoding, b⃗, which is a 2^N × 1 vector, is encoded into a log_22^N-qubit state (for brevity, we shall hereafter denote 2^N as n). Thus, when the HHL approach is applied to `killer applications' of quantum computing such as quantum chemistry, there is a natural exponential suppression in qubit number relative to the other known approaches such as the quantum phase estimation (QPE) and the variational quantum eigensolver (VQE) algorithms <cit.>, thus opening new avenues to apply HHL for calculating properties of large molecules. 


Despite possessing the aforementioned advantages, the HHL algorithm relies upon the probability of successfully post-selecting 1 on the HHL ancilla qubit, that is, P(1) <cit.>. It is known that there is a substantial decrease in P(1) with increase in the condition number of the problem matrix A, which we shall denote by κ. In other words, the fraction of usable shots in the HHL algorithm becomes very low when large κ input matrices are involved. Unfortunately, it is not uncommon to encounter the issue while employing HHL for some interesting problems (for example, see Ref. <cit.> for application of the algorithm to hydrological systems), including quantum chemistry, which we shall discuss in detail in our Results section. Several works in literature have addressed the issue of handling large κ matrices either directly (where κ of the A matrix itself is altered via approaches such as preconditioning) or indirectly <cit.>. For a thorough survey, we direct the reader to Section 1.2 of Ref. <cit.>. 
 

In this work, we propose a new method to address the issue of handling A matrices with large condition numbers, by introducing the post-selection-improved HHL (Psi-HHL) algorithm, which relies upon the intuition of extracting the correct signal by subtracting a wrong signal from a mixed signal (that contains both the correct and wrong signals). A signal, in the context of our work, refers to a scalar that is a feature of |x⟩. For the purposes of this work, we pick the overlap between the input and output states of HHL, which in the case of molecules corresponds to their correlation energies (up to some normalization constants), as our signal, although it is to be noted that Psi-HHL itself is applicable to features as general as transition matrix elements. The wrong signal is obtained by successfully post-selecting outcome bit `0' instead of `1' (we call this approach HHL_1). The mixed signal is generated in an HHL quantum circuit by introducing an appropriately chosen 2 × 2 unitary just before measurement of the HHL ancilla (we name this procedure HHL_2). Figure <ref> presents schematic diagrams outlining the workings of the HHL and the Psi-HHL algorithms. 
 

This simple modification involving execution of two HHL routines enables one to carry out calculations for treating systems of linear equations involving large-κ matrices without having to contend with a large number of shots. In fact, we argue that the condition number scales linearly with system size with our approach for large κ situations, while still staying in the realm of a typical HHL computation. This compares favourably with several other notable works in literature, including but not limited to the original HHL work: 𝒪(κ^3) and 𝒪(κ^2) without and with amplitude amplification respectively <cit.>, 𝒪(κlog ^3κ) with variable time amplitude amplification <cit.>, and works by Childs et al: 𝒪(κpolylog (κ)) <cit.>, Wossnig et al: 𝒪(κ^2), and Costa et al's discrete quantum adiabatic approach based quantum linear solver algorithm <cit.>: 𝒪(κ). In particular, the last approach also achieves linear scaling in condition number, but we do so without invoking any additional algorithmic machinery  beyond the scope of the traditional HHL algorithm. In order to provide a very qualitative perspective, we note that for condition number of about 1 million, κ^3 : κ^2 : κlog^3κ : κ is ∼ 10^18: 10^12: 7.9 × 10^9: 10^6. For completeness, we add that the best known classical approach, conjugate gradient, scales as 𝒪(√(κ)) <cit.>, but its scaling with respect to system size is, of course, exponentially more expensive than that for HHL. 
 

The paper is organized as follows: Section <ref> presents the theory and methodology involved in our work by introducing the Psi-HHL approach, including a discussion on the complexity theoretic aspects (in κ) of Psi-HHL. Section <ref> presents our results for Psi-HHL applied to (a) suitably chosen sets of 4 × 4, 8 × 8, 16 × 16, 32 × 32, and 64 × 64 toy matrices, as well as for (b) molecular calculations (64 × 64 and 256 × 256 A matrix sizes) in the linearized coupled cluster (LCC) HHL and Psi-HHL frameworks. We conclude in Section <ref>. 
 

 



§ THEORY AND METHODOLOGY




 §.§ The HHL algorithm


We briefly sketch the relevant details of the HHL algorithm (we refer the readers to Refs. <cit.> for detailed descriptions of the algorithm) in Figs. <ref>(a) and <ref>(b), where an amplitude-encoded input b⟩ = ∑_i b_i ν_i ⟩, such that ∑_i |b_i|^2 = 1 where {ν_i ⟩} refers to the eigenbasis of A, is fed into the HHL algorithm along with n_r number of clock register qubits and one HHL ancillary qubit. The HHL algorithm itself involves QPE, followed by a non-trivial controlled-rotation module, and then a QPE^† module. At the end of the circuit, we have the following state vector,


    ∑_i b_i (√(1-C^2/λ̃^2_i)|0⟩ + C/λ̃_i|1⟩) ⊗|0^n_r⟩⊗|ν_i⟩.
 

In the above expression, {λ̃_i} refers to the estimates of the actual eigenvalues of A (1 ≥λ̃_i ≥ 1/κ) obtained via the QPE module of HHL. Typically, C is chosen to be the minimum eigenvalue of A, but it is worth noting that incurring such a large classical overhead can always be circumvented via approaches such as AdaptHHL <cit.>. When one post-selects the outcome `1' on the HHL ancilla qubit, one can think of it as choosing the `correct' signal. 
 

The solution vector |x⟩∝ A^-1 |b ⟩, and the input vector |b⟩ can be written as,
  
    |x⟩=|x⟩_ un/|x⟩_ un and |b⟩=|b⟩_ un/|b⟩_ un
 
respectively, where subscript `un' refers to unnormalized states— |b⟩_ un is the unnormalized input state after amplitude encoding and |x⟩_ un=∑_i b_i C/λ̃_i|ν_i ⟩, with ∑_i |b_i|^2 = 1. The probability of successfully post-selecting outcome `1' is given as P(1)=|x⟩_ un^2.
Finally, when one wants to predict a feature of the solution vector, say, the correlation energy of a molecular system (proportional to the overlap between |x⟩ and |b⟩), it can be done by appending an additional circuit (the Hong-Ou-Mandel module <cit.>), abbreviated as HOM in Fig. <ref>(b) that inputs the output, x ⟩, of the HHL algorithm and b ⟩, to output the overlap between the two quantities. The correlation energy is then calculated as 


    E_c^ HHL   =    - |b⟩_ un^2  |x⟩_ un  |⟨ b | x ⟩|  
       =    - |b⟩_un^2 ∑_i |b_i|^2 C/λ̃_i.
 



For the sake of simplicity, we set this quantity hereafter as the feature of |x⟩ that is of interest to us, for the rest of this work, although Psi-HHL can be applied to more general features, such as transition matrix elements involving |x⟩. Therefore, even for non-quantum chemistry examples that we shall later consider, we will use the term `correlation energy' to refer to the quantity from Eq. <ref>. The number of trials needed to successfully post-select outcome `1' is 𝒪(κ^2). Thus, in the event of large κ, the success probability falls quickly, making it practically difficult to carry computations using large κ matrices. In this work, we demonstrate a new approach— `Psi-HHL', towards handling large κ matrices by leveraging the probability of successfully post-selecting the `wrong' outcome `0', P(0), in HHL.   



 §.§ The Psi-HHL approach


In this section, we discuss the two essential ingredients— HHL_1 and HHL_2, which make up Psi-HHL as illustrated in Fig. <ref>(c). The HHL_1 module depicts the case where the `wrong' signal is post-selected from the traditional HHL given in Fig. <ref>(a) and Fig. <ref>(b). The HHL_2 entails  traditional HHL module appended with a mixing 2 × 2 unitary, given as R_Y(2α) and post-selecting 1 from the mixed signal. The role of this unitary is to mix the coefficients of |0⟩ and |1⟩ in the traditional HHL (see Eq. <ref>), thereby outputting a `mixed' signal. The core idea behind Psi-HHL is to subtract the correlation energies associated with HHL_1 and HHL_2 thus leaving us with the correlation energy corresponding to the `correct' signal. The gain in taking this route is the efficiency that we achieve in the number of shots expended. We summarize this in the following proposition: 

*proposition*Proposition

Psi-HHL yields the same correlation energy as traditional HHL through two implementations of HHL, but is more efficient in the number of shots than HHL, especially when dealing with A matrices with large condition numbers. 
 

We expound on the first part of the proposition in the following paragraphs, and the second in sub-section <ref>. 



  §.§.§ HHL_1: Computing the wrong signal


The first HHL module, HHL_1, serves to post-select the `wrong' signal of the HHL ancilla qubit, that is, the state attached to |0⟩ in Eq.<ref>. 
We re-write the state at the output of all the registers |ψ_ out⟩ except the clock qubits register, as follows, 
    |ψ_ out⟩   =   ∑_i b_i (√(1-C^2/λ̃^2_i)|0⟩⊗|ν_i⟩ + C/λ̃_i|1⟩⊗|ν_i⟩)
       =   ( |0⟩⊗  |x_ w⟩+ |1⟩⊗ |x_r⟩ ).
 
Note that here |x_w⟩ and |x_r⟩ are the unnormalised vectors in the superposition. The subscripts w and r refer to `wrong' and `correct' respectively, where for the latter, we use r since c is used already to denote `correlation'. The normalised solution vector |x_0⟩ which appears at the output register after successfully post-selecting outcome `0' is given by,

    |x_ 0⟩   =   |x_w⟩/|x_w⟩
       =   ∑_i b_i √(1 - C^2/λ̃_i^2)|ν_i⟩/ |x_w⟩,

where,

    |x_ w⟩ = √(P(0))= (∑_i |b_i√(1- C^2/λ̃_i^2)|^2)^1/2.
 Here, P(0) is the probability of successfully post-selecting outcome `0'. The associated correlation energy is then given as,

    E_ c^ w = -|b⟩_ un^2  |x_ w⟩  |⟨ b|x_0⟩| .

Here, |b⟩_ un is the unnormalized input vector after amplitude encoding, and

    |⟨ b|x_0⟩|= ∑_i  |b_i|^2 √(1- C^2/λ̃_i^2).
 



  §.§.§ HHL_2: Computing the mixed signal


The second HHL execution involves attaching the rotation gate R_Y(2α) on the HHL ancilla before the measurement as illustrated in the lower panel of Fig. <ref>(c). This seemingly trivial addition to the algorithm introduces a new degree of freedom via the angle α, enabling one to boost the probability of successful matrix inversion in the case of large κ. We recall that R_Y(2α) is a rotation about the Y axis on the Bloch sphere as given below

    R_Y(2α) = [  cos(α) -sin(α);  sin(α)  cos(α) ].
 
The state vector at the output of HHL_2 module |ψ'_ out⟩, ignoring the clock qubits register, after applying the unitary  in Eq. <ref> is given below,


    |ψ'_ out⟩   =   ∑_i b_i (√(1-C^2/λ̃_i^2)cos(α) - C/λ̃_isin(α) ) |0⟩⊗|ν_i⟩
       +     b_i ( √(1-C^2/λ̃_i^2)sin(α) + C/λ̃_icos(α)) |1⟩⊗|ν_i⟩
       =   (|0⟩⊗|x_ m⟩+|1⟩⊗|x'_ m⟩ ).


Note that in Eq. <ref>, |x_m⟩ and |x'_m⟩ are unnormalized vectors. The subscript `m' denotes `mixed', and while the unprimed vector accompanies |0⟩, the primed one accompanies |1⟩. The solution vector at the output after successfully post-selecting outcome `1' is given below


    |x'⟩   =   |x'_ m⟩/|x'_ m⟩
       =   ∑_i b_i ( √(1-C^2/λ̃_i^2)sin(α) + C/λ̃_icos(α)) |ν_i⟩/ |x'_ m⟩,


where 


    |x'_ m⟩   =   √(P'(1))
       =   (∑_i | b_i (√(1-C^2/λ̃_i^2)sin(α) +  C/λ̃_icos(α)) |^2)^1/2.


Here, P'(1) is the probability of successfully obtaining outcome `1' in HHL_2. The correlation energy is then obtained as 

    E_c^ m = -|b⟩_ un^2   |x_ m'⟩ |⟨ b|x'⟩|,

where

    |⟨ b|x'⟩| = ∑_i |b_i|^2 ( √(1- C^2/λ̃_i^2)sin (α) +C/λ̃_icos(α) ).
 
We choose α such that sin(α) and cos(α) are  positive and P'(1)≫ P'(0). P'(0) refers to the probability of post-selecting outcome `0' in HHL_2. 



  §.§.§ Psi-HHL: Subtracting signals
 

In order to prove this statement, we subtract Eq. <ref> and Eq. <ref>, thus obtaining 


    E_c^Psi-HHLcot(α)   =   E_c^m/sin (α) - E_c^w
       =     - |b⟩_ un^2 ∑_i |b_i|^2 C/λ̃_i  cot(α).

By comparing the right hand sides of Eq. <ref> and Eq. <ref>, we see that the expressions for E_c^ HHL and E_c^Psi-HHL differ only by a factor of cot(α) as shown below.


    E_c^Psi-HHL  =   E_c^m/ sin(α) - E_c^w/ cot(α) = E_c^ HHL.
 

Although theoretically, we obtain the same correlation energy from Psi-HHL, we observe that this technique is superior to traditional HHL in the way it handles large κ matrices, due to the optimal probability of success that one achieves while post-selecting, as we explain in the subsequent section.
Therefore, E_c^Psi-HHL can predict the true correlation energy, E_c, to a reasonable level of precision with fewer shots even for values of κ where HHL procedure fails to do so, which we illustrate in the Results section with examples. We find through our examples that for up to fairly large values of κ that are well beyond those that HHL can handle, Psi-HHL predicts the correlation energy with fewer shots. It is also important to note that throughout the derivation discussed above, we can replace |b⟩ with O|y⟩, where O is a suitably block-encoded observable <cit.>, and |y⟩ an arbitrary state that can be an output of another algorithm, and our results from Psi-HHL still hold. We also add that in problems where it is not immediately obvious a priori that the condition number is large, one cannot determine whether to employ HHL or Psi-HHL. However, this is not a problem as Psi-HHL intrinsically involves the regular HHL procedure, and hence if P(1) is significant, one can choose not to opt for the Psi-HHL procedure. 



 §.§ Complexity aspects
 

We now demonstrate the complexity (in κ) of the Psi-HHL algorithm. Since for the purposes of deriving condition number behaviour at the measurement-post-selection part of HHL, the distinction between the estimated eigenvalues of A, that is, λ̃_i, and the  actual eigenvalues, λ_i, does not matter, we denote all estimated eigenvalues in this section as simply λ_i for the sake of convenience. 
 

In order to elaborate on the second part of our Proposition, we invoke the probability with which one post selects the desired outcome successfully in terms of κ. 
 

a) Success probability of the post-selection process in HHL_1:  The probability of successfully post-selecting the outcome `0', P(0), in HHL_1 as given in Eq. <ref> can be expanded as <cit.> 


    P(0)   =   ∑_i |b_i√(1- C^2/λ_i^2)|^2 
       ≈   √((𝕀-C^2 A^-2))|b⟩^2,

where 𝕀 is the identity operator written in the eigenbasis of operator A, constant C= O(1/κ), where the condition number κ is taken as λ_ max/λ_ min  and |b⟩ is the input to the HHL circuit. The set of values {√(1-C^2/λ_i^2)} are the eigenvalues of the operator √(𝕀-C^2 A^-2), with the largest eigenvalue being √(1-C^2/λ_ max^2), and therefore 


    P(0)   ≤   √((𝕀-C^2 A^-2))_ op^2|b⟩ ^2 
    
    P(0)   ≤    1-C^2/λ_ max^2
       ≤    1-1/κ^2.

 
The first line uses the property of sub-multiplicativity of matrix norms, that is, A B≤A B. In  Eq. <ref>, we recall that we set C=λ_ min, ._ op is the operator norm, and |b⟩=1. Note that the smallest eigenvalue of this operator is √(1-C^2/λ_ min^2)=0, and corresponds to P(0)=0, which leads to expending a large number of shots in HHL_1. One could circumvent this issue by appropriately choosing a scaled C = γλ_min, where γ∈ 
( 0,1), leading to P(0)≥ 1-γ^2. This idea has been illustrated via an example as shown in Fig. <ref>, where we choose the scaling γ=0.5. 
 



b) Success probability of the post-selection process in HHL_2: A similar expression for the probability of successful post-selection in HHL_2,  P'(1), can be obtained by further expanding Eq. <ref> as follows, 


    P'(1)   ≈    [√((𝕀-C^2A^-2)) sin(α) 
       +    (C A^-1) cos(α) ]|b⟩^2.

The operator B √(𝕀-C^2A^-2)sin(α) + (C A^-1) cos(α) in its eigenbasis {|ν_i⟩} could be given in the diagonal form as 


    diag( ⋯, √(1-C^2/λ^2_max)sin(α)+(C/λ_max)cos(α), ⋯, 
    √(1- C^2/λ^2_min)sin(α)+(C/λ_min)cos(α),  
    ⋯ ),
 

where the entries that occur in the parentheses after `diag' are the diagonal elements of the matrix. Note that the largest and smallest eigenvalue of the matrix given in Eq. <ref> depend on the choice of α. When α=π/2, the component of that eigenvalue attached to sin(α) remains, and we do not extract the desired solution. However, when α is too small, say 0, the component attached to cos(α) remains and we get the same complexity as the traditional HHL approach. Therefore, α is chosen to satisfy the following properties: 


  * The success probability, P'(1) is large enough.

  * The solution component that matters to us does not vanish in the superpositions that occur along the diagonal entries in Eq. <ref>.


We demonstrate the case where α is close to π/2 and C=λ_min, in which case the operator in Eq. <ref> could be written in the form where the largest eigenvalue is the first entry and the smallest eigenvalue is the last entry, as given below,


    diag(⋯, √(1-1/κ^2)sin(α)+(1/κ)cos(α), ⋯, 
    cos(α)).


Using once again the sub-multiplicativity of matrix norms in Eq. <ref>, we can write 


    P'(1)   ≤    [√(𝕀-C^2A^-2) sin(α) 
       +    CA^-1cos(α) ] ^2_ op |b⟩^2 ,


where ._ op is the operator norm yielding the largest eigenvalue of the operator, and |b⟩=1. Then P'(1) is bounded the following way, 


    cos^2(α)≤ P'(1)   ≤   |[(√(1- 1/κ^2)) sin(α) + 1/κ cos(α) ]  |^2 
    
       =   (1- 1/κ^2) sin^2(α) + 1/κ^2 cos^2(α) 
       +    2(√(1- 1/κ^2))1/κsin(α)cos(α) 
       ≈    sin^2(α) + 1/κ sin(2α) +  1/κ^2cos(2α) 
       -   1/κ^3sin(α)cos(α).


In this bound, the left hand side is the square of the smallest eigenvalue of the matrix given in Eq. <ref>, while the right hand side is the square of the  largest eigenvalue of the matrix in Eq. <ref>. 

For the choice of C=γλ_min, where γ∈ ( 0,1), one can push the lower bound for the choice of given α. This implies that fewer measurements are required to invert matrices with large condition numbers, as compared to the traditional HHL technique illustrated in Fig. <ref>(a). We show this in Fig. <ref>, where for a given α=60^∘, we obtain 


    |[(√(1- γ^2/κ^2)) sin(α) + γ/κ cos(α) ]  |^2   ≤    P'(1) 
       ≤   |[(√(1- γ^2)) sin(α) 
       +   γcos(α) ]  |^2 .


While the A  and B matrices share the same basis {|ν_i⟩}, an eigenstate of B matrix that achieves the lower bound for P'(1) may not often correspond to the largest/smallest eigenvalue  of A matrix, for a given choice of angle α. 

Note that for an arbitrary input |b⟩ which is a nontrivial superposition of all the eigenvectors {|ν_i⟩} of A, we have the following condition,

    P'(1)   =   (∑_i | b_i (√(1-C^2/λ_i^2)sin(α) +  C/λ_icos(α)) |^2) 
       =   (∑_i | b_i|^2λ_i'^2).
 

Here, {λ_i'} are the eigenvalues of the operator √(𝕀-C^2A^-2) sin(α) + CA^-1cos(α). Eq. <ref> is a convex combination of the square of eigenvalues, λ_i'^2, therefore, an arbitrary input |b⟩ will yield a P'(1) satisfying the condition given in Eq. <ref>. We augment our findings with the simulations in the subsequent Sec. <ref>. 
 


§ RESULTS AND DISCUSSIONS
 

In this section, we consider two sets of examples to compare the performance of Psi-HHL with that of HHL: 



  * Toy A matrices of sizes 4 × 4, 8 × 8, 16 × 16, 32 × 32, and 64 × 64: For practical reasons associated with computational resources, we do not choose matrices whose κ grows with the A matrix size. Instead, we generate a sequence of A matrices with systematically increasing condition numbers for every system size. For example, for 4 × 4 matrix size, we begin with a sparse and diagonally dominant matrix (the latter is not a requirement but simplifies matters for our illustrations) such that its smallest diagonal entry is 2^-(n_r-1) and the largest 1. Recall that n_r is the number of QPE clock register qubits. We generate the sequence by varying n_r from 2 to 20. In such a sequence, κ grows exponentially with n_r, thus enabling us to reach the failure point of HHL rather rapidly and with existing computational resources. We also note that changing the position of the minimum diagonal element of A along the diagonal does not affect our analysis. 

  * Quantum chemistry: Since quantum chemistry is one of the main applications of interest with quantum computers, it is worth exploring this domain for our analysis <cit.>. However, since we no longer are considering toy examples, we begin by qualitatively assessing the applicability of HHL to the LCC problem by: 
 

  * Checking how the A matrices grow with system size (and along the way, defining the system size), thereby providing a handle on the classical hardness of the problem, and 

  * Checking how κ itself grows as a function of what we set to be the system size. Only after this qualitative analysis do we move to our main results for this problem domain. 
 

  * We carry out the following analyses: 


  * For the toy matrices, we compare HHL and Psi-HHL, and demonstrate the ability of the latter in predicting correlation energies across a range of condition number values. As we shall explain later in this section, we find that since κ grows slowly for the quantum chemistry case, it is not computationally feasible to carry out the analysis. We quantify the quality of our results for the correlation energies through percentage fraction difference, abbreviated as PFD, given by E_c-E_c^HHL/E_c× 100 for HHL and E_c-E_c^Psi-HHL/E_c× 100 for Psi-HHL. We denote the correlation energy obtained on a traditional computer by E_c, whereas E_c^HHL and E_c^Psi-HHL are the values for the quantity obtained from HHL and Psi-HHL respectively. 

  * For both the toy matrices as well as the quantum chemistry matrices, we study the PFDs of HHL and Psi-HHL approaches versus the number of shots. 

 

 



 §.§ Toy matrices: 4 × 4 through 64 × 64
 

We begin by discussing the details of our A matrices and b⃗ vectors. We pick our 4 × 4 matrix to be non-diagonal, and is given by  


    A=[  2^-(n_r-1)        0.00        0.00        0.00;        0.00        0.75 0.10× 10^-3        0.00;        0.00  0.10×10^-3        0.50        0.00;        0.00        0.00        0.00        1.00 ],

and b⃗=[ 0.10; 0.01; 0.20; 1.00 ]. The sparsity, s, of A is defined as the number of non-zero entries in the row with the most number of non-zero elements and is set to 2. Our choices for the other matrices, 8 × 8 through 64 × 64 can be found in Figure <ref> of the Appendix. We have set the following values for sparsity: 3 for 8 × 8, 4 for 16 × 16, 17 for 32× 32, and 7 for 64 × 64 matrices. Throughout, we have ensured that the eigenvalues of the A matrix lies between 1/κ and 1. We provide more examples for the 4 × 4 matrix size in Section <ref> of the Appendix. 
 

Figures <ref>(a) through <ref>(e) provides our findings for 4 × 4 through the 64 × 64 examples considered. Each data point in the sub-figures is the average result obtained over 10 repetitions, with each repetition involving 1 million shots. We pick α = 60^∘ for the Psi-HHL calculations. The corresponding data is given in Table <ref> of the Appendix. Our results obtained using other choices of angles, 70^∘ and 80^∘, are provided in Tables <ref> and <ref> respectively. We now return to Figure <ref>, where each sub-figure provides two sets of data: one which compares the probabilities, P(0), P(1), P'(0), and P'(1), obtained by executing HHL and Psi-HHL, and the other providing the PFDs for the two methods. We choose condition numbers all the way till about 1 million. Figures <ref> and <ref> of the Appendix shows these results for α of 60^∘, 70^∘, and 80^∘, with 10^5 and 10^6 shots respectively. The figures show that our data fits for HHL align with the expected behaviour of P(1) ∼𝒪(κ^-2). We also find that P'(1) scales as ∼𝒪(1) for large condition numbers (with the fit parameter a=0.75 matching with the Y-axis intercept for large condition number values). For completeness, we verify that the fit probabilities add to one (see Table <ref> of the Appendix). 
 

 

We now comment on the PFDs. We immediately see that HHL fails at some value of κ for the chosen number of shots (indicated by `nan' values in Tables <ref>, <ref>, and <ref> of the Appendix). This is due to the fact that as P(1) becomes very small, the fraction of the total number of shots that yield 1 upon measurement tend to very small values. Therefore, the number of shots available for the HOM module become too small to give any reasonable result beyond a certain value of κ. The bottom panel of each sub-figure of Figure <ref> shows that the error bars, in which the upper and lower points of the bars reflect the largest and the smallest PFD values obtained using HHL, increases as the condition number increases, till it reaches a critical value of κ, beyond which HHL fails. In contrast, we see that for Psi-HHL, even at κ ≈ 1 million, the length of the error bar is about 17 percent. 
 

We now verify that HHL does not fail upon increasing the number of shots, but the 
number requires to be about 1 billion in order to get a comparable range of PFDs as Psi-HHL, while Psi-HHL only incurs 10 million shots, as Figure <ref> shows. In particular, HHL gives a maximum PFD of 1.51 percent and a minimum of -2.98 percent with a billion shots, whereas Psi-HHL gives a maximum and minimum PFD of 2.33 and -2.97 percent respectively with only 10 million shots. 
 

We finally comment on our results from Figure <ref> on the larger size matrices (8 × 8 through 64 × 64), where for Psi-HHL, we consider α=60^∘. Figures <ref> through <ref> provides the data for 70^∘ and 80^∘ cases, with Figure <ref> also giving the information for 10^5 shots. We observe that broadly, the trends are similar to what we find for the smaller 4 × 4 cases. Figure <ref> of the Appendix summarizes the situations where HHL fails for all of the toy matrices considered, as a heat map. We find that for 1 million shots, HHL starts returning `nan' values between κ of 2^8 and 2^10. In view of the computational cost involved, we did not go all the way up to a condition number of 2^20 for matrices above 4 × 4 size, and we stopped much earlier for the 64 × 64 case. 
 



 §.§ Quantum chemistry matrices
 

Quantum chemistry is regarded as one of the killer applications of quantum computing <cit.>. Therefore, it is interesting to consider the performance of Psi-HHL in this scenario. In particular, we consider quantum chemistry problems via the LCC approach <cit.>, for this purpose. Quantum chemistry on quantum computers can involve different approaches to selecting systems of interest depending on the application domain, including but not limited to performing calculations on numerous candidate molecules of progressively increasing size, picking one molecule and systematically improving the precision in one's calculations via suitable active space choices (as well as better quality single particle bases, incorporating more physical effects in the molecular non-relativistic Hamiltonian, etc), etc. For the purposes of this work, since we only want to compare performance of the HHL and Psi-HHL approaches, we opt for the substantially simpler second approach. 
 

 



  §.§.§ The A matrix dimension


Since we are now going beyond toy examples, we begin with some preliminary qualitative remarks. We begin with the growth of the LCC A_mol matrix. We make a distinction between the dimension of the LCC matrix, A_mol, which is m × m, and the dimension of the associated A matrix, n × n, where we recall that n=2^N. We go from A_mol to A via padding, which we explain in Section <ref> of the Appendix. 

 

  * The number of rows (or columns), m, of the A_mol matrix, which we shall hereafter think of as the system size, is the number of particle-hole excitations that one considers for a calculation, which in turn is decided by choice of single particle basis and active space size. We denote the number of occupied orbitals as n_o and virtuals by n_v. It is common to employ the LCC approach by only considering single and double excitations (termed LCCSD), in which case A_mol matrix grows as m ∼ n_o^2n_v^2 for fixed values of n_o and n_v. As we include more excitations systematically, such as triples, quadruples, etc, while keeping the number of orbitals fixed, we observe that the A_mol matrix grows as ∼ n_o^ℰn_v^ℰ, which is exponential. ℰ refers to the excitation rank. 

  * For a given excitation rank, ℰ, as we increase the number of orbitals, the system size, m, grows further polynomially, for example, see Figure <ref> in the Appendix. For the purposes of this work, we restrict ourselves to LCCSD approximation (ℰ=2), and study the size of A_mol with increase in number of unoccupied orbitals, n_v, in a molecule, while keeping n_o fixed at a reasonable number. 
 

For our analyses, we do not attempt to consider a molecule in a given active space size to study the variation of m with choice of single particle basis, and defer the study for a future work. We choose three examples: the light Li_2, the moderately heavy RbH, and the heavy CsH neutral diatomic molecules, all in their equilibrium bond lengths (Li_2: 5.06 Bohr, RbH: 4.4726 Bohr, and CsH: 4.7135 Bohr)  <cit.>. The adjective `light' and `heavy' refer to the atomic number of the heavier atom in a diatomic molecule. We pick the Sapporo double zeta basis sets for RbH and CsH (Rb: Sapporo-DZP, Cs: Sapporo-DKH3-DZP-2012, and H: Sapporo-DZP) <cit.>, while for Li_2, we choose the STO-6G basis <cit.>. We employ the C_2v point group symmetry for RbH and CsH computations, while for the case of Li_2, we pick the D_2h symmetry. We fix n_o to 3, 13 , and 18 for Li_2, RbH, and CsH respectively. The technical details involved in the generation of A matrix as well as the b⃗ vector are discussed in Section [SM: chemistry-matrices-A]A2.A of the Appendix. 
 



  §.§.§ Condition number of the A matrix
 

Although HHL offers an exponential advantage in terms of system size, it is worth checking if for the LCCSD problem, the condition number itself scales favourably with system size, m, for the systems that we considered. We point out towards the end of Section [SM: chemistry-matrices-A]A2.A of the Appendix that it is sufficient to analyze κ versus m; this curve upper bounds κ versus n. We increase the active space, and thus m, in a systematic manner by carrying out an orbital character analysis, whose details we explain in Section [SM: chemistry-matrices-B]A2.B of the Appendix. Furthermore, for this analysis, we only consider RbH and CsH, and not Li_2 as the system is too small (in the STO-6G basis) to carry out condition number versus system size analysis. For RbH, we consider n_o=13 for our calculations, and vary the number of virtuals from 1 to 21, which corresponds to A_mol matrix size of 40 × 40 to 9065 × 9065 respectively. For CsH, n_o=18 and virtuals are varied from 1 to 16, while the A_mol matrix size goes from 67 × 67 to 11058 × 11058. 
 





Our results shown in Figure <ref> of the Appendix indicate that for RbH as well as for CsH, as we go from m of 40 to 9065 and 67 to 11059 respectively, κ grows polylogarithmically (fourth and sixth order respectively). This behaviour is also consistent with what one may expect from chemistry intuition: we begin by referring the reader to Table <ref> of the Appendix, where we list the largest and smallest eigenvalues of the A matrix for each value of m, for both RbH and CsH. The data shows that the increase in κ as we add virtual orbitals is driven by the decrease of the lowest eigenvalue of the A matrix. We note that every matrix element of the A matrix can be expressed as ⟨Φ_p|H|Φ_q ⟩, where H is the Hamiltonian and |Φ_p⟩ and |Φ_q⟩ are the excited determinants built from particle-hole excitations arising from the Hartree-Fock configuration. The smallest eigenvalue of the sparse and diagonally dominant A matrix, which we saw as being important in determining the growth of κ, 
can be qualitatively thought of as corresponding to the energy of the first excited state (assuming that the wave function of the first excited state has small contribution from the HF configuration). Thus, we expect that the decrease in the lowest eigenvalue of A saturates as we keep adding virtual orbitals, and eventually, the lowest eigenvalue of the A matrix will converge to a certain value. Thus, the increase of κ saturates too, thus leading one to expect that the condition number would vary as a polylogarithmic function of system size. We add at this juncture that while considering only two molecules does not allow any quantitative inference on the behaviour of condition number with system size, it gives a qualitative indication that quantum chemistry using HHL-LCCSD may a reasonable direction to consider. However, further analysis would be required to assess what one would expect for κ(m) in general for an HHL-LCC computation. We also note that we carried out a similar analysis for sparsity, and found that it too grows polylogarithmically for RbH and CsH (see Figure <ref> of the Appendix). 
 



  §.§.§ HHL and Psi-HHL results
 

We now discuss our results for the performance of the HHL and Psi-HHL algorithms for the molecules considered in this work.  
 

We begin by conducting a study similar to the one shown in Figure <ref> for the toy matrices. It is not practical to carry out an analysis similar to that in Figure <ref>, since unlike the toy matrices case, the condition number grows much slower and the computations become too expensive too soon. For the purposes of our analysis in Figure <ref>, we vary the number of shots and compare the performance of HHL and Psi-HHL in predicting the PFD, as well as the range across which the PFDs are spread, across 5 repetitions. For this analysis, we consider Li_2, RbH and CsH. The details are: 

 

  * Li_2: 21-qubit circuit with an A matrix size of 64 × 64 (built out of single and double excitations arising from 3 occupied and 8 virtual orbitals) in the STO-6G basis, with κ=64.07. 

  * RbH: 25-qubit circuit with an A matrix size of 256 × 256 (built out of single and double excitations arising from 13 occupied and 3 virtual orbitals) in the Sapporo double zeta basis, with κ=153.17. 

  * CsH: 26-qubit calculations with an A matrix size of 256 × 256 (built out of single and double excitations arising from 18 occupied and 2 virtual orbitals) in the Sapporo double zeta basis, with κ=466.59. Although RbH and CsH have the same A matrix size, the latter involves a 26-qubit calculation because we set an additional ancilla for CsH due to its larger atomic number. 


We note that for RbH, the same calculation would have incurred more qubits for VQE-UCCSD (32-qubit circuit) and QPE-CASCI (with n_r=8, 40-qubit circuit) computations, while we carry out a 25-qubit circuit evaluation in the case of HHL-LCC. Similarly, for CsH, VQE-UCCSD: 40 qubits, QPE-CASCI: 48 qubits, and HHL-LCC: 26 qubits. 
 

Our results are presented in Figure <ref>, with the accompanying data given in Table <ref> of the Appendix. For Li_2, the condition number is very small, and hence we only study the performance of HHL and Psi-HHL between 500 and 10000 shots. The data indicates that for Li_2, the Psi-HHL approach gives better average PFDs, especially in the regime of fewer number of shots (500 and 1000 shots). In the same regime, the spread in PFDs is also lesser for Psi-HHL. For the cases of RbH and CsH, as we increase the number of shots, the error bars for both HHL and Psi-HHL are seen to become much smaller. 
 


We now move to our next set of results. From a practical point of view, it is computationally not feasible to pick the larger system sizes from Figure <ref>. For potential energy curves using both HHL and Psi-HHL approaches, we consider: 



  * Li_2: 21-qubit circuit with an A matrix size of 64 × 64 (3 occupied and 7 virtual orbitals) in the STO-6G basis. 

  * RbH: 21-qubit circuit with an A matrix size of 64 × 64 (4 occupied and 4 virtual orbitals) in the Sapporo double zeta bases. 

  * CsH: 26-qubit calculations with an A matrix size of 256 × 256 (built out of single and double excitations arising from 18 occupied and 2 virtual orbitals) in the Sapporo double zeta bases. 
 

Furthermore, for each molecule, we also consider geometries in the neighbourhood of the equilibrium bond length (±1 Bohr in steps of 0.25 Bohr). With such a choice, the correlation energies required to be captured ranges from ∼ 10s of mHa (milli-Hartree) for Li_2 to RbH's ∼ 0.1-1 mHa to CsH's ∼ 10s of μHa, thus allowing us to check the performance of HHL and Psi-HHL across a range of correlation energy values. Calculating energies across many bond lengths further allows us to check across a (limited) range of condition numbers the behaviour of HHL and HHL. We note that the range of condition numbers corresponding to the considered geometries range from about 59 to 121 for Li_2, 14 to 19 for RbH, and 455 to 505 for CsH. Each calculation involves a million shots and our reported results are averaged over 5 repetitions. Figure <ref> shows the results when we plot Δ E, which is the difference between HHL (Psi-HHL) and the LCC correlation energy evaluated on a traditional computer, against the geometry. We find that Psi-HHL mostly performs better than HHL in predicting correlation energies. However, ... . This is not immediately obvious for the case of CsH, since the differences occur typically at μHa level. 
 




§ CONCLUSION
 

We propose the Psi-HHL algorithm to efficiently handle problems involving A matrices with large condition numbers. The approach involves post-selecting 0 in HHL (the `wrong signal'), followed by post-selecting 1 from an HHL-like circuit that contains an additional mixing 2 × 2 unitary (the `mixed signal'), and taking the difference between the wrong and the mixed signals to extract the correct signal. 
  

We present arguments to show that the complexity of Psi-HHL is optimal in condition number for large κ regimes; the complexity of the algorithm scales as κ. This is an improvement over earlier approaches in literature, such as HHL with amplitude amplification (scales quadratically, and incurs several additional gates due to its iterative nature) and HHL with variable time amplitude amplification (scales as κlog^3κ). It is also important to note that while the optimal κ behaviour can be recovered using a discrete quantum adiabatic approach based quantum linear solver algorithm <cit.>, we achieve the same result using Psi-HHL, which is a simple modification to the HHL algorithm. 
 

We apply our method to different toy matrices, where we systematically vary the condition number all the way to about 1 million while keeping the number of shots fixed to 1 million, and show that while HHL fails beyond a certain value, Psi-HHL continues to predict good results. We also demonstrate the performance of our proposed Psi-HHL approach for quantum chemistry examples, where we once again find that Psi-HHL yields better quality results in much fewer number of shots. En route, we also carry out a qualitative analysis to check suitability of treating LCC equations using the HHL algorithm. 
 

We anticipate that our work will pave the way for development of non-trivial Psi-HHL variants that would allow it to be integrated as an algorithmic primitive in larger algorithmic frameworks that use traditional HHL as a subroutine. 
 


We acknowledge National Supercomputing Mission (NSM) for providing computing resources of `PARAM Siddhi-AI’, under National PARAM Supercomputing Facility (NPSF), C-DAC, Pune and supported by the Ministry of Electronics and Information Technology (MeitY) and Department of Science and Technology (DST), Government of India. VSP thanks Prof. Aram Harrow for a very fruitful discussion on HHL as well as HHL-LCC. VSP and AJ thank Dr. Shibdas Roy for nice discussions on complexity. VSP and PBT thank Prof. Debashis Mukherjee for useful discussions on the quantum chemical sections of the work. KS acknowledges support from Quantum Leap Fragship Program (Grant No. JPMXS0120319794) from Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan; Center of Innovations for Sustainable Quantum AI (JPMJPF2221) from Japan Science and Technology Agency (JST), Japan; and Grants-in-Aid for Scientific Research C (21K03407) and for Transformative Research Area B (23H03819) from Japan Society for the Promotion of Sciences (JSPS), Japan. 










§ APPENDIX







 
 
 




§ DETAILS ON CALCULATIONS INVOLVING TOY MATRICES FROM SIZE 4 × 4 THROUGH 64 × 64
  

 

In our Results section in the main text, we discussed the case of A not diagonal and the amplitude encoded state corresponding to b⃗ in an unequal superposition. We also provide our 4 × 4 through 64 × 64 matrices that we considered in the main text in Figure <ref>. 

We now provide additional data on two more choices for our 4 × 4 A matrices: 



  * A is diagonal and the amplitude encoded state corresponding to b⃗ is in an equal superposition. In particular, 	A=[ 2^-(n_r-1)       0.00       0.00       0.00;       0.00       0.75       0.00       0.00;       0.00       0.00       0.50       0.00;       0.00       0.00       0.00       1.00 ], b⃗=[ 1.00; 1.00; 1.00; 1.00 ]. The sparsity, s, of A is set to 1. 

  * A is diagonal and the amplitude encoded state associated with b⃗ is in an unequal superposition. In particular, A is the same as in case 1 given above, and b⃗=[ 0.10; 0.01; 0.20; 1.00 ]. 
 

We also present these results for α set to 60, 70, and 80 degrees (the main text only discussed the first case). Furthermore, while the main text considers 1 million shots, we also perform the same analyses for 10^5 shots. All of these results are presented in Figures <ref> and <ref> for the cases considering 10^5 and 10^6 shots respectively. The accompanying data is provided in Tables <ref>, <ref>, and <ref>, for 60, 70, and 80 degrees examples respectively. 









 



 

 

 

 

 













§ DETAILS ON CALCULATIONS INVOLVING QUANTUM CHEMISTRY MATRICES
 



 §.§ A matrix and b⃗ vector generation
 

We begin by explaining the procedure for generating the A matrix of size n. We use the configuration state function-based Graphical Unitary Group Approach-Configuration Interaction Singles and Doubles (GUGA-CISD) method available in the GAMESS program <cit.> to construct the Hamiltonian for a given molecule in a specified single particle basis. GUGA-CI ensures that the Hamiltonian is constructed in an orthogonal basis; hence, it is Hermitian. Also, because it is a spin-adapted CI method, it avoids spin contamination of the wave function, thereby giving accurate results. We remove the first row and the first column of the GUGA-CISD Hamiltonian matrix and subtract the diagonal entries of the resulting matrix with the energy obtained from Møller-Plesset theory to second order in perturbation (also called MP2 in literature). The latter step is carried out for the purpose of normal ordering, which in turn ensures that HHL would output the correlation part of the ground state energy directly instead of the total ground state energy (which can be written as a sum of the Hartree-Fock (HF) energy and the correlation energy, with the latter being much lesser than the former). This is desirable when one executes HHL-LCC calculations on NISQ era quantum hardware, as the error in the total ground state energy can be much greater than the correlation energy itself. We also note that unlike the traditional approach where the HF energy is subtracted from the diagonal entries, we take away the MP2 energy, in order to make sure that A is positive definite. In a sense, subtracting MP2 energy could be thought of as a level shift over and beyond subtracting the HF energy. 
 

We now focus our attention on the elements of the A matrix. The size of an A matrix, m, generated from a quantum chemical calculation, which we can denote as  A_mol, is not necessarily of the dimension n × n. In such cases, we embed the A_mol matrix of dimension m× m as a diagonal block in a larger matrix of dimension n × n, with the other diagonal block being an identity matrix of size (n-m) × (n-m), denoted as ℐ. In such a case, we solve a system of linear equations given by 


    A = [ A_mol     0;     0     ℐ ][ b⃗;  0 ] = [ x⃗;  0 ].


This approach ensures that the eigenvalues of A_mol remain unaffected, as long as its maximum eigenvalue is ≈ 1. We choose our matrices carefully so that this condition is satisfied. In the event that the maximum diagonal entry of A is less than 1, it is recommended to pad with the largest (or the smallest) diagonal entry of A so that the number of shots expended is reduced. As an example, we picked a 3 × 3 sparse matrix such that the largest diagonal entry is 0.70: [ 0.25 0.10 0.00; 0.10 0.45 0.20; 0.00 0.20 0.70 ], and considered two cases: padding with 1 and with 0.70. We found that kappa is larger in the latter (5.16) than in the former (4.21). 
 

Since for the problems we consider, κ increases with system size monotonically, and further, going from an unpadded to a padded matrix always right shifts a data point, it is sufficient to carry out κ versus m analysis for the unpadded case, as it upper bounds the complexity of the padded scenario of κ versus n. 

  



 §.§ Orbital character analysis
  

We now explain the procedure that we adopted for selecting orbitals, based on orbital character. We begin with RbH, and then proceed to explain our procedure for CsH (with accompanying data provided in Table <ref>). 
 

We fix the number of core orbitals to 13. We now describe the method adopted in choosing virtual molecular orbitals (MOs). Instead of taking the approach of ordering the virtual MOs by energy in ascending order and then imposing an energy cut-off to define our active space, we pick those MOs that are dominated by important atomic orbitals (AOs). We select the important AOs, and for the purposes of this work, we pick those AOs in the order of the principal quantum numbers. For example, in the case of RbH, we can decide the important AOs from the set that arise from our choice of basis, namely the Sapporo double zeta basis: 5s through 7p. For the purposes of our analysis on condition number versus system size, we pick all the AOs that the basis allows in order to obtain more data points for fitting (details given in Table <ref>). We now move to some specifics for RbH: for the smallest system size considered (1 virtual MO), we pick the first virtual from our sorted MO list, which is the MO dominated by the Rb 5s-5p_z hybrid, and thus our active space that is built out of 13 occupied MOs and 1 virtual one has a system size, m, of 40 × 40, and subsequently upon padding, an A matrix size, n, of 64 × 64. For the next system size, we add the next MO in the sorted list, which is dominated by Rb 5p_y, leading to a 119 × 119 matrix, which, after padding, becomes a matrix of size 128 × 128. We continue to construct A matrices in this fashion until we reach m=9065, for which κ is about 348. 
 

For the case of CsH molecule, we fix the number of occupied orbitals to 18 while progressively increasing the number of valence orbitals. The order in which we add virtual MOs as well as each such virtual MO's details are mentioned in Table <ref>. ]

 Paper Title: Decomposition of Nonlinear Collision Operator in Quantum Lattice Boltzmann Algorithm.


Spatio-temporal patterns in Growing Bacterial Suspensions: Impact of Growth dynamics
    Shradha Mishra^1
    August 5, 2024
====================================================================================






§ INTRODUCTION



The potential for simulating fluid flows on quantum computers has generated significant interest due to the unique properties of qubits, such as superposition and entanglement. By encoding flow variables onto the probability amplitudes of qubit states, we can achieve significantly reduced memory requirements and a potentially exponential increase in computing speed. However, quantum operations are inherently linear and unitary, while the equations of fluid mechanics exhibit nonlinearity. As a result,  existing computational fluid dynamics (CFD) algorithms need to be redesigned and adapted to fully utilize the power of quantum processor units (QPUs).
Traditional CFD methods solve the discretized Navier-Stokes equation (NSE), and commonly used algorithms are implicit in time-stepping, which requires linear system solvers (LSS)
<cit.>. The Lattice Boltzmann Method (LBM) has been considered an alternate approach to NSE-based CFD solvers and has been studied extensively for various flow problems. Due to locality in space and explicit time-stepping, LBM does not require LSS and has proven to be efficient on graphic processor units (GPUs) <cit.>.
Compared to NSE, where the nonlinear term (𝐮·∇𝐮) is also non-local, the LB equation has nonlinearity (𝐮·𝐮) in the equilibrium function and is local. This nonlinearity does not pose any difficulty in classical computers, as  𝐮 will be saved as a floating point number followed by a square operation <cit.>. However, if we mimic the classical procedure on QPUs, the quantum advantage may be lost, especially when amplitude encoding is chosen. 

There are several existing works on Quantum Lattice Boltzmann (QLB) for fluid flow simulation, but most of them ignore nonlinearity and are only used for very low Reynolds number cases <cit.>. In the absence of nonlinearity, the particle distribution functions (PDF) f can be encoded directly as probability amplitudes of qubits. 
However, evaluating the nonlinear term involves the vector f⊗f. <cit.> highlighted that it is impossible to have a linear operator that performs the squares of probability amplitudes, as the resulting state vector may not be a unit vector. Furthermore, the no-cloning theorem states that it is impossible to copy an arbitrary quantum state to create an identical quantum state <cit.>.
To address this, the Carleman linearization (CL) technique can be utilized which converts the finite-dimensional nonlinear system into a infinite-dimensional linear system which is then truncated for implementation <cit.>. Previously, CL has been successfully applied to LBM up to second-order truncation for low to moderate Reynolds number cases. When CL is applied to LBM, the resultant state vector becomes infinite, i.e., (f, f⊗f, f⊗f⊗f, …). 
In addition, another approximation has to be made for the term 1/ρ≈ 2-ρ, where ρ is the density, which is valid for weakly incompressible regimes. Even after successive simplifications, the Carleman LB (CLB) algorithm still requires repeated encoding and read-out of state vectors for each time step <cit.>.
In the present work, instead of the CL algorithm, we propose the decomposition of the LB operator as a product of multiple operators derived from the PDFs from the previous time step. We retain the first-order Taylor approximation for 1/ρ and accept that the encoding and read-out process is required for each time step <cit.>. Our approach does not involve any truncation of infinite sets. Instead, the state vector is a finite set, specifically, four sets of f appended sequentially. 
In the following, we briefly describe the LB equations, followed by deriving the quadratic formulation of the LB collision operation. Later, we will present the quantum algorithm and the computational complexity of two-qubit gates.  




§ LATTICE BOLTZMANN METHOD



LBM describes the evolution of flow field through PDFs in a uniform Cartesian grid with the chosen lattice model , where m and n denotes the number of spatial dimensions and velocity directions, respectively. The single relaxation time LB model for the flow field is given by

    f_i(𝐱 + 𝐞_iΔ t, t+Δ t)  
    	= 
    	f_i (𝐱,t) - Δ t/τ[ 
    	f_i(𝐱,t) - f_i^eq(𝐱,t)
    	]

where f_i represents the PDF along the i^th direction, 
𝐞_i is the lattice velocity, and 
the kinematic viscosity ν is related to the relaxation parameter  
τ = 3ν + 0.5. The equilibrium distribution function (EDF) is given by

    f_i^eq = w_iρ[ 1+𝐞_i·𝐮/c_s^2 + (𝐞_i·𝐮)^2/2c_s^4 - 𝐮·𝐮/2c_s^2]

where w_i is the lattice weight, c_s the sound speed, ρ the fluid density, and 𝐮 is the flow velocity. 
The zeroth and first moment of PDFs yields the macroscopic flow variables density(ρ) and momentum (ρ𝐮),

    ρ(𝐱, t)    = ∑_i f_i(𝐱, t)
    		
    ρ(𝐱, t)𝐮(𝐱, t)    = ∑_i  𝐞_i f_i(𝐱, t)

Typically, LB algorithm is split into collision and streaming steps. In the collision step, 
the interaction of particles and their relaxation to equilibrium state is modelled. 
Later the particles move to their neighbouring site according to the lattice velocity, a process termed streaming.
Thus, Eq. (<ref>) can be written as 

    f_i^*(𝐱, t)    = 
    	f_i (𝐱,t) - Δ t/τ[ f_i(𝐱,t) - f_i^eq(𝐱,t) ]
    	
    
    f_i(𝐱, t+Δ t)     = f_i^*(𝐱 - 𝐞_iΔ t, t)

where f_i^* is the post-collision state. 
In the presence of a wall boundary, we use half-way bounce back scheme to reconstruct PDFs coming from the wall node (f_-i), 

    f_-i(𝐱_b, t+Δ t)     =  f^*_i(𝐱, t)





§ FORMULATION





 §.§ Quadratic form for collision step




First, we rewrite Eq. (<ref>) and 
using Eqs. (<ref>) and (<ref>), we get the quadratic form for EDF as,

    f_i^eq   = 1/ρ w_i[ ρ^2 + ρ𝐞_i·ρ𝐮/c_s^2 + (𝐞_i·ρ𝐮)^2/2c_s^4 - ρ𝐮·ρ𝐮/2c_s^2] 
       = 1/ρ∑_j ( ∑_k 
     j ≤ kα_ijk f_k  ) f_j

where the coefficient α_ijk is given by








    α_ijk = w_i(1/2)^δ_jk[ 2 + 1/c_s^2( 𝐞_i·𝐞_j  + 𝐞_i·𝐞_k  - 𝐞_j·𝐞_k  )
         .
    .
      +  1/c_s^4 (𝐞_i·𝐞_j)(𝐞_i·𝐞_k) ]

and δ_jk denotes the Kronecker delta function. Next, the expression for collision in Eq. (<ref>) can be rewritten as, 

    f_i^*    = 1/ρ( 1- Δ t/τ) ρ f_i + Δ t/τ f_i^eq

Using Eqs. (<ref>) and (<ref>), the quadratic form of the collision step can be written as 


    f_i^*    = 1/ρ[   f_1;   f_2;     ⋮; f_n_𝐞 ]^𝐓[     β_i11     β_i12         …   β_i1n_𝐞;               β_i22         …   β_i2n_𝐞;        2c         ⋱         ⋮;                               β_in_𝐞n_𝐞 ][   f_1;   f_2;     ⋮; f_n_𝐞 ]



where β_ijk = γ__ijk·( 1- Δ t/τ) + _jk·Δ t/τα_ijk and 𝐓 denotes the transpose. The indicator functions γ and  for  i^th lattice direction  is given by,

    γ__ijk   = δ_ij + δ_ik - δ_ij·δ_ik
    _jk   =
    	
    		1    if  j ≤ k 
    
    		0    otherwise





 §.§ Matrix formulation for weakly compressible LBM



In case of weakly compressible flows, ρ^-1≈ 2-ρ, we then express

    (2-ρ)    = (-1,-1,⋯,-1,2) · (f_1,f_2,⋯,f_n_𝐞,1)^𝐓
  
where the value 1 appended at the end of PDFs can be treated as a auxiliary constant.
Therefore, the collision step can be written as

    [ f^*_i;     1 ]   = 
    [ f; 1 ]^𝐓
    [ β_i   0;   0   1 ]
    [ 𝖣𝗂𝖺𝗀(f)       0;       0       1 ]𝐖
    [ f; 1 ]


where f=(f_1,f_2,⋯,f_n_𝐞), and 
β_i represent the upper triangular matrix in Eq. (<ref>).
The term 𝖣𝗂𝖺𝗀(f) represents the diagonal matrix with entries from f. 
The matrix 𝐖 is given by

    𝐖   =
    [ -1 -1  ⋯ -1  2;  ⋮  ⋮     ⋮  ⋮; -1 -1  ⋯ -1  2;  0  0  ⋯  0  1;    ]





§ QUANTUM ALGORITHM





 §.§ Encoding



Let n_g denote the total number of grid points and the number of PDFs will be n_f = n_𝐞 n_g.
According to the lattice directions, we arrange and order the PDFs in a vector form,
df = (f_1, f_2, …, f_n_𝐞,1), where the suffix refers to the direction, and each f_i is of length n_g. Since PDFs are encoded as a quantum state-vector, we define 
ϕ = (df, df, df, df).
A computational register containing 
n_q = log_2(n_f) qubits along with two ancilla qubits will be required to encode the PDFs. 
The initial state to encode will be 

    |ϕ^0⟩ = |0⟩_a |0⟩_a ∑_i=1^2^n_qϕ_i/‖ϕ_i ‖|i⟩_q












 §.§ Collision



Before we express matrix version of Eq. (<ref>), we define the following matrices:

    𝐖   =
    [     -𝐈^^g_11     -𝐈^^g_12            ⋯   -𝐈^^g_1n_𝐞       2^^g_1;     -𝐈^^g_21     -𝐈^^g_22            ⋯   -𝐈^^g_2n_𝐞       2^^g_2;            ⋮            ⋮                         ⋮            ⋮;   -𝐈^^g_n_𝐞1   -𝐈^^g_n_𝐞2            ⋯ -𝐈^^g_n_𝐞n_𝐞     2^^g_n_𝐞;            0            0            ⋯            0            1;              ]
    
    
    𝐃   = 𝖣𝗂𝖺𝗀(df)


    𝐁_i    =
    [     β̃_1i1     β̃_1i2          ⋯   β̃_1in_𝐞     0^^g_1;     β̃_2i1     β̃_2i2          ⋯   β̃_2in_𝐞     0^^g_2;          ⋮          ⋮                     ⋮          ⋮;   β̃_n_𝐞i1   β̃_n_𝐞i2          ⋯ β̃_n_𝐞in_𝐞   0^^g_n_𝐞;          0          0          ⋯          0          1 ]
    
    
    𝐃_i    = 𝖣𝗂𝖺𝗀(f_i)


    𝐅_i    =
    [     (𝐃_i)_11                                       ;                  (𝐃_i)_22                                                    ;                                      ⋱                          ;                                        (𝐃_i)_n_𝐞n_𝐞             ;                                                             δ_i1;              ]

where 𝐈^^g_rc denotes the identity matrix of size n_g × n_g, 
2^^g_i = (2, ⋯, 2) and 0^^g_i = (0, ⋯, 0) are the vectors of size n_g.
The term β̃_ric = _icβ_ric𝐈^^g_rc, where 
β_ric is defined in Eq. (<ref>), with r and c represent row and column 
indices, respectively. 








Based on the above set of matrices, we further define 

    𝐖   = 
        [ 𝐖      ;   𝐖    ;     𝐖  ;       𝐖;   ] ;
        
    𝐁_1    = 
        [ 𝐁_1            ;       𝐈        ;           𝐈    ;               𝐈;     ] ;
        
    𝐁_i   = 
        [   𝐈   0   0   0;       0 𝐁_i   0;           𝐈   0;               𝐈;     ] ;
    𝐃   = 
        [ 𝐃      ;   𝐃    ;     𝐃  ;       𝐃;   ]
    𝐅_1    = 
        [ 𝐅_1            ;       𝐈        ;           𝐈    ;               𝐈;     ]
    𝐅_i   = 
        [   𝐈 𝐅_i   0   0;       0   𝐈   0;           𝐈   0;               𝐈;     ]

Therefore, the collision operator is given by 

    |ϕ^*⟩   = 𝐅_n_𝐞𝐁_n_𝐞⋯𝐅_2𝐁_2𝐅_1𝐁_1𝐃𝐖|ϕ^0⟩

In Eq. (<ref>), when the SV is multiplied by 𝐖 yields the value 2-ρ, which then multiplied by 𝐃 results in the first order approximation of f_i/ρ, i.e., the column vector defined on the right side of 
Eq. (<ref>). Upon successive multiplication by 𝐅_i𝐁_i, yields the PDFs in the i^th direction. Thus, for n_𝐞 velocity directions, the collision operation is 
decomposed into 2n_𝐞+2 operations. It is important to note that the state vector has been initialized with four sets of df. Among these, one will ultimately lead to the final outcome, while another is designated for storing temporary values. The remaining two sets remain constant, providing the initial state vector values during intermediate computations. Consequently, the matrices defined in Eq. (<ref>) are not unique.  In particular, the following is a valid choice:


    𝐁_i   = 
        [   𝐈   0   0   0;       0   𝐈   0;     𝐁_i   0   0;               𝐈;     ]
    ;
    𝐅_i   = 
        [   𝐈   0 𝐅_i   0;       0   0   𝐈;           𝐈   0;               𝐈;     ]




 §.§ Streaming and Boundary Conditions


The streaming matrix S consists of n_𝐞× n_𝐞
blocks with the size of each block n_g × n_g. The matrix S is square binary matrix 
where each row/column contains one entry equal to 1. Since every row index i can be decomposed as 
x + y n_x + i_e n_x n_y, and the corresponding column index j such that s_ij=1, can be computed by replacing the co-ordinate 𝐱 by 𝐱 - 𝐞_iΔ t. In the present study, we utilized the 
wall boundary condition, accordingly the column index will be computed as 
x_Δ + y_Δ n_x + i_-e n_x n_y, where 
(x_Δ, y_Δ) = 𝐱 - 𝐞_i and i_-e is the reflection of the direction i_e. 
Based on the matrix S, we define the streaming operator as

    𝐒   = 
    	[ 𝐒      ;   𝐈    ;     𝐈  ;       𝐈;   ] ; where  𝐒   = 
    			[ S  ;   1;   ]

From Eq. (<ref>), it is clear that streaming is performed only on the computational qubits without any involvement from ancilla qubits.
Thus, the PDF for the next time step can be obtained from the quantum state,

    |ϕ^n⟩   =  𝐒|ϕ^*⟩





§ RESULTS AND DISCUSSION




 §.§ Verification - Case 1: 1D discontinuity flow

The classical shock tube discontinuity problem in one dimension is used to verify the present the algorithm. Because of the weak compressibility assumption, we refer it as a discontinuity rather than a shock. The one dimensional domain of size n_g=500 grid points and the D1Q3 lattice has been chosen. At the initial time, the velocity is set to zero everywhere and density is defined as, 

    ρ (x,0)    =
        
            1.0 + Δρ   if  x ≤n_g/2
    
            1.0    otherwise

where Δρ=5×10^-5 is chosen to satisfy the weak incompressibility assumption made earlier. 
Bounce-back boundary condition is applied at the both ends of the domain. After simulating
200 time steps, we obtained a good comparison with the exact Riemann solution for normalized pressure (p^* = (p-p_s)/p_s; p_s=Δρc_s^2) 
and velocity (u) <cit.> and is given in Fig. <ref>. The relative difference between exact and QLB results is to found to be 5%. 




 §.§ Case 2: 2D Kolmogorov flow

In order to compare the accuracy of present algorithm with that of Carleman linearized - LBM presented in 
<cit.>, we performed the two-dimensional simulation of Kolmogorov-like flow on a 32×32 grid. The D2Q9 lattice is
used.
Initially, the PDFs are defined as,

    f_i(x,y) = 
    w_i [ 
            1 + A_x cos( 2π k_x/N_y y ) 𝐞_i ·𝐞_x +
                 .
    .
                A_y cos( 2π k_y/N_x x ) 𝐞_i ·𝐞_y
        ]








where 𝐞_x=(1,0) and 𝐞_y=(0,1). The parameters in Eq. (<ref>) are taken as: A_x=0.3, A_y=0.2, k_x=1, and k_y=4.
We choose different viscosity values ν ranging from 1/6 to 0.0088, and for each ν we ran the simulation up to 100 time steps.  
After simulating 100 time steps, we compute the mean value of Root Mean Squared Error (RMSE) between the distribution functions obtained from the present algorithm (f^p_i) and the exact LBM simulation (f^e_i),

    <RSME> = ∑_i=1^n_𝐞1/n_𝐞√(∑_j=1^n_g1/n_g( f^p_i_j- f^e_i_j/f^e_i_j) )
 
Since CL transforms a finite set to an infinite dimensional set of equations, <cit.> 
analysed two different approaches, truncation (neglecting higher order terms) and 
closure (approximate the product of functions into products of a function and a constant). Fig. <ref> shows the comparison of RSME with the 
results of Carleman second order truncation and closure approaches presented in <cit.>. Since the present approach does not require any order of truncation
except the assumption on weak compressiblity, the <RSME> value is below 10^-5 compared to 10^-2 from Carleman closure approach. 






 §.§ Computational Complexity


In the present approach, the number of qubits required will be 2+log_2(n_f), whereas the Carleman truncated system of k^th order would require 1+log_2(n_f + n_f^2 + … + n_f^k) qubits. Fig. <ref>a shows the number of qubits required for the grid size ranging from 10^1 to 10^20 using D2Q9 lattice. For larger grid sizes, the CL- second (CL2) and third (CL3) order truncation approach requires almost twice and thrice the qubit resource than the present approach.  
The number of two-qubit gates is one of the ways to estimate the complexity of the present quantum algorithm. Since PDFs are encoded as amplitudes of the qubit states, the state preparation step is required before performing LB operations. According to <cit.>, O(2^n_q) CNOT gates are required for state preparation. 
Based on Eq. (<ref>) and (<ref>), the number of LB operators per time step is 2n_𝐞+3, and each operator would require O(2^n_q-1(2^n_q-1)) CNOT gates. In the second-order Carleman truncation, the qubit requirement will be n^c_q = 1+log_2(n_f + n_f^2). Correspondingly, the O(2^n^c_q-1(2^n^c_q-1)) CNOT gates will be required. Fig. <ref>b shows the number of CNOT gates required for different grid sizes. For smaller grids of size 10^3, the gate counts of CL2 are a few orders higher than the present approach; however, for the larger grid, it is exponentially larger.









§ SUMMARY

We have developed a quantum algorithm to deal with the non-linearity in the Lattice Boltzmann collision operator. By decomposing the collision operator, we significantly reduced the circuit width by half and circuit depth in exponential order when compared to the Carleman linearization technique. While there are still challenges to be addressed, such as the encoding and read-out process for each time step and the construction of quantum gates, this work aims to minimize the quantum resources required by the CL technique and prevent the formation of an infinite system. Due to the weakly compressible assumption and the first-order approximation for 1/ρ, this work is limited to moderate Reynolds numbers of O(100).



§ ACKNOWLEDGMENTS

This research was kindly supported by the Quantum Computing Consortium that has been funded by the MAGNET program of the Israel Innovation Authority (IIA).




eplbib





 Paper Title: On the use of calibration data in error-aware compilation techniques for NISQ devices.









On the use of calibration data in error-aware compilation techniques for NISQ  devices
    Handy Kurniawan


Universidad Complutense de Madrid

Madrid, Spain 

handykur@ucm.es
Laura Rodríguez-Soriano


Universitat Politècnica de València

Valencia, Spain 

lrodsor1@disca.upv.es
Daniele Cuomo


Universitat Politècnica de València

Valencia, Spain 

cuomo.daniele@outlook.com
Carmen G. Almudever


Universitat Politècnica de València
Valencia, Spain 

cargara2@disca.upv.es
Francisco García Herrero


Universidad Complutense de Madrid
Madrid, Spain 

francg18@ucm.es

    August 5, 2024
================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================



§ ABSTRACT 
Reliably executing quantum algorithms on noisy intermediate-scale quantum (NISQ) devices is challenging, as they are severely constrained and prone to errors. Efficient quantum circuit compilation techniques are therefore crucial for overcoming their limitations and dealing with their high error rates. These techniques consider the quantum hardware restrictions, such as the limited qubit connectivity, and perform some transformations to the original circuit that can be executed on a given quantum processor. Certain compilation methods use error information based on calibration data to further improve the success probability or the fidelity of the circuit to be run. However, it is uncertain to what extent incorporating calibration information in the compilation process can enhance the circuit performance. 
For instance, considering the most recent error data provided by vendors after calibrating the processor might not be functional enough as quantum systems are subject to drift, making the latest calibration data obsolete within minutes. 

In this paper, we explore how different usage of calibration data impacts the circuit fidelity, by using several compilation techniques and quantum processors (IBM Perth and Brisbane). To this aim, we implemented a framework that incorporates some of the state-of-the-art noise-aware and non-noise-aware compilation techniques and allows the user to perform fair comparisons under similar processor conditions. Our experiments yield valuable insights into the effects of noise-aware methodologies and the employment of calibration data. The main finding is that pre-processing historical calibration data can improve fidelity when real-time calibration data is not available due to factors such as cloud service latency and waiting queues between compilation and execution on the quantum backend.















Quantum compilation, error-aware circuit mapping, algorithm fidelity 




§ INTRODUCTION


Quantum systems are susceptible to many sources of noise that introduce errors and degrade the fidelity of the algorithms that are run <cit.>. These error sources permeate every stage of circuit execution. For example, in IBM superconducting processors <cit.>, qubit initialization is associated with error rates typically in the range of 10^-3 to 10^-2. In addition, quantum gates exhibit error rates ranging from 10^-4 to 10^-2 for single-qubit (1Q) gates and from 10^-3 to 10^-2 for two-qubit () gates. Furthermore, qubit information leaks over time because of decoherence, and the process of measuring qubits introduces noise, with error rates typically in the range of 10^-3 to 10^-2. Also, note that these error rates are not uniform across the processor, but they vary in space (i.e. variability between qubits and links) and time (i.e. between calibration cycles). Due to this error variability in the chip, the fidelity of the executed algorithm may differ depending on the selection of the physical qubits, the operations employed, and the degree of gate parallelism.

The error-proneness of current quantum processors, together with the limited qubit count, positions quantum computing in the Noisy Intermediate-Scale Quantum (NISQ) era, where quantum error correction (QEC) schemes cannot be applied. One promising approach to mitigate noise in NISQ devices is through the use of error-aware quantum circuit compilation techniques. In literature, we can find extensive research on integrating error data from quantum processor's calibration into the compilation process <cit.>. Some works focus on finding low error rates by using calibration data only for the initial qubit mapping of qubits <cit.>, whereas others consider error information for the initial qubit mapping as well as for the qubit routing <cit.>. Note that both processes may significantly contribute to diminishing errors during compilation by selecting the most connected and most reliable qubits and the shortest paths with the lowest error rates. 


  

However, it is uncertain to what extent incorporating calibration information in the qubit allocation and routing steps can enhance the circuit performance due to the error variability across the processor. More precisely, the potential improvement will depend on how calibration data is processed and included in the circuit compilation. Most of the infrastructures employed to access the quantum devices are usually based on execution queues within cloud services, which introduce non-negligible delays in the execution of the quantum circuits. For instance, in the case of IBM devices, the queue time —time elapsed since the circuit is sent until the execution starts— can be up to eight hours <cit.>. For this reason, the error information provided by the vendor may be outdated within minutes after calibration due to the quantum system drifting <cit.>. 

In this paper, we explore different methods to process the error information provided in several calibration cycles (historical data) to incorporate it in state-of-the-art noise-aware compilation techniques. The aim is to improve the overall fidelity of the executed circuits even when no real-time error data is available. To conduct the experiments, we developed a framework that allows the user to compare the performance of the subject compilation methods under similar conditions in real backends. The experiments were implemented on two IBM processors ( and ), showing that using historical calibration data is more beneficial than using the latest available calibration data whenever this is not provided in real time. Specifically, it was observed a circuit fidelity enhancement that increases with the size of the device. For instance, on  there was an average improvement of 0.08% (up to 4.09%) and 8.32% (up to 41.76%) on .









This paper is structured as follows. In Sec. <ref>, we provide the background on error-aware compilation techniques and device calibration information. The methodology used is detailed in Sec. <ref> along with the designed framework to execute the noise-aware compilation techniques under fair conditions for comparison. In Sec. <ref>, we derive some observations for the results of the conducted experiments. We conclude with Sec. <ref>, summarizing the main findings and outlining future directions.



§ BACKGROUND
 














 §.§ Quantum circuit compilation process
 

Quantum circuit compilation transforms a quantum circuit into an equivalent one that can be executed on a quantum device accordingly with hardware limitations. Fig. <ref> illustrates some steps of the compilation process with an example: let us consider the circuit depicted in Fig. <ref> (referred to as the original circuit) intended for execution on a device with the topology represented in Fig. <ref>; where physical qubits are represented by circles and denoted with Q_n (n corresponds to the qubit index), and the edges represent links to perform  gates. The initial qubit placement, also known as qubit allocation or assignment, involves mapping each virtual qubit in the circuit (q_n) to a physical qubit on the device (q_n  Q_n), as illustrated in Fig. <ref>. This allocation enables the execution of the first two CNOT gates as there is an edge connecting the respective qubits (Q_0 and Q_2, Q_1 and Q_3). However, to execute the third CNOT gate (between q_0 and q_3), the information of the involved qubits must be moved to adjacent ones. To do this, it is necessary to add SWAP[For two qubits A and B, SWAP(A,B) := {CNOT A,B; CNOT B,A; CNOT A,B;}.] gates. This is shown in Fig. <ref>, where a SWAP between q_2 and q_3 allows the execution of the third CNOT gate in the initial circuit. 

Other compilation steps not illustrated in the example are i) operation scheduling, to optimize the computation according to a specific metric (e.g. circuit depth), and ii) gate nativization, which decomposes the gates into the native set of gates the device supports. 







 §.§ Prior work
 

As can be seen from Fig. <ref>, the compilation steps usually result in an increase in the number of gates and circuit depth, which generally affects negatively the overall circuit fidelity. Therefore, it is important to use efficient initial qubit mapping and routing strategies that add the fewest SWAP gates; but this is an NP-hard problem <cit.>. For this reason, most of the proposed (and scalable) compilation techniques use heuristics for initial qubit mapping and/or routing <cit.>. For instance, with the reverse traversal method employed in <cit.>, the circuit is first executed with a random initial placement strategy. Then the final mapping from this execution is chosen as the initial qubit mapping to run the reversed circuit, and the final qubit mapping is used as the updated initial qubit mapping. An alternative approach for the initial qubit allocation involves using recursive graph-isomorphism search <cit.>. This method searches for a subgraph in the topology that is isomorphic to the qubit interaction graph. Regarding the routing of qubits, <cit.> and <cit.>, in addition to looking for the shortest path, they use a cost function with a look-ahead window to consider future gates instead of considering the entire circuit. This reduces the search space and allows to make informed decisions and add operations that have minimal impact on subsequent gates. The work in <cit.> analyzes the possibility of using bridge gates (which consist of four operations without swapping qubit information) instead of swap gates to lower gate overhead and consequently reduce noise.

Other quantum circuit compilation techniques have also been proposed to further enhance the reliability of quantum algorithms by taking into account the error characteristics of the quantum device. Some methods consider the measurement error of each qubit to perform the initial qubit mapping when virtual qubits can be placed in different areas of the chip that have the same connectivity, as described in <cit.>. Additionally, during qubit routing, the fidelity of each qubit coupling link (i.e. two-qubit gate fidelity) is taken into account, ensuring that quantum information is moved through the most reliable links <cit.>.

Some vendors provide the necessary error information to apply these compilation techniques after calibrating the quantum devices. However, as mentioned in Section I, this information is outdated shortly after calibration takes place due to the drift that processors suffer from. Obtaining real-time calibration data poses a challenge. The vendor provides data measured several times a day, e.g., twice a day according to <cit.>, and due to the existence of an execution queue, accurately predicting the time between compilation and the moment in which the algorithm will be executed on the real backend remains a limitation, highlighting how hard it is to align experimental conditions with real-time quantum processor operations. An alternative approach was proposed in <cit.> to obtain the latest calibration data through a micro-benchmark and use it to compile the circuit. However, in <cit.>, it is argued that micro-benchmarks are sub-optimal for quantum processors with more than one way of implementing  gates. In contrast, <cit.> presents a real-time framework to re-calibrate unused qubits while running a quantum circuit on the processors. But it is important to note that the current open-source cloud quantum machines do not support flexible and user-defined calibration, which limits the framework's usage.



 §.§ Calibration data
 

Noise-aware compilation techniques consider error rate information coming from calibration routines, which provide an error map of the quantum chip at a specific time. This serves as a comprehensive static snapshot of the quantum computer's state. However, the calibration data varies across qubits and in time, meaning there is a spatiotemporal variability in the fidelity of different operations such as 1Q and  gates or measurements <cit.>. Fig. <ref> displays an example of the reliability of performing a  gate between any pair of qubits on the  chip (see Fig. <ref>). The graph illustrates both the spatial (y-axis) and temporal (x-axis, 28 days) variability of the calibration data. For instance,  the fidelity of the  gate between qubits 1 and 3 remains consistently high over time, while the  gate between qubits 0 and 1 shows greater temporal variability, with some days exhibiting low fidelity (purple) and others showing a significant increase (yellow). Considering the spatial variability, it can be observed that a  gate between qubits 5 and 6 generally exhibits lower fidelity than a  gate between qubits 1 and 3. Aware of this variability, we investigate how to process and integrate calibration data within error-aware compilation techniques. To this end, error information has been gathered every time there was a new calibration cycle  for two different IBM quantum processors.









§ METHODOLOGY
 

We conduct experiments on various IBM devices using several compilation techniques, including noise-aware and non-noise-aware methods. We consider different approaches for processing calibration data we collected for several months, with daily acquisitions. As a result, each original circuit is compiled using different qubit mapping and routing strategies. These form a Qiskit job[A Qiskit job is formed by a collection of circuits that will be executed on the same backend when the user's turn comes up after waiting in the execution queue.] and submitted to the execution queue of the given backend. This protocol helps to work under similar conditions, as circuits of the same job are run consecutively. Finally, different state-of-the-art circuit reliability metrics are computed based on the obtained device measurements.






 §.§ Designed simulation and execution framework


To conduct the experiments, we developed a framework that aims to compare, under similar conditions, the existing noise-aware compilation methods described in Sec. <ref>. Previous research, conducted at different times and conditions, lacks cohesive benchmarks and metrics, making it difficult to perform fair and effective performance assessments of compilation. The framework seeks to unify comparisons by providing a consistent tool to evaluate various techniques under the same conditions. It consists of three main components, as shown in Fig.  <ref>: (i) the client application, (ii) the cloud server, and (iii) the vendor's backend. The user inputs an algorithm into the client, selects the quantum processor or noisy simulator, and chooses the compilation strategy to compile the algorithm from the corresponding QASM files. The cloud server submits a quantum job, consisting of all quantum circuits, to the chosen vendor's backend waiting queue. The results from execution are then processed and used to calculate the metrics defined in Sec. <ref>. Note that in this work, we focused on IBM's quantum hardware (see Sec. <ref>), but the methodology and outcomes may be ported to other vendors and/or compilation techniques and even to quantum error mitigation techniques <cit.>.





 §.§ Quantum circuit compilation techniques
 

Several quantum algorithm compilation techniques can be found in the scientific literature and industry (see Sec. <ref>). In this study, we focus on two noise-aware techniques: (i) TriQ <cit.>; and (ii) Noise-Adaptive (Q-NA) <cit.>, implemented in Qiskit. TriQ considers errors in both the initial qubit mapping and routing stages. Noise-Adaptive, on the other hand, only considers them in the initial qubit mapping stage. These are state-of-the-art noise-aware compilation techniques. Note that other methods like NASSC <cit.> faced some problems like incompatibility with newer versions of Qiskit, and therefore it was not included in this work. 

To further explore noise-aware routing algorithms, we upgraded TriQ – named TriQ+ – by changing the path-finding method. More precisely, instead of using Dijkstra's <cit.>, which seeks the shortest path from only one side (moving source to target qubit or vice versa), we use Floyd-Warshall's <cit.>, which considers paths from both source and target at the same time, finding the position of qubits within the shortest path that results in the highest reliability. 




The noise-aware approaches were also compared against non-noise-aware compilation techniques. Specifically, Qiskit with optimization level 3 combined with SABRE <cit.> (Q-3) <cit.>. At the time of writing, this is the default setting in Qiskit and the one that considers a bigger search space when making decisions about qubit routing. Note that there are other non-noise-aware compilation strategies, such as MIRAGE <cit.>, that we opted to not include in the paper due to obtaining lower fidelity than SABRE. A summary of the aforementioned quantum circuit compilation techniques is shown in Table <ref>. 



All experiments were conducted using Python 3.11.5 within a Docker system equipped with 13 GB of main memory, 8 CPUs, and Ubuntu 20.04. It follows the list of software specifics we used: Qiskit versions 0.44.3 <cit.>, qiskit-aer versions 0.12.3 <cit.>, qiskit-ibm-runtime versions 0.12.2 <cit.>, Mirage versions 0.2.0 <cit.>, and updated TriQ package <cit.>.



 §.§ Historical calibration data
 






Motivated by the fact that the latest calibration data is often outdated at the time of executing a job (i.e., the waiting time can be up to eight hours <cit.>), we explore alternatives to exploit the error information available. Specifically, we collected and analyzed calibration data from IBM backends, searching for a methodology to better exploit all this information to enhance the fidelity of circuits when real-time calibration data is unavailable.

Our proposal is to process historical calibration data and apply noise-aware compilation techniques accordingly. The data collection spans over several months, with a daily basis extraction. We classify the error information as follows: i) Latest (lcd) that considers the calibration data that is provided by the vendor at the moment of sending the quantum circuit to the backend waiting queue; ii) Average (avg) in which the mean of all the stored error rates is used; iii) Mix (mix) that combines the last calibration data with the average for those qubits that have not been calibrated the same day that the experiment is sent to the backend waiting queue; and iv) Window-n (w-n)  calculates the average fidelity of a window size of n days prior to the day the circuit is added to the queue. On top of this, taking into account temporal variability in the calibration data, we calculate the standard deviation (std) of the collected data and apply it to the above methods (listed in Table <ref>). This processing method is referred to as -adj in the result section. For instance, lcd-adj means the latest calibration data considering the standard deviation. This involves generating a new metric which is a linear combination of the error rate and the standard deviation derived from historical calibration data for each pair of gates and qubits High deviation can make the qubit less reliable, with a fluctuating error rate. Conversely, low deviation can provide a more stable and reliable qubit. The goal is to penalize qubit pairs with high deviation and select the more reliable one.





 §.§ Benchmarks
 





To assess the accuracy of our proposed calibration processing methods, we selected a set of quantum circuits with a high percentage of  gates, as summarized in Table <ref>. It is important to note that these circuits have been used as benchmarks in previous studies <cit.>. This study aims to analyze the impact of noise-aware compilation techniques on the overall performance of quantum algorithms under standardized conditions. 





 §.§ Quantum processors








IBM has several quantum chips based on superconducting technology that differ in qubit size, topology, set of native gates, and processor architecture. In this work, we focused on two (openly available) IBM devices that exemplify this diversity. Firstly, we used  (Fig.  <ref>), which has 7 qubits and is powered by the Falcon r5.11 processor. Its supported  native gate is the controlled-NOT (CNOT) <cit.>. Secondly, we used  (Fig. <ref>), which has 127 qubits and employs the Eagle r3 processor with Echoed Cross-Resonance (ECR)[ECR is equivalent to a CNOT up to single-qubit pre-rotations <cit.>.] gate as the  native gate. This choice of quantum processors was made based on availability and size. Larger devices offer more options for the initial qubit placement and routing leading to different results. 

After selecting the backend, the circuit was run for 8192 shots[When we started designing the framework under evaluation, the maximum number of shots was 8192. Now the number of shots has increased, but for consistency, we keep the same number.]. To increase precision, each circuit was replicated four times, and the results were compiled into one job. This job was submitted to the selected backend daily for several days, ranging from 7 to 14 days. Subsequently, from the executed circuits, we then derived different performance metrics as explained in Sec. <ref>.







 §.§ Noisy simulator
 

To emulate the effect of having access to real-time error data, we used the  <cit.>, building models that replicate the topology of real backends and load all the calibration data from the device. With this method, we can emulate the behaviour of a quantum device that behaves exactly in the same manner as the real device at the moment that it was calibrated. It is important to highlight that the simulator does not consider a phenomenological noise model with just a single physical error rate parameter, but the exact calibration data of each of the parts that compose the device, including readout circuits, individual quantum gates, etc. 





 §.§ Performance metrics
 






To evaluate the performance of different quantum circuit compilation techniques, various metrics are typically used, including 1Q and  gates overhead, circuit depth, and a fidelity F or Estimated Success Probability ESP. In this work, we chose  gate overhead, circuit depth, and fidelity as performance metrics. Note that to compute the circuit fidelity, we use three different equations based on  <cit.>, which are shown in Table <ref>.








































































§ RESULTS
 

The results presented in this section are based on experiments conducted on two real backends,   and  (refer to Sec. <ref>), as well as a noisy simulator (Sec. <ref>). Each of the techniques described in Sec. <ref> was evaluated using the circuits listed in Table <ref>. For details on the number of runs and shots, refer to Sec. <ref>.



By performing an exhaustive analysis, we derived several observations that could help to further understand the performance of different combinations of quantum circuits, noise-aware compilation techniques, and quantum devices. We expect that these initial observations, together with the developed framework, can assist in getting a deeper knowledge of the use of calibration information for mitigating quantum errors and enhancing the performance of quantum algorithms when executed on NISQ devices. 


    
  * Noise-aware compilation techniques based on calibration data can enhance the fidelity of the quantum algorithms when calibration data is properly considered. 
    
    
  * In addition, the closer the calibration data is to the real-time data, the larger the improvement of the circuit fidelity when using noise-aware compilation techniques. 

    
  * In the absence of real-time calibration data, noise-aware compilation techniques that rely solely on the latest calibration data can result in greater deviation from the theoretical results. In contrast, utilizing historical calibration data can prevent this deviation.
    
    
  * Compilation strategies based on calibration data can lead to higher circuit fidelities than those only optimizing the number of added  gates. In fact, using more gates (i.e., SWAPs) may still result in a higher circuit fidelity if these are properly selected according to calibration data.

    
  * Noise-aware compilation techniques can lead to higher fidelity enhancements when applied to larger devices.

    
  * Among the noise-aware techniques, no strategy emerges as superior in enhancing circuit fidelity. All of them provide improvements whose magnitude depends on the structure of the quantum circuit under study and the targeted quantum device.










































Some of these observations were made in previous works. However, they were somehow scattered or even sometimes contradictory, and no clear conclusions could be derived regarding the use of calibration data in different noise-aware compilation methods. Therefore, in our case, we have performed a thorough evaluation in which all experiments have been conducted under the same system conditions and for different devices and circuit compilation techniques. Authors in <cit.> highlighted the importance of using calibration data in noise-aware compilation techniques, especially during the initial qubit mapping and routing stages. This statement aligns with observation (1). However, we found, and it is also supported by <cit.>, that the latest calibration data may not accurately capture device error characteristics due to frequent device drift. On the other hand, it has been observed that the overall circuit fidelity increases as its execution gets closer to the calibration cycle <cit.>. This observation is consistent with our findings in (2). Furthermore, studies <cit.> have shown that frequent and rapid machine drift can cause inaccurate calibration data, causing a compiler to follow deceptive information. This observation is consistent with (3), which shows that relying on the latest calibration data without real-time error information is sub-optimal. Moreover, <cit.> demonstrate that using higher fidelity qubits, even if it results in longer paths, might lead to better circuit fidelity than simply minimizing circuit size and depth. This finding is consistent with (4). Other studies <cit.> show that higher qubit connectivity and qubit count can enhance circuit fidelity through compilation techniques, which aligns with our observation (5). As expected and mentioned in (6), the level of improvement in circuit fidelity using different noise-aware solutions under the same conditions depends on the algorithm characteristics and targeted processor <cit.>.  
Lastly, to the best of our knowledge, observation (3)  has not been reported before. We conclude that using historical calibration data in noise-aware compilation techniques results in a higher reliability improvement. 



In the the following subsections, we will answer a series of questions, based on our experiments, which led to the previous summarized observations.





 §.§ Do noise-aware compilation techniques improve the fidelity of the algorithm to be executed?





To analyze how the use of calibration data impacts the potential improvements of noise-aware compilation techniques, we conducted experiments on both  and  (see Fig. <ref>) using noise-aware TriQ and TriQ+ with lcd and avg methods of processing calibration data as the initial exploratory, and non-noise-aware Q-3 as the baseline. However, for , TriQ and TriQ+ were excluded as they do not scale in finding the best initial qubit mapping with the Z3 solver <cit.>. As an alternative, we applied Q-NA. The results on , shown in Fig. <ref>, indicate that noise-aware compilation techniques (TriQ, TriQ+) generally outperform the non-noise-aware compilation technique (Q-3). In bv4, noise-aware compilation techniques have an overhead of 100% in the number of  gates, as shown in Fig. <ref>. However, the circuit fidelity is still higher by 10% compared to the non-noise-aware compilation technique. To achieve higher fidelity, it is important to select the most reliable (according to measurement fidelity) qubits for initial qubit mapping, even if it results in an increase in the number of  gates. The Fredkin circuit exhibits similar behaviour, with TriQ+ having the highest number of  gates among the three, yet still achieving higher fidelity. 
As shown in Fig. <ref>, Q-3 outperforms TriQ and TriQ+ when the number of  gates in the compiled circuit is higher than 40% with respect to the baseline circuit. Finally, Fig. <ref> presents results from  that support the previously mentioned trend. The noise-aware compilation technique (Q-NA) results in a higher fidelity for all circuits except bv6, where the increased number of  gates is twice as high compared to Q-3. However, the observation still holds for circuit adder_n4 and Fredkin gate, where the increase of  gates is just around 20%. The difference between Q-3 and Q-NA is due to the initial qubit mapping process. It is important to note that a higher number of  gates can still lead to an increased fidelity, as shown in Fig. <ref>, with the exception of bv6. The results of the algorithms that do not need routing (bv4, bv6, and hs4) emphasize the significance of implementing noise-aware techniques during the initial qubit mapping process. The experiment concludes that the noise-aware compilation technique has the potential to improve overall fidelity across various algorithms and topologies. Additionally, it suggests that the number of  gates is important, but it does not completely determine reliability.






 §.§ Is including the latest calibration data in the compilation method the most efficient way to improve circuit fidelity when real-time error information is not available? 













The objective is to evaluate how incorporating the latest calibration data provided by the vendor in the noise-aware compilation techniques affects the fidelity of the executed circuits. Reference <cit.> analyzed the inter-day and intra-day calibration drift of qubits and <cit.> demonstrated that a rapid drift occurs within minutes, resulting in a 2% frequency drift and an average 18%±5% increment in single-qubit gate errors <cit.>. 

Performing this evaluation is challenging because the job submitted to the quantum processor will be added to the queue, and its execution time cannot be controlled, resulting in a waiting time of up to eight hours <cit.>. A delay in executing the job may cause outdated calibration data for the circuit compilation process. It is hypothesized that the use of obsolete calibration data will not improve the overall fidelity, while the use of calibration data that is closer to real-time error information will increase the overall fidelity.

The top part of Fig. <ref> (Real) shows that, for a real backend, the fidelity of the quantum circuit under test (adder_n4) drops more than 30% (day 6) when only the latest calibration data is considered in the error-aware compilation technique (TriQ-lcd). On the bottom (simulator), we can observe that the noise-aware compilation technique exhibits a better and more stable performance with the latest calibration data when simulating the quantum circuit. This is because the noisy simulator uses exactly the same (latest) calibration data provided to the compiler, which emulates having real-time error information, as described in Section <ref>.



To tackle the challenge of not having access to real-time calibration information, we explore several techniques for processing historical calibration data as outlined in Table <ref>. They focus on the  gate and readout error rates, as they are the main sources of noise that considerably affect circuit fidelity <cit.>. The experiments will compare the latest calibration data with different processing methods (See Sec. <ref>). Fig. <ref> demonstrates that the use of historical calibration data can reduce the impact of unexpected changes on the real device. Observe that the circuit fidelity when using TriQ-w-15 (orange line) does not experience a sharp decrease
on Day-6. However, on the bottom graph (Simulator), TriQ-w-15 shows a slight decrease of around 10% compared to TriQ-lcd on Day-9 and Day-10. The same observation can also be seen from other circuits (Fig. <ref>), where historical calibration data has a smaller deviation (error bar) compared to the latest calibration data.

Based on these experiments
, in which we use the noisy simulator to emulate real-time calibration data, we can conclude that acquiring calibration data closer to real-time error conditions of the device provides the best overall circuit fidelity, as expected<cit.>. However, since this configuration can be time and resource-intensive and is not available in most cloud-based services, it is important to have an alternative that mitigates the performance loss. For this reason, we examined the impact of different historical calibration data models.



 §.§ Does using historical calibration data always lead to higher circuit performance (fidelity) than utilizing the latest calibration data? Does the size of the device impact on the circuit fidelity enhancement?









The purpose of using historical calibration data is to compensate for any machine drift that may occur on the device. To further investigate this, we compute additional calibration processing methods according to the classification outlined in Table <ref>. It is important to highlight that there may be multiple approaches to processing this information that can be explored in the future. For this work, the aim is to demonstrate that introducing historical error information in the compilation process might be beneficial. These proposed methods will be applied to the compilation techniques shown in Table <ref>. 

























Fig. <ref> presents a summary of the experiments. It shows the advantage of noise-aware compilation techniques over non-noise-aware; it also compares several noise-aware methods when different historical calibration data is considered for both quantum processors. For example, Q-NA outperforms TriQ, particularly for the circuit adder_n4, due to its more efficient routing technique compared to TriQ's reverse SWAP. However, with a circuit in which no qubit routing is needed (bv4), TriQ achieves the highest mean fidelity because it chooses higher fidelity qubits through the use of a reliability matrix in the initial qubit mapping stage. This kind of experiment indicates that there is no outperforming noise-aware solution, and all of them can be improved to increase fidelity in different quantum circuits.

Moreover, to assess the scalability of noise-aware compilation techniques to larger devices, we also conducted experiments on , which has 127 qubits. We took into account the differences in qubit count and processor types (Falcon r5.11 with CNOT in , Eagle r3 with ECR in ). Our findings align with  <cit.>, indicating that larger devices offer more diverse mapping and routing configurations, enhancing the potential benefits of noise-aware compilation techniques. Fig. <ref> illustrates circuit fidelity improvements between 2% and 100% on , with enhancements in five out of six algorithms. In contrast,  demonstrates more modest improvements (0.44% to 48%) in four out of six algorithms. This highlights the impact of device size on noise-aware techniques and emphasizes their potential to enhance performance in larger quantum processors. From the results, it can also be concluded that the choice of approach for processing the calibration data is important as it influences the magnitude of the fidelity improvement.



Finally, we also performed a more detailed analysis of the daily fidelity obtained with each of the techniques for the different circuits from Table <ref>. As an example, Fig. <ref> shows the daily fidelity of an adder_n4 circuit with various calibration models on both  and  backends. Q-3 shows noticeable fluctuations, while Q-NA-lcd demonstrates stability, consistently approaching the mean. However, relying solely on the latest calibration data may expose vulnerabilities. For example, on Day-13, Q-NA-lcd experienced a significant drop in contrast to Q-NA-avg, which maintained fidelity. Results from , which has more options for qubit mapping and routing, show increased variability. Fig. <ref> highlights the benefits of noise-aware calibration, with configurations like Q-NA-avg exhibiting stability and the highest mean. So we can conclude that in the absence of real-time calibration data, noise-aware compilation techniques that rely solely on the latest calibration data can result in greater deviations. In contrast, utilizing historical calibration data can prevent sharp deviations and result in a higher average fidelity.







 §.§ What is the impact on the algorithm's fidelity as the circuit size increases?









Moving from our in-depth examination of small quantum circuits, we shift our focus to larger quantum algorithms (higher circuit width and depth).
Fig. <ref> demonstrates that using noise-aware compilation techniques yields greater benefits for larger circuits compared to the non-noise-aware ones, like Q-3. For the experiments conducted on , this improvement varies greatly, ranging from 4.83% (bv_2) to a significant 127.79% (bv_7) depending on the circuit and method of processing calibration data. A trend can be seen, indicating a decrease in fidelity as the circuit size increases. Nevertheless, certain methods of processing calibration data demonstrate improved performance on larger circuits. For instance, Q-NA-w-15-adj exhibits better fidelity in larger circuits despite its suboptimal performance in smaller circuits. While noise-aware compilation techniques enhance fidelity as the circuit size increases, no technique consistently outperforms the others.







 §.§ What is the correlation between circuit depth and the total number of  gates with the algorithm's fidelity?






    




Most of the compilation techniques aim to minimize circuit depth and/or the overhead of  gates to mitigate issues related to decoherence and instability arising from  gates <cit.>. In this set of experiments, we explore the impact of circuit depth and the total number of  gates (after compilation) on the circuit fidelity when using several noise-aware compilation techniques.



Fig. <ref> displays the total number of  gates based on the circuits presented in Fig. <ref>. The data indicates that, in general, as the total number of  gates increases, the circuit fidelity decreases. However, Q-NA-w-15-adj and Q-NA-avg have the highest fidelity despite having a higher number of  gates. It is suggested that using historical calibration data in noise-aware compilation techniques may result in a higher number of  gates, but it does not necessarily mean having lower circuit fidelity. In addition, as shown before, it exhibits higher performance than the latest calibration data methods.



The relationship between circuit depth and fidelity is well-established: the higher the circuit depth, the lower the circuit fidelity is. However, as shown in Fig. <ref> (circuits from Fig. <ref>), historical calibration data can lead to a high circuit depth but still show an increased fidelity. 
Similar to  gates, it can also be observed that using historical calibration data may result in a higher circuit depth. Despite the increase in depth, the benefits of this approach outweigh them.

Based on the previous two figures related to  gates and circuit depth, we can conclude that using only these metrics is insufficient to represent overall fidelity. 


















§ CONCLUSIONS
 

Noise-aware circuit compilation techniques are promising to further improve the overall circuit fidelity in NISQ devices as a result of selecting qubits with small readout errors for the initial qubit mapping and reliable paths for the qubit routing stage. In this paper, we have developed a framework that supports running multiple circuits with various calibration techniques under the same conditions within a single quantum job on a vendor processor. This framework also allows for the execution of circuits on both noisy simulators and real devices. With the help of this framework, we examine the benefits of using noise-aware compilation techniques compared to non-noise-aware techniques. The study found that the overall fidelity achieved with noise-aware compilation techniques is greater when the calibration data is closer to real-time data compared to non-noise-aware techniques. When comparing the results of the noisy simulator and the real device, it was observed that noise-aware compilation techniques relying solely on the latest calibration data can yield
 greater deviation in the absence of real-time calibration data. This led to the definition of different approaches for processing calibration data. The experiments demonstrated that incorporating historical calibration data can prevent deviation and lead to a more accurate average.  Additionally, the results indicated that the use of high-fidelity qubits in  gates and measurements is more crucial than the number of circuit depth and  gates employed. Therefore, increasing the depth and the number of gates may still provide a better final fidelity when using high-fidelity qubits. It can be concluded that using only the metrics of circuit depth and  gates is insufficient to represent overall fidelity.

All experiments compared two different topologies in  (7 qubits) and  (127 qubits). The results demonstrate that for noise-aware compilation techniques to work efficiently, a larger device is needed as it provides more options and chances to increase overall fidelity. Based on the experiments conducted, it was concluded that no approach for processing calibration data was found to be a universal strategy among the noise-aware techniques that is optimum. The degree of improvement provided varied depending on the specificities of the quantum circuit being run. 

In future work, we will investigate how to characterize calibration data in a more sophisticated manner and combine error information with QEM. Additionally, we will examine the impact of applying these techniques to other quantum computing platforms.



§ ACKNOWLEDGMENT


This work is supported by the QuantERA grant EQUIP with the grant numbers PCI2022-133004 and PCI2022-132922, funded by Agencia Estatal de Investigación, Ministerio de Ciencia e Innovación, Gobierno de España, MCIN/AEI/10.13039/501100011033, and by the European Union “NextGenerationEU/PRTR”. CGA acknowledges funding from the Spanish Ministry of Science, Innovation and Universities through the Beatriz Galindo program 2020 (BG20-00023) and from the  European ERDF under grant PID2021-123627OB-C51. We acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Quantum team.
ieeetr









