Paper Title: Engineering a multi-level bath for transmon with three-wave mixing and parametric drives.


AIP/123-QED


Chemical Potential]Engineering a multi-level bath for transmon with three-wave mixing and parametric drives


These authors contributed equally
[Now at: ]Department of Physics, University of Illinois Urbana-Champaign, Champaign, IL, USA


These authors contributed equally





[email: ]hatridge@pitt.edu
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA, 15260, USA




§ ABSTRACT 
A photonic system with a tunable bath environment provides an extra degree of freedom for quantum simulators.
Such a system can be realized by parametrically modulating the coupling between the system and bath.
In this letter, by coupling a transmon qubit to a lossy Superconducting Nonlinear Asymmetric Inductive eLement (SNAIL) mode, we experimentally create a tunable chemical potential for the qubit mode. 
We show that the qubit can be thermalized to equilibrium with different population distributions under different parametric pumping conditions. 
We further extend our method to the third level of the transmon, showing its practical use beyond the simple two-level case. 
Our results provide a useful tool that can be readily integrated with quantum simulators that would benefit from a non-trivial photon population distribution. 


[
    Michael Hatridge
    August 5, 2024
====================






A quantum simulator <cit.> can address complex problems in various domains of physics, including quantum materials simulation <cit.>, quantum chemistry <cit.>, quantum transport <cit.>, many-body dynamics <cit.>, and even black holes <cit.>, by mapping a problem that is otherwise difficult to probe or track in its original setting to an engineered quantum system over which experimenters have good control. 
Photonic excitation provides an important degree of freedom in many quantum simulating platforms, such as microwave superconducting circuits <cit.>, trapped ions <cit.>, and optical photon-based quantum machines <cit.>.
Therefore, the ability to engineer different properties of a photonic system becomes essential in these quantum simulators. 
A tunable chemical potential, or more generally, a controllable photonic bath, is of great interest.
A photonic system with a well defined chemical potential has a fixed occupation number and can thus be used to simulate entities with a conserved particle number, such as electrons and atoms <cit.>.
However, a well-defined occupation number, and thus the concept of a chemical potential, does not arise naturally in a photonic system. 
In Planck's initial study on the thermodynamics of blackbody radiation <cit.>, a chemical potential term is absent due to the photons' absorption and emission by the walls of their container. 
Later it came to be understood that in systems without such walls, photons can acquire a non-zero chemical potential. 
For example in semiconductor junctions, the chemical potential of photons arises due to interactions between electrons and holes <cit.>. 
More recently, there have been demonstrations of non-zero chemical potential in a dye-filled optical cavity that forms a Bose-Einstein condensate <cit.>.

Superconducting circuits and circuit quantum electrodynamics (cQED) <cit.> provide a promising platform for the investigation of this topic in the microwave domain.
The strong coupling between superconducting circuits and their environment, and controllable parametric interactions between different circuit modes, enable the manipulation of mode populations with bath engineering techniques. 
Demonstrations include: fast qubit reset by forcing it to its ground state with appropriate drives <cit.>, stabilizing a qubit to an arbitrary state <cit.>, and in particular, stabilizing a photonic system with a non-trivial photonic state <cit.>.
Based on these remarkable results, it is natural to consider the implementation of a tunable chemical potential in a photonic system, or a tunable photonic bath.
 
In this letter, we experimentally demonstrate a multi-level bath for a transmon qubit. 
We start by coupling it to a lossy mode via a mode conversion process and a two-mode squeezing process  within its |g⟩-|e⟩ manifold. 
The transmon is treated as a two-level system whose dissipation is engineered by tuning the strengths of these two parametric processes.
This tunable dissipation sets the average photon number in the qubit mode, and by constraining this photonic mode in a two-level setting, we are effectively realizing a population statistic in the form of Fermi-Dirac distribution with a chemical potential. 
We verify this behavior by stabilizing the qubit to various equilibrium states with different population distributions, including a population inverted state which cannot be achieved by coupling to a natural bath at positive temperatures. 



Furthermore, the anharmonic multi-level nature of the transmon allows us to go beyond the simple two-level model.
We extend our parametric bath engineering technique to the third energy level of the transmon, creating a non-trivial photonic state in a three-level system.
This allows us to enter the regime of Gentile statistics <cit.>, a generalization of Fermi-Dirac and Bose-Einstein statistics where the mode maximum occupation number (here photons in a qubit) can be set between one (Fermions) and infinity (Bosons). 
Our work demonstrates a promising method to create an effective in situ tunable chemical potential for a photonic qubit system, as well as generate arbitrary multi-level state configurations in the scope of certain intermediate statistics beyond Bose–Einstein and Fermi–Dirac statistics. 









Theory: We begin by constructing a model that reflects our objective: we aim to create a tunable multi-level bath for a photonic system.
Inspired by a proposal from Hafezi et al. <cit.>, this can be realized by parametrically coupling a microwave mode to another lossy mode. 
We consider a system that consists of a transmon qubit q <cit.> capacitively coupled to a SNAIL-mon mode s <cit.> (a SNAIL element shunted by a large capacitor) as shown in the top panel of Fig. <ref>(a).
We aim to create a tunable bath for the qubit mode, and the SNAIL-mon is used both as a three-wave mixing element and the lossy mode. 
The qubit is also coupled to a 3D cavity for dispersive readout <cit.>.
We note that while our experiment is implemented with cQED, the general model of the design is fundamentally platform-independent.

Central to our protocol are the parametric interactions between the qubit and the SNAIL-mon.
In particular, we are interested in two types of interactions: two-mode squeezing and mode conversion, whose Hamiltonian are given by <cit.>: 

    H_tms   = g_Σ (s^† q^† + h.c.), 
    
        H_bs   = g_δ (s^† q + h.c.),

where s and q are the annihilation operators of the SNAIL-mon and qubit mode respectively, g_Σ and g_δ are the interaction strength of the corresponding processes, which can be controlled via external drives. 
The subscripts are labelled Σ and δ due to a pump at summation/difference frequency of the SNAIL-mon (ω_s) and qubit (ω_q), i.e., ω_p = ω_s±ω_q, is needed for each process respectively.
A detailed derivation of above equations can be found in Appendix <ref>. 

Equations <ref> and <ref> give a general description of the two processes between the linear modes.
In our case, however, the anharmonicity of the transmon qubit needs to be considered. 
Let us take the two-mode squeezing case as an example.
As shown in Fig. <ref>(b), the two-mode squeezing pump creates a coherent process between the |0, g⟩ and |1, e⟩ (the joint ket notation follows the convention of |SNAIL-mon (number) , qubit (letter)⟩ throughout the paper). 
When both modes are linear, the pump will also drive the similar processes, such as |1, e⟩↔|2, f⟩, |2, f⟩↔|3, h⟩ and etc.
However, the transmon's large anharmonicity makes the pump off-resonance for these processes and thus forbids (at least to the leading order) them to appear in the rotating frame.
It is then necessary to re-write Eqs. <ref> and <ref> for individual qubit levels. 
For two adjacent qubit levels |i⟩ and |i + 1⟩, the respective interaction Hamiltonian terms are given by:

    H_Σ_s, i, i+1   = g_Σ_s, i, i+1  s^†|i+1⟩⟨i| + h.c., 
     
        H_δ_s, i, i+1   = g_δ_s, i, i+1  s^†|i⟩⟨i+1| + h.c.,

where g_Σ_s, i, i+1 and g_δ_s, i, i+1 are the effective coupling rates of the two individual processes (Appendix <ref>), with the corresponding pump frequencies: ω_Σ_s,i, i+1  = ω_s + ω_i, i+1 and ω_δ_s,i, i+1 = ω_s - ω_i, i+1.
We have chosen to replace the usual transmon raising/lowering operators in these equations with ket-bra notation to make clear that each parametric drive frequency links only a single pair of transmon levels.
Because of the frequency selective nature of the parametric processes, multiple parametric interactions can be activated at the same time by simultaneously applying multiple pumps. 


To understand how to engineer the transmon bath, let us first consider a simple case where the transmon is treated as a two-level system.
We implement an effective tunable chemical potential for the transmon mode within this two-level manifold.
The crucial step is to constrain the mode occupation number, which is implemented by combining the parametric processes and SNAIL-mon dissipation. 
As shown in Fig. <ref>(b), a pump at ω_p = ω_Σ_s, ge coherently brings the system from |0, g⟩ to |1, e⟩.
When the SNAIL-mon decay rate is much larger than the other relevant rates, i.e., κ_s ≫ g_Σ_s, ge, 1/T^q_1, the system quickly falls into |0, e⟩, effectively constraining the transmon occupation number at 1. 
Similarly, as shown in Fig. <ref>(c), a pump at ω_p = ω_δ_s, ge will effectively constrain the transmon occupation number at 0. 
We refer to these two processes as parametric “heating” and “cooling”, as they push the transmon to a higher and lower energy state respectively. 

The tunable chemical potential can be realized by applying both pumps simultaneously. 
By controlling the effective rates of the two processes, we can thermalize the transmon to an arbitrary equilibrium state, as shown in Fig. <ref>(d).
We show that (Appendix <ref>), the density matrix of the equilibrium state can be described by a grand canonical ensemble:

    ρ = Z(β, μ)^-1 e^- β (ω_s/2σ^z_q - μ/2σ^z_q),

where Z(β, μ) = Tr[e^-β (ω_s/2σ^z_q - μ/2σ^z_q)], σ^z_q is the Pauli z operator for the qubit, and β = 1/k T where k is the Boltzmann constant, and T is the SNAIL-mon temperature. 
The chemical potential μ is given by:

    μ = 1/βln(g^2_Σ_s, ge e^βħω_s + g^2_δ_s, ge/g^2_Σ_s, ge e^-βħω_s + g^2_δ_s, ge),

which is a function of the parametric interaction strengths – both quantities that can be externally controlled. 
However, we would like to point out two important caveats. 

First, when deriving Eq. <ref> and <ref>, we assume that the qubit only exchanges energy with environment via the SNAIL-mon mode.
In practice, the qubit can also thermalize through other channels at different temperatures, whose rates will add to the parametrically induced ones. 
We minimize this effect by using a qubit that is much more coherent that the SNAIL-mon. 
Second, each decay event is heralded by a photon in the SNAIL-mon mode, and so we must take into account the effects of distinguishability of these photons when several processes create these SNAIL-mon excitations at the same time. 
For example, some engineered bosonic decay processes <cit.> require photon indistinguishability, as the decay operator should protect a manifold of states. 
In this work, however, we focus on distinguishable photons in our finite temperature bath so that each heating/cooling event is uncorrelated with others. 
One method to ensure distinguishability is to use unique lossy modes for each process, but this requires greater hardware. 
Instead, we simply must take care to tune our drives to use the SNAIL-mon's broad bandwidth to create deliberately different SNAIL-mon excitations for each parametric process. 






Experiment: Our experiment scheme is shown in the lower panel of Fig. <ref>(a). 
The lossy SNAIL-mon has a bandwidth of 12.98 MHz, and transmon's natural decay and heating rates are: Γ^N_ge/2π = 1.22 kHz, Γ^N_eg/2π = 8.28 kHz, Γ^N_ef/2π = 2.20 kHz, and Γ^N_fe/2π = 12.27 kHz. 
Other relevant circuit parameters are given in Appendix. <ref>.
A Traveling Wave Parametric Amplifier (TWPA) <cit.> is used for single shot readout and transmon state differentiation. 
We perform a three-part division on the readout data, and label all states from |f⟩ and beyond as |f^+⟩. 

We first demonstrate the effect of the “heating” and “cooling” drives.
In the heating experiments, the qubit is initialized in its |g⟩ state. 
We apply the Σ_s, ge drive while sweeping its strength and duration before the qubit measurement, as shown in Fig. <ref>(a). 
The measurement results of the heating process for different pump strengths are shown in Fig. <ref>(b), where the qubit state population is plotted as a function of pump duration. 
The pump strength increases from left to right, resulting in a rise of the heating rate (Γ^p_ge/Γ^N_ge = 9.9, 29.6, 205.4 respectively) as expected. 

The rate is extracted from a semi-classical model (solid line, see Appendix <ref>).
The qubit also populates the |f^+⟩ state to some extent, which we attribute to the qubit's finite (undriven) temperature that excites the qubit from the |e⟩ to |f⟩ state. 
In addition, a strong enough pump could also lead to extra parametric heating due to its finite spectral weight on the transmon's higher transition. 

We emphasise that, for strong heating drives, the qubit population inverts, rather than spreading to states |f⟩ and higher, the scenario typically achieved by physically heating it.
This indicates that our scheme indeed creates a finite chemical potential rather than simply changes the effective temperature of the qubit. 
More, the system, from the point-of-view of joint SNAIL-qubit states, forms a traditional lambda structure with a trapped excited state, making it suitable for single-“atom” masing experiments <cit.>. 
Unlike previous implementations, which for instance use a lossy cavity mode and Purcell decay to invert a qubit <cit.>, we require no special mixing frequencies, are compatible with a wide range of qubit frequencies, and can readily access higher excitation states of the transmon.  


In cooling experiments, the qubit is initialized in the  |e⟩ state. 
Then the δ_s, ge pump is applied to the system with various pump strengths and pump durations, as shown in Fig. <ref>(a).
The measurement results of the cooling process are shown in Fig. <ref>(d-f). 
For each panel from left to right, the transmon thermalizes to its equilibrium state faster as the pump strength increases (Γ^p_eg/Γ^N_eg = 1.6, 3.9, 10.6 respectively).
In contrast to <cit.> where both the qubit and cavity are driven to force the qubit fully into the |g⟩ state, our method allows us to determine the |g⟩ probability and only requires one drive on the SNAIL-mon. 
Unlike the heating experiment, there is not a rise in the |f^+⟩ state because we are actively pushing the qubit to the ground state and there is no parametric processes that can bring the transmon to the |f^+⟩ state at a frequency near the δ_s, ge pump. 





We have demonstrated our control over each single parametric process with the heating and cooling results by establishing the equilibrium states with different chemical potentials.
We note that the difference in the final distribution for different single pump strengths does not conflict with Eq. <ref> for the case with g_Σ = 0 or g_δ = 0, as the qubit also directly couples to the environment weakly in practice.
To extend the utility of our setup as a tunable photonic bath in a larger system, a demonstration of the full control over the transmon's thermalization process is necessary.
That is, we need to be able to set both the final equilibrium state distribution as well as the thermalization rate.

Therefore we next consider the case where both pumps are applied.
As shown in Fig. <ref>(a), when both the Σ_s, ge and δ_s, ge drives are on, the two processes happen at the same time with different rates set by their respective pump strengths.
The system is initialized in the state |g, 0⟩, then both pumps are applied for a certain duration, followed by a dispersive transmon measurement, as shown in Fig. <ref>(b). 
In Fig. <ref>(c), we show the transmon population as a function of pump length for two sets of heating and cooling rates (top: Γ^p_ge/2π = 54 kHz, Γ^p_eg/2π = 54 kHz, and bottom: Γ^p_ge/2π = 24 kHz, Γ^p_eg/2π = 22 kHz).
The pump conditions are chosen such that the ratio between the two rates are close to 1 (Γ^p_ge/Γ^p_eg∼ 1).
As expected from Eq. <ref> and <ref>, the transmon thermalizes to the same mixed 50%|g⟩ - 50%|e⟩ state (up to the residual population in the |f^+⟩ state).
However, the thermalization rates are different due to the difference in the parametric drive strengths.

In addition, the data in the supplementary Fig. <ref> shows that by tuning the ratio between the two rates, we are able to set the transmon equilibrium with arbitrary population distribution.
This confirms our full control over the transmon bath, in the equilibrium state as well as the thermalization rates, which extends its potential utility as an exotic bath in larger quantum simulators. 


We note that because the SNAIL-mon is very lossy, and thus has a large bandwidth, it is easy to create distinguishable photons. 
This is the general case unless great care is taken to match the two pump frequencies to create identical excitation frequencies in the SNAIL-mon. 
In this latter case, certain states of the qubit become `dark' to the parametric process, and the overall behavior changes dramatically. 
We have separately investigated this in a similar sample <cit.>.
However, experimental observation of these effects is greatly hindered by our ability to maintain a stable set of microwave phases among the qubit and parametric tones; attempts to measure these dark states with conventional few-hundred MHz bandwidth AWGs yielded inconclusive results due to rapid phase drifts; after the conclusion of this experiment we have recently realized fully stable drives which allow us to observe these dark states using high-speed RFSoC electronics and the QICK platform <cit.>.



Finally, we extend our protocol beyond the two-level manifold and bring the system into even more exotic  states by accessing the {|g⟩, |e⟩, |f⟩} manifolds. 
As an example, we aim to thermalize the transmon to a mixture of roughly 50%|g⟩ and 50%|f⟩, as indicated in the bottom right of Fig. <ref>(d). 
This is achieved by simultaneously pumping the |e⟩→|g⟩ cooling process at ω_δ_s, ge and the |e⟩→|f⟩ heating process at ω_Σ_s, ef with matched transition rates away from |e⟩, as shown in Fig. <ref>(a).
The transmon is initialized in the ground state, then both pumps are turned on and the qubit state is readout at the end of the pumps, as shown in Fig. <ref>(b).
The transmon population as a function of time is shown in Fig. <ref>(c).
We note that there is a residual of 15%|e⟩ state which is due to the qubit decay from the |f⟩ state and thermal excitation from the |g⟩ state, as well as the finite measurement fidelity.  
The simple semi-classical model cannot fully capture the dynamics of this process; we attribute this to the large frequency detuning of the two pump tones in the rotating frame and finite transmon thermalization rate. 
By including these factors, we are able to numerically simulate the system's behavior using QuTiP (Appendix <ref>).
By going beyond the simple two-level case, we are effectively accessing a certain type of intermediate statistics  – Gentile statistics <cit.>, where the maximum occupation number of particles in one state is neither infinity (as for Boson) nor one (as for Fermion), but a finite integer n (in our case, n ≥ 2).







In conclusion, we have experimentally created a controllable multi-level photonic bath for a transmon qubit via parametric interactions between the transmon mode and another cold, lossy bath mode.
In the two-level setting, this is equivalent to a tunable chemical potential for the transmon that allows for thermodynamical adjustments, a useful control knob that is typically missing for photons.
This will allow  us to explore new physics by integrating these controllable bath qubits into larger quantum simulators based on combinations of qubits and cavities arrays, which would benefit from some of its sites having a non-trivial population distribution <cit.>.
On the other hand, this scheme also forms a traditional lambda structure with a trapped excited state, which enables the generation of highly coherent microwave signals <cit.>.
Further, we extended our protocol to the higher levels of the transmon, creating exotic equilibrium states that cannot be realized with a natural thermal bath.
This brings interesting perspectives on the experimental study of Gentile statistics, offering opportunities for simulating certain condense matter systems, such as magnons and spin waves <cit.>. 


We acknowledge contributions to earlier versions of this experiment from Guanyu Zhu, Mohammad Hafezi, Mattias Fitzpatrick, Neereja Sundaresan, Basil Smithham, Christie Chou, and Andrew Houck. 
The TWPA used for amplified readout in this experiment was provided by MIT Lincoln Laboratory. 
This work was supported by the  Army Research Office under Grants No. W911NF15-1-0397 and M. Hatridge’s NSF CAREER grant (PHY-1847025). 
The views and conclusions contained in this document are those of the authors and should be be interpreted as representing official policies, either expressed or implied, of the Army Research Office or the US Government. 
The US Government is authorized to reproduce and distribute reprints for government purposes not withstanding any copyright notation herein. 














§ CIRCUIT PARAMETERS AND EXPERIMENT SETUP






The qubit frequency is ω_q/2π = 4.520GHz, anharmonicity α/2π = -197.39MHz, and its natural decay and heating rates given in the main text are measured by initializing it in different states (Fig. <ref>).
The SNAIL-mon frequency (with no external flux applied) is ω_s_0/2π = 9.136GHz and its linewidth κ_s/2π = 12.98MHz. 
To minimize the undesired higher order effects of the SNAIL-mon, we operate at its fourth-order Kerr cancellation point <cit.>, at which the SNAIL-mon frequency is ω_s_op/2π = 8.01GHz. 
The hybridization strength (g/Δ)_sq =  1e-2, while (g/Δ)_sc =  7e-4. 
The readout cavity has bandwidth κ_c/2π = 0.761MHz and is followed by a Traveling Wave Parametric Amplifier (TWPA) <cit.> provided by MIT Lincoln Laboratory through IARPA (see Fig. <ref>), which allows for single shot readout and clear state differentiation.

The entire experiment was cooled to approximately 20mK on the base stage of a dilution refrigerator, see Fig. <ref>. Each input drive line is attenuated and filtered with lossy filters made of Eccosorb CR-110 for high frequency light. The external flux threaded through the SNAIL-mon's superconducting Josephson junction loop is generated by DC signals of a current source (YOKOGAWA GS200) through superconducting wires.
Qubit, pump, and readout pulses are generated by mixing a local oscillator tone with finite single sideband frequency (generated by a SignalCore SC5511A rf signal generator) with the pulses made by an arbitrary waveform generator (Tektronix AWG5014C) with a sampling rate of 1.0 GSa/s using an IQ-mixer (Marki IQ0618).



We measure the reflected signal phase shift from the readout cavity to dispersively infer the qubit's state  <cit.>. 
The reflected signal is amplified at the base stage by a TWPA, then moves through an Eccosorb filter to prevent high frequency stray photons.
It next passes through microwave isolators on the output lines and is amplified using a Low Noise Factory HEMT amplifier at the 4K stage. 
Once outside of the fridge, the signal is mixed down to 50 MHz using an IR mixer (Marki IR4509) and a local oscillator signal from the SignalCore. The signal is then recorded using a fast ADC card (AlazarTech).

The sample housing, shown in Fig. <ref>, is comprised of two parts. On one side (left in photo) is a 6061 aluminum coaxial, λ/4 cavity <cit.> that acts as our readout resonator. An opening on the side of the coaxial cavity connects to an OFHC copper tube <cit.> that the sapphire spans to allow for flux biasing the SNAIL-mon's Josephson junction loop. The qubit and SNAIL-mon are fabricated out of aluminum, with Al/AlOx/Al Josephson junctions, on the same chip of sapphire, via nano-scale lithography. The qubit is positioned between the aluminum and copper housing sections to optimize inter-element couplings. Microwave coupling pins were inserted into the housing, one each per element. 










§ SNAIL-QUBIT HAMILTONIAN


The transmon-SNAIL-mon Hamiltonian can be written as:

    H_0ħ   = ω_q q^†q + c^q_4 (q + q^†)^4 
       + ω_s s^† s +  c^s_3(s + s^†)^3 
       + g(s^†q + s q^†),

where ω_q is the transmon frequency, ω_s is the SNAIL-mon frequency, g is the capacitive coupling strength between the transmon and SNAIL-mon, c^q_4 is the coefficient of the transmon's fourth order nonlinearity, and c^s_3 is the coefficient of the SNAIL-mon's third order nonlinearity. 
Note here we neglect the fourth order term from the SNAIL-mon mode, as it can be largely suppressed at its optimal working point. 
In addition the Hamiltonian for the SNAIL-mon pump is given by:

    H_drive  = ϵ_d ( e^i(ω_d t + ϕ_d)s^† + e^ -i(ω_d t + ϕ_d) s),

where ω_d, ϵ_d, and ϕ_d are the driving frequency, amplitude, and phase respectively.


In the dispersive regime, i.e., g ≪Δ = ω_s - ω_q, the system can be re-diagonalized under the Bogoliubov transformation <cit.>.
Note that all operators and coefficients are transformed from their original basis, but marked as such for simplicity:

    H_sysħ   =  ω_q q^†q + c^q_4 (q + g/Δ s + h.c.)^4 
       + ω_s s^† s + c^s_3 (s + g/Δ q + h.c. )^3 
       + ϵ_d ( e^i(ω_d t + ϕ_d)s^† + e^ -i(ω_d t + ϕ_d) s)

The drive terms can be removed by moving to a displacement frame with the transformation: H' = O^† H O - iO^†Ȯ, where the transform operator is chosen to be O_D = exp(β s^† - β^*s), with β = ϵ_d e^-i ϕ_d / (ω_d - ω_s).
The system Hamiltonian is then given by:

    H_sysħ   =  ω_q q^†q + c^q_4 (q + g/Δ (s + β e^i(ω_d t + ϕ_d)) + h.c.)^4 
       + ω_s (s + β e^i(ω_d t + ϕ_d))(s^† + β^* e^-i(ω_d t + ϕ_d)) 
       + c^s_3 (s + β e^i(ω_d t + ϕ_d) + g/Δ q + h.c.)^3

This can be further simplified by moving into a rotating frame in a similar fashion by choosing the transform operator to be O_R = e^i ω_s s^† s t + i ω_q q^† q t.
We get:

    H_sysħ   = 24 c^q_4 |β|^2 q^† q - α q^†qq^†q + c^3_s( s e^-i ω_s t
       + g/Δ q e^-i ω_q t + β e^-i ω_d t + h.c.)^3.

Here the first term is the qubit Stark shift due the pump, and α = -6 c^q_4 is the qubit anharmonicity.
The effects of the frequency-selected parametric pumps are shown in the third term.
Expanding this term <ref> produces all possible combinations of SNAIL-mon and qubit operators; most of them can be ignored due to their fast rotations. 
The pump frequency provides a degree of freedom for selecting the desired processes by matching the frequency of it.
In particular, when the SNAIL-mon is driven at ω_d = ω_s + ω_q, the qubit-SNAIL-mon Hamiltonian takes the form

    H_Σħ   = 24 c^q_4 |β_Σ|^2 q^† q - α q^†qq^†q 
       + g_Σ(β_Σ s^† q^† + β_Σ^* sq),

where g_Σ = 6 c^s_3 g/Δ. 
Likewise for the difference frequency pump ω_d = ω_s - ω_q:

    H_δħ   = 24 c^q_4 |β_δ|^2 q^† q - α q^†qq^†q 
       + g_δ(β_Σ s^† q^† + β_Σ^* sq),

where g_δ = 6 c^s_3 g/Δ.
Both of these expressions feature pump-induced Stark shifting, the transmon's anharmonicity, and the respective effective 2-body parametric process. 
Specifically for this application, we restrict the system to only neighboring transitions, allowing us to write Eqs. <ref> and <ref>.



§ TUNABLE CHEMICAL POTENTIAL OF A PARAMETRICALLY DRIVEN QUBIT


We next derive the steady state population distribution of the qubit in the system shown in Fig. <ref>(a) and the expression of its chemical potential. 
We start with a quick review on a simple model: a qubit coupled to a thermal bath at temperature T_Bath. 
The master equation of the system is given by <cit.>:

    ρ̇ =    i/ħ [ρ, H] 
       + γ/2(N(Ω) + 1) (2 σ^- ρσ^+ - ρσ^+ σ^- + σ^+ σ^- ρ) 
       + γ/2N(Ω) (2 σ^+ ρσ^- - ρσ^- σ^+ + σ^- σ^+ ρ),

where H = ħΩ/2σ^z is the single qubit Hamiltonian, γ is the bandwidth of the qubit mode, Ω is the qubit frequency, N(Ω) = 1/(e^ħΩ/kT_Bath - 1) is the average photon number for the thermal state at temperature T_Bath. 
The density matrix at equilibrium is then given by solving the above equation with ρ̇ = 0:

    ρ (∞) = [ 1 + N(Ω)/1 + 2 N(Ω)                   0;                   0     N(Ω)/1 + 2 N(Ω) ].

By plugging in the expression of N(Ω), we get:

    ρ (∞) = Z(β)^-1 e^-β H,

where β = 1 / k_B T_Bath, and Z(β) = Tr(e^-β H).
The steady state of the qubit is a thermal state at the bath temperature as we expected.
Note that in this simple example, the equilibrium state is described by a canonical density operator without a chemical potential. 

In our experiment, the setting is more complicated: the qubit is parametrically coupled to the SNAIL-mon, which is also in contact with a thermal bath at temperature T_Bath. 
The Hamiltonian of the system is given by:

    H_sys/ħ =    ω_q/2σ^z_q + ω_s/2σ^z_s
       + (g_Σ e^iω_Σ tσ^+_sσ^+_q + g_δ e^-i ω_δ tσ^+_sσ^-_q + h.c.),

where the SNAIL-mon is also treated as a two-level system under the assumption that its population does not exceed 1 during the qubit thermalization process. This assumption is valid when the bath is cold (T_Bath∼ 0) and SNAIL-mon thermalization is much faster than the parametric processes (γ_s≫ g_Σ, 
 g_δ).
With pump frequencies ω_Σ = ω_s + ω_q and ω_δ = ω_s - ω_q, we get the Hamiltonian in the rotating frame:

    H^R_sys/ħ = g_Σσ^+_sσ^+_q + g_δσ^+_sσ^-_q + h.c.



The master equation of this coupled system is given by:

    ρ̇ =    i/ħ [ρ, H^R_sys] 
       + γ_s/2(N(ω_s) + 1) (2 σ^- ρσ^+ - ρσ^+ σ^- + σ^+ σ^- ρ) 
       + γ_s/2N(ω_s) (2 σ^+ ρσ^- - ρσ^- σ^+ + σ^- σ^+ ρ).

For simplicity, we assume that only the SNAIL-mon sees a thermal bath.
Similarly, the system density matrix at equilibrium is then given by solving the master equation with ρ̇ = 0.
Let us first look at the SNAIL-mon mode, its density matrix can be obtained by tracing off the qubit degree of freedom ρ^s = Tr_q (ρ). 



    ρ^s_00   = 8 g^2_δ g^2_Σ + (g^2_δ + g^2_Σ) (1 + N(ω_s)) (1 + 2N(ω_s))γ_s^2/16 g^2_δ g^2_Σ + (g^2_δ + g^2_Σ)(1 + 2 N(ω_s))^2 γ_s^2
    ρ^s_01   = 0  
    ρ^s_10   = 0 
    ρ^s_11   = 8 g^2_δ g^2_Σ + (g^2_δ + g^2_Σ) N(ω_s) (1 + 2N(ω_s))γ_s^2/16 g^2_δ g^2_Σ + (g^2_δ + g^2_Σ)(1 + 2 N(ω_s))^2 γ_s^2.

Given that γ_s≫ g_Σ,  g_δ,  

    g^2_δ g^2_Σ/(g^2_δ + g^2_Σ)  γ_s^2≪ 1,

the density matrix simplifies to:

    ρ^s = [ 1 + N(ω_s)/1 + 2 N(ω_s)                       0;                       0     N(ω_s)/1 + 2 N(ω_s) ],

This is a thermal state at the bath temperature, as expected. 


The density matrix for the qubit can be obtained in a similar fashion, ρ^q = Tr_s (ρ):

    ρ^q_gg  = 8 g^2_δ g^2_Σ + (1 + 2 N(ω_s))[g^2_ΣN(ω_s) +g^2_δ(1 + N(ω_s))] γ_s^2/16 g^2_δ g^2_Σ + (g^2_δ + g^2_Σ)(1 + 2 N(ω_s))^2 γ_s^2
       ρ^q_ge = 0  
       ρ^q_eg = 0  
       ρ^q_ee = 8 g^2_δ g^2_Σ + (1 + 2N(ω_s))[g^2_δN(ω_s) +g^2_Σ(1 + N(ω_s))] γ_s^2/16 g^2_δ g^2_Σ + (g^2_δ + g^2_Σ)(1 + 2 N(ω_s))^2 γ_s^2.

We note that the qubit state is a function of the thermal population of the SNAIL-mon mode N(ω_s), this is because we assume that the qubit does not see any other environment other than the SNAIL-mon.

As a sanity check, let us examine the cooling case (g_Σ = 0), where the qubit has a weak photon exchange interaction with the SNAIL-mon.
The qubit density matrix is then reduced to: 

    ρ^q = [ 1 + N(ω_s)/1 + 2 N(ω_s)                       0;                       0     N(ω_s)/1 + 2 N(ω_s) ],

The qubit reaches the same thermal state as the SNAIL-mon.
This is reasonable, as the qubit will eventually thermalize with the SNAIL-mon through their energy exchanging interaction. 

Similarly for the heating case (g_δ = 0), the qubit density matrix is:

    ρ^q = [     N(ω_s)/1 + 2 N(ω_s)                       0;                       0 1 + N(ω_s)/1 + 2 N(ω_s) ],

where the population inversion is realized as expected <cit.>. 
Note here the thermal property of the SNAIL-mon mode (N(ω_s)) remains the same, the non-trivial qubit equilibrium is indeed a consequence of parametric bath engineering, rather than simple changing the temperature of the environment. 


For a more complicated case, where both pumps are on, we again use the relation in Eq. <ref>, and get:

    ρ^q_gg   = g^2_ΣN(ω_s) +g^2_δ(1 + N(ω_s)) /(g^2_δ + g^2_Σ) (1 + 2 N(ω_s)) , 
    ρ^q_ee   = g^2_δN(ω_s) +g^2_Σ(1 + N(ω_s))/(g^2_δ + g^2_Σ)(1 + 2 N(ω_s)).

Again using the expression of N(ω_s), and rearranging to get:

    P^q_g   = ρ^q_gg = 1/1 + e^-β (ħω_s  - μ  )
    
        P^q_e   = ρ^q_ee = e^-β (ħω_s - μ  )/1 + e^-β (ħω_s - μ  ),

where 

    μ = 1/βln(g^2_Σ e^βħω_s + g^2_δ/g^2_Σ e^-βħω_s + g^2_δ).





Equations <ref> and <ref> result in a grand canonical density matrix:

    ρ^q (∞) = Z(β, μ)^-1 e^- β (ω_s/2σ^z_q - μ/2σ^z_q)

where Z(β, μ) = Tr[e^-β (ω_s/2σ^z_q - μ/2σ^z_q)], and μ given in Eq. <ref> can be identified as the chemical potential of the qubit. 
The emergence of this chemical potential is a result of two parametric couplings between the qubit and the SNAIL-mon.
We can thus tune the chemical potential of the qubit in situ, and by extension its equilibrium state, by adjusting the ratio between the two processes' parametric rates, as shown in Fig. <ref>, as well as Fig. <ref> in the main text. 







§ DYNAMICS OF QUBIT THERMALIZATION

 
In the previous section, we discussed the equilibrium state of the qubit when parametrically coupled to a lossy mode. 
However, the result in Appendix <ref> does not give a description of the dynamics of the process. 
In order to extract the rates for the heating and cooling processes from the data, we need to have a model to match our experimental results. 
We start with a simple semi-classical phenomenological model for the two level system. 
Let P_g and P_e be the probability of the qubit being in the |g⟩ and |e⟩ state, respectively, and the rate jumping from |g⟩ to |e⟩ is Γ_ge, from |e⟩ to |g⟩ is Γ_eg.
The equations for the probabilities are given by:

    Ṗ_̇ġ = -Γ_ge P_g + Γ_eg P_e,

with the condition P_g + P_e = 1. 
The solution to this equation group is:

    P_g = c_0 e^-(Γ_ge + Γ_eg) t + Γ_eg/Γ_ge + Γ_eg,

where c_0 is a constant to be determined by initial conditions. 
This model demonstrates the exponential features seen in the heating and cooling data and can be used to extract the relevant rates.  


However, to have a full quantum understanding of the process, we need to consider the master equation of the qubit-SNAIL-mon system.
In order to obtain an analytical solution, we assume both the SNAIL-mon and qubit can each be described as two-level systems.
The assumption is valid for the qubit because of the transmon's non-zero anharmonicity and for the SNAIL-mon because the large decay rate κ_s reduces the likelihood of higher energy level occupation. 
We also assume that the two modes are at a physical temperature low enough that their natural heating rates can be ignored. 
The system Hamiltonian can be written with Pauli matrices (under the rotating wave approximation):

    H = g_Σ (σ_s^+σ_q^+ + σ_s^-σ_q^-) + g_δ (σ_s^-σ_q^+ + σ_s^+σ_q^-).

Again, these two terms are the effective two-body terms for the heating and cooling processes respectively. 
The dynamics of the system's density matrix ρ are given by the Lindblad master equation:

    ρ̇ = -i [H, ρ]  + κ_s D(σ_s^-) ρ + κ_q D(σ_q^-) ρ

where D(a) ρ = a ρ a^† - 1/2{ a^†a , ρ} is the term that models the interaction between a mode and its environment, and κ_i is the coupling rate of the i^th decay channel.


As an example, let us consider the single heating process. We keep only the SNAIL-mon mode decay term (since κ_s≫κ_q). 
Now we may solve a simpler Lindblad master equation: 

    ρ̇ = -i [H, ρ] + κ_s D(σ_s^-)ρ

with H = g_Σ (σ_s^+σ_q^+ + σ_s^-σ_q^-). 
We are interested in the dynamics of the qubit mode ρ_q = Tr_s(ρ). 
An analytical solution for the qubit population is available in this case, (for initial state ρ(0) = |0, g⟩⟨0, g|):

    P_g =    c_1 e^-κ_s/2 t + c_2 e^-κ_s+√((-16 g_Σ^2 + κ_s^2 ))/2
       + c_3 e^-κ_s - √((-16 g_Σ^2 + κ_s^2 ))/2,

where the coefficients are:

    c_1    = 8 g_Σ^2/16 g^2 - κ_s^2
    
        c_2    = 8 g_Σ^2 + κ_s(-κ_s+√(-16 g_Σ^2 + κ_s^2))/32 g^2 - 2 κ_s^2
    
        c_3    = 8 g_Σ^2 - κ_s(-κ_s+√(-16 g_Σ^2 + κ_s^2))/32 g^2 - 2 κ_s^2

In the limit g_Σ/κ_s≪ 1, this solution can be further reduced to a simpler form:

    P_g = e^- 4 g^2_Σ/κ_s t .

This agrees with Eq. <ref> found in the semi-classical case when Γ_eg = 0 and P_g(0) = 1 (since we ignore the qubit decay and start in the ground state).
We also note that these solutions give results that are consistent with Eqs. <ref> and <ref> at t=∞ for N(ω_s) = 0 (since we treat the bath to be cold).
We verify this by sweeping the drive voltages in both the heating and cooling experiment and extract the corresponding rate. 
In Fig. <ref>, we show the heating and cooling rate as a function of drive voltages. Both data fit nicely to a quadratic model, thus validating the use of the semi-classical model for rate extraction. 
We also note that, for the cases where we also see a non-trivial |f^+⟩ populations, a slightly more complicated semi-classical model that involves the qubit's third level is used for obtaining the rate:

    Ṗ_̇ġ   = -Γ_ge P_g + Γ_eg P_e, 
    Ṗ_̇ė   = Γ_ge P_g - (Γ_ef +  Γ_eg) P_e + Γ_fe P_f,

with P_g + P_e + P_f = 1.







§ NUMERICAL FITTING OF A THREE-LEVEL SYSTEM

 

The semi-classical model agrees with the data when the pumps only involves the |g⟩↔|e⟩ manifold.
However, the dynamics of the processes that involve pumps connected to all three levels cannot be nicely captured by it, as shown in Fig. <ref>.
Here we go beyond the theoretical model and numerically study our system to understand its behavior in the three-level manifold. 
The effective Hamiltonian under the rotating wave approximation (in the frame rotating at ω_ge) is:

    H    = -α q^† q q^† q +  (g_Σ_s, ef e^iα t s^† q^† + h.c.) 
       + (g_δ_s, ge s^† q + h.c.)

We note that in the ω_ge frame, there will always be time dependent terms that rotate at anharmonicity frequency α for the kete↔|f⟩ manifold drives.
The full Lindblad master equation is given by:

    ρ̇ = -i[H, ρ] + κ_s D(s) ρ + κ^↓_q D(q) ρ + κ^↑_q D(q^†) ρ.

Here in the dissipator terms, we consider both the decay of the SNAIL-mon and qubit (with collapse operator s and q at rate κ_s and κ^↓_q) and also the natural heating of the qubit due the finite temperature (with collapse operator q^† at rate κ^↑_q).
This equation is then simulated using QuTiP. 
We also take into account the up-pumped qubit thermalization during the measurement process, this is modeled by turning off the pump tone in the Hamiltonian and letting the system evolve for the time of measurement.



The parameters used for the simulation we show in Fig. <ref>(b) are as follows: α/2π = 197 MHz, κ_s/2π = 12.98 MHz, κ^↓_q/2π = 0.029 MHz, κ^↑_q/2π = 0.006 MHz and measurement time τ_msmt = 1.2 μs.
We note that the qubit decay rate used in the simulation is equivalent to a qubit with T_1 ∼ 4.5 μs, which is faster than the transmon's natural decoherence rate.
We attribute this to the faster decay rate of higher transmon levels and possible qubit coherence time reduction due to large pump photon numbers, which could invalidate the assumptions we used for the semi-classical model.









apsrev4-1 








 Paper Title: Real-time chiral dynamics at finite temperature from quantum simulation.


Methodological Foundations of Modern Causal Inference in Social Science Research
    Guanghui Pan[Department of Sociology, University of Oxford. 41 Park End Street, Oxford, the United Kingdom, OX1 1HH. Email: guanghui.pan@sociology.ox.ac.ukguanghui.pan@sociology.ox.ac.uk. Manuscript in progress, do not cite or redistribute without permission.]
    August 5, 2024
========================================================================================================================================================================================================================================================================





§ INTRODUCTION

In recent years, the exploration of quantum simulations for field theories <cit.> has garnered significant attention across many subfields of physics <cit.>. This growing interest is largely driven by the advancements in quantum computing technology, which promises to revolutionize our ability to tackle complex and computationally intensive problems. Traditional classical computers, despite their power, often fall short when dealing with the intricate dynamics and properties of quantum systems due to the exponential growth of computational resources required. On the other hand, quantum simulations hold the potential for breakthrough discoveries in areas such as high-energy physics, material science, and chemistry, aiding in understanding phase transitions <cit.>, solving quantum many-body problems <cit.>, and discovering new materials with unique properties <cit.>.

To explore these potential developments, we focus on the Schwinger model <cit.>, quantum electrodynamics (QED) in (1+1) dimensions. 
The Schwinger model provides an excellent testing ground for studying various phenomena, including confinement, dynamical mass generation, and anomaly-induced symmetry breaking. It offers the simplicity of fewer degrees of freedom while still encapsulating the rich physics of gauge theories, making it an ideal benchmark model for quantum simulations on both quantum and classical hardware <cit.>.


In this study, we explore the real-time dynamics of the chiral magnetic effect (CME) at finite temperature within the massive Schwinger model. The CME refers to the generation of an electric current along a magnetic field in systems where there is an imbalance between left-handed and right-handed fermions <cit.>. This imbalance is typically produced by the presence of a chiral chemical potential. To investigate the CME within the context of the Schwinger model, we employ a quench protocol as outlined in our previous study <cit.>. By introducing a chiral chemical potential through this quench process, we drive the system out of equilibrium and analyze the resulting vector currents and how they evolve over time. The Hamiltonian is adjusted to include the time-dependent chiral chemical potential, enabling the investigation of the CME within a quantum computing framework.

The interest in the study of the temperature dependence of the real-time CME stems from the following consideration. The CME is intrinsically a non-equilibrium, real-time phenomenon. However, when the external fields and the chiral chemical potential vary slowly with time, the CME response is driven by the fermion zero modes, and should thus be temperature-independent <cit.>. This independence of CME current on the temperature has been established in analytical computations <cit.>. However, under a rapid quench, the CME current receives contributions from excited fermion modes <cit.>, and can thus develop a dependence on the temperature. This temperature dependence  has not been evaluated before from first principles. In this paper, we will evaluate it using the methods of quantum simulation in massive Schwinger model at finite temperature.  

We employ the quantum imaginary time evolution (QITE) algorithm to study the thermal states and utilize the Suzuki-Trotter decomposition for real-time evolution. The QITE algorithm is essential for efficient quantum simulation on quantum circuits for several reasons, and various related algorithms have been developed <cit.>.
It enables the simulation of thermal states, crucial for studying phenomena like phase transitions at finite temperatures <cit.>. Since imaginary time evolution involves non-unitary operations, which cannot be directly implemented on quantum hardware, QITE approximates these operations with unitary ones, making it feasible on quantum circuits. Additionally, traditional classical algorithms struggle to prepare thermal states of many-body systems, while QITE allows for efficient exploration of these state spaces. Its versatility makes it applicable to a wide range of quantum systems and models, bridging theoretical models and practical implementations on quantum hardware. By iteratively applying small imaginary time steps and recalibrating the system, the QITE ensures the system gradually reaches the desired state where subsequent measurements are performed to extract physical observables.

Our motivation follows the successful quantum simulations performed on the (1+1)-dimensional Nambu–Jona-Lasinio (NJL) model <cit.>, where quantum algorithms provided insights into the chiral phase transition at finite temperature and chemical potential. Extending this line of research, our previous work on the quantum computational study of field theories has demonstrated the feasibility and efficacy of these simulations in delivering significant theoretical insights. This work on the Schwinger model bridges the study of simple models accessible by current quantum hardware and the more complex theories underpinning the standard model of particle physics. By exploring the unique properties and dynamics of the Schwinger model, we aim to provide insights into chiral dynamics in low-dimensional quantum field theories.

The rest of this article is organized as follows. In Sec. <ref>, we present the model and the quench protocol that we study in this work. In Sec. <ref>, we describe our algorithm of the quantum imaginary time evolution to implement the thermal states and subsequent expectation evaluation to extract physical observables at finite temperature. In Sec. <ref>, we show the results from our quantum simulations, where we discuss the real-time chiral dynamics of the current at finite temperatures, the phase diagram of the Schwinger model, and the temperature dependence of the electric charge (axial current). Finally, the article is concluded in Sec. <ref>. 




§ THE MASSIVE SCHWINGER MODEL WITH THE TOPOLOGICAL TERM
 


 §.§ The Schwinger Hamiltonian

The action of the massive Schwinger model <cit.> with θ term in (1+1)-dimensional Minkowski space is

    S = ∫ d^2x[-1/4 F^μν F_μν + gθ/4πϵ^μνF_μν + ψ̅(iD-m)ψ],

with D=γ^μ(∂_μ + i gA_μ).
Here, A_μ is the U(1) gauge potential, E=Ȧ_1 is the corresponding electric field, ψ is a two-component fermion field, m is the fermion mass, and γ^μ are two-dimensional γ-matrices satisfying the Clifford algebra. When g≠0, the θ-term induces the classical background electric field, which breaks the parity symmetry. The gauge field A_μ and the coupling constant g have mass dimensions 0 and 1, respectively.
By a chiral transformation, ψ→ e^iγ^5θψ and ψ̅→ψ̅e^iγ^5θ, the action is transformed to 

    S = ∫ d^2x[-1/4 F^μν F_μν  + ψ̅(iγ^μ D_μ-me^iγ^5θ)ψ].

The action is invariant under this transformation only up to the boundary term.
It is evident from eq. (<ref>) that the massive theory with a positive mass (m>0) at θ=π is equivalent to the theory at θ=0 but with a negative mass (-m).  

The Hamiltonian in the temporal gauge A_0=0 is given by

    H=∫ dz [    
        E^2/2
        -ψ̅(iγ^1∂_1 - gγ^1A_1 - me^iγ^5θ)ψ]  ,

where the space-time coordinate is labeled by x^μ=(t,z). We denote the Pauli matrices as X, Y, and Z, and use the following convention for the Dirac matrices: γ^0 = Z, γ^1 = i Y, γ^5=γ^0 γ^1 = X.
In (1+1) dimensions, the axial charge density Q_5(x)≡ψ̅(x)γ^5γ^0ψ(x) 
and the vector current density J(x)≡ψ̅(x)γ^1ψ(x) are related 
by Q_5(x) = -J(x). Moreover the vector charge density Q(x)≡ψ̅(x)γ^0ψ(x) and the 
axial current density J_5(x)≡ψ̅(x)γ^5γ^1ψ(x) are related by Q(x) = J_5(x).



 §.§ Real-time chiral dynamics at finite temperature


As we have mentioned in the introduction, the chiral magnetic effect refers to the generation of an electric current along a magnetic field in a system with an imbalance in the number of left-handed and right-handed fermions, induced by the presence of a chiral chemical potential. In the Schwinger model, when the chiral quench is applied, the system is driven out of equilibrium, creating a non-zero chiral chemical potential μ_5. This chiral chemical potential represents an asymmetry between left-handed and right-handed fermions.

Here, we consider the μ_5-quench <cit.> protocol: The system Hamiltonian with θ=0 is prepared at time t<0. Then, starting at t=0, the Hamiltonian rotates the θ angle according to θ=-2μ_5t corresponding to constant chiral chemical potential (i.e. θ̇ = -2μ_5.). The Hamiltonian after the quench is modified as

    H'= dz [    
        E^2/2
        -ψ̅(iγ^1∂_1 - gγ^1A_1-γ^1θ̇/2 - me^iγ^5θ)ψ]  .

The real-time evolution of an operator O can be evaluated by the time-ordered integral 

    O(t)=𝒯[e^i∫_0^tdt'H'(t')]O𝒯[e^-i∫_0^tdt'H'(t')],

and the thermal expectation value of O evaluated at time t at finite temperature T is

    ⟨ O(t)⟩_β=[e^-β HO(t)]/ [e^-β H],

where H is the unquenched Hamiltonian in eq. (<ref>) at θ=0 and β=1/T is the inverse of the temperature of the system.

Notably, this evolving θ = -2μ_5 t after the quench at t = 0 indicates a dynamical chiral chemical potential, driving the system into a non-equilibrium state where left-handed and right-handed fermions are imbalanced. In the presence of this chiral chemical potential, a vector current J (which is typically zero before the quench) is induced. The vector current's time evolution at a finite β, influenced by the quench, aligns with the characteristic behavior observed in the CME at a finite temperature.





 §.§ The lattice Hamiltonian on the qubits


To discretize our Hamiltonian, we use staggered fermions <cit.> on a lattice such that

    ψ(x≡ na) = 1/√(a)(
        [          χ_2n;        χ_2n+1 ]) ,

where a is the finite lattice spacing. Then the lattice Hamiltonian corresponding to eq. (<ref>) is <cit.> 

    H    = a g^2/2∑_n=1^N-1 L_n^2
     -i/2a∑_n=1^N-1[χ^†_n+1 U_nχ_n-χ^†_nU_n^†χ_n+1]
    + mcosθ∑_n=1^N (-1)^n χ^†_nχ_n
       + imsinθ/2∑_n=1^N-1 (-1)^n [χ^†_n+1 U_nχ_n-χ^†_nU_n^†χ_n+1]-θ̇/4a∑_n=1^N-1[χ^†_n+1χ_n+χ^†_nχ_n+1] ,

where L_n is the electric field operator satisfying the Gauss' law constraint

    L_n-L_n-1 =  χ_n^†χ_n-1-(-1)^n/2.

Using the Gauss' law constraint, we eliminate the link fields U_n by the gauge transformation,

    χ_n→ g_nχ_n,  χ^†_n→χ^†_n g_n^†,  
        U_n → g_n+1U_n g_n^†,

with

    g_1=1,    g_n = ∏_i=1^n-1U^†_i.


For the purpose of quantum simulation, we rewrite the lattice Hamiltonian in the spin representation using the Jordan–Wigner transformation <cit.>:

    χ_n=X_n-iY_n/2∏_i=1^n-1(-i Z_i).

The full Hamiltonian in the qubit representation considered in this study is

    H=   ∑_n=1^N-1(1/4a-m/4(-1)^nsinθ)[X_n X_n+1+Y_n Y_n+1]
         +mcosθ/2∑_n=1^N(-1)^n Z_n
       +a g^2/2∑_n=1^N-1L^2_n-θ̇/8a∑_n=1^N-1[X_n Y_n+1-Y_n X_n+1].

Moreover, the local vector and axial charge densities are, respectively <cit.>, 

    Q_n ≡    ψ̅γ^0ψ = Z_n+(-1)^n/2a,
    
        Q_5,n≡    ψ̅γ^5γ^0ψ = X_nY_n+1-Y_nX_n+1/4a .

For later convenience, we introduce the average vector and axial charge operators that sum over all lattice sites,

    Q≡a/N∑_n=1^NQ_n,     
        Q_5≡a/N-1∑_n=1^N-1Q_5,n.

Notably, Q commutes with the Hamiltonian while Q_5 does not. With the boundary condition L_0=0, the Gauss' law constraint, eq. (<ref>), leads to the solution L_n =  a∑_j=1^n Q_j.

Notably, one can project the Hamiltonian to different charge sectors to work with states with specified quantum numbers. In the case of the Schwinger model, the projection operator P to the charge neutral sector (Q=0) can be written as P_Q=0 = ∑_n|n⟩⟨n| where n are all the bitstrings with even number of 1s and 0s. On the qubits, P_Q=0 can be always written as the sum of Pauli-Z operators such that H_Q=0=P_Q=0 H P_Q=0^† is the projected Hamiltonian to the Q=0 sector[For example, at N=4, the projection operator is

    P_Q=0 = P_Q=0^† = 0.375 (1 + Z_1 Z_2 Z_3 Z_4) - 0.125 ( Z_3 Z_4 + Z_2 Z_4 + Z_2 Z_3 + Z_1 Z_4 + Z_1 Z_3 + Z_1 Z_2 )
]. Here the coefficients are determined in such a way that P_Q=0^2=P_Q=0. For the numerical simulation results presented in this paper, we always work in the Q=0 sector. This is because 
the Schwinger model spectrum does not have charged states due to confinement at all scales.
Therefore, when we say Hamiltonian, in what follows, we always mean the charge neutral Hamiltonian H=H_Q=0. 



§ QUANTUM SIMULATION OF QED_2 AT FINITE TEMPERATURE



 §.§ Quantum imaginary time evolution

Quantum imaginary time evolution (QITE) is an efficient way to investigate eigenstates and thermal states on a quantum computer <cit.>. Here, we briefly review the QITE algorithm used in this work for preparing the thermal state and extracting the finite temperature observable.

In QITE, one attempts to approximate the action of the imaginary time evolution
operator on a state | Ψ⟩ with a parameterized unitary
operation:

    c(Δβ)^-1/2 e^-ΔβĤ | Ψ⟩≈ e^-i ΔβÂ(a⃗)| Ψ⟩,

where Δβ is a small time step and c(Δβ)^-1/2 is a normalization coefficient approximated
by 1-2Δβ⟨Ψ | Ĥ | Ψ⟩. The real-time evolution operator Â(a⃗)
is given by a linear combination of N_μ Pauli string:

    Â(a⃗) = ∑_μ^N_μ a_μP̂_μ.

Here, P̂_μ=∏_lσ_μ_l,l is a Pauli string and the subscript μ of a_μ labels the various Pauli strings, and their coefficients a_μ are solved from the linear equation ( S+ S^T)  a= b, where the matrix S and vector b are defined by

    S_μν   =⟨Ψ(β)|P̂_ν^†P̂_μ|Ψ(β)⟩ ,
    
    b_μ   =-i/√(c(Δβ))⟨Ψ(β)| (ĤP̂_μ+P̂_μ^†Ĥ)|Ψ(β)⟩ .

The QITE procedure is repeated for a specified
number of time steps to reach a target total evolution time. Using this iterative approach, we are able to evolve an initial quantum state |Ψ_k(0)⟩ under the unitary operator e^-iΔβ A to any imaginary time β by trotterization:

    |Ψ_k(0)⟩→ |Ψ_k(β)⟩ +𝒪(Δβ) .

Note the sum of |Ψ_k(β)⟩ over all basis states approximates e^-βĤ/(e^-βĤ) at finite temperature T=1/β in the sense that it gives the thermal expectation of an observable Ô:

    1/Z∑_k⟨Ψ_k(β/2)|Ô|Ψ_k(β/2)⟩≈⟨Ô⟩_β=1/Z (e^-βĤÔ),

where Z= (e^-βĤ)=∑_k⟨Ψ_k(β/2)||Ψ_k(β/2)⟩ is the canonical partition function, which is also the sum of the normalizations over all the quantum states.





 §.§ Real-time evolution at finite temperature


Studying the dynamics of the quantum system at finite temperatures is usually complicated. In this work, we are interested in the real-time evolution of the thermal mixed state at finite temperatures of the Schwinger Hamiltonian. Using another evolution Hamiltonian H' that is not equal to the Hamiltonian H of the thermal system, we define a quench process by the real-time evolution of the operator. See eq. (<ref>) for our quench protocol.

To implement this protocol on the quantum circuit, we first use the QITE algorithm to obtain the collection of the |Ψ_k(β/2)⟩ states to represent the thermal state at finite temperature T=1/β. Then, we simply evolve each |Ψ_k(β/2)⟩ in real time as usual:

    |Ψ_k(t,β/2)⟩=𝒯[e^-i∫_0^tdt'Ĥ'(t')]|Ψ_k(0,β/2)⟩.

The final thermal expectation at real-time t is

    ⟨Ô(t)⟩_β=1/Z∑_k⟨Ψ_k(t,β/2)|Ô|Ψ_k(t,β/2)⟩.

Compared with eq. (<ref>), we see that thermal expectation evaluated on the quantum circuit is associated with the states instead of observables. Nevertheless, the two methods are entirely equivalent, which can be seen in our comparison between the quantum simulation and exact diagonalization results. To perform the real-time evolution, we trotterize the time into small time steps using second-order Suzuki-Trotter product formula <cit.>, which gives good convergence.




§ NUMERICAL RESULTS




 §.§ Thermal state preparation

To study the real-time evolution of physical properties at finite temperatures, we must first prepare the initial thermal state at the desired temperatures. In this work, we use the QITE algorithm developed using Qiskit <cit.> in a proceeding work <cit.> for the thermal medium preparation. Specifically, the unitary evolution is implemented using PauliEvolutionGate library with SuzukiTrotter method, and the thermal expectation is evaluated with the Sampler class on the statevector quantum simulators. In particular, we use the QITE to prepare thermal states with four qubits from β/a=0 (aT=∞) to β/a=4.0 (aT=0.25) for the three masses ma=0.25, 0.5, 4.0, and at coupling g=a^-1. A universal trotter step size Δβ = 0.01 is used.



To measure the validity of the quantum simulation, we use the fidelity between two density matrices σ and ρ:

    F(σ,ρ)=(√(√(σ)ρ√(σ)))^2.

Here, we take σ=e^-β H/Z to be the exact density matrix calculated via exact diagonalization and ρ=∑_k|Ψ_k(β/2)⟩⟨Ψ_k(β/2)|/Z for the thermal mixed state prepared by the QITE algorithm. In Fig. <ref>(left), we show the fidelity of these two density matrices as a function of the QITE imaginary evolution parameter β (i.e., inverse temperature). 
For all three cases of different masses, the fidelity is between 0.999 and 1.0, very close to 1, suggesting that the thermal state prepared by QITE is a good practical approximation of the corresponding thermal equilibrium state. In particular, we see a comparatively slight decrease of fidelity in the early β (high temperature) region for the heaviest mass ma=4.0, which is due to the insufficiently small step size Δβ = 0.01 used in the simulation for the heavy mass.

In Fig. <ref>(right), we show the purity of the mixed quantum state ρ at different temperatures and masses. Purity is defined by [ρ^2] and is an index used to determine how much a quantum state is mixed, with values equal 1 indicating a pure state, and values approaching 0 signifying a mixed state. For every mass, we expect that the purity to approach 1 as the temperature approaches 0 (equivalently, as β approaches infinity), since the state is pure at zero temperature (i.e., the ground state). For smaller mass values, such as ma = 0.25, the curve shows that the purity decreases more slowly with temperature, indicating smaller mass states transition to mixed states more gently as the temperature rises. In constrast, as the mass ma increases, the purity drops more rapidly. Despite these differences, all curves will eventually approach a lower purity limit as the temperature continues to increase, showing the natural tendency of quantum systems to shift towards mixed states under a finite temperature. 




 §.§ Real-time evolution of the charge and the current at finite temperature


Now that the thermal state is prepared using the QITE protocol, we study real-time evolution of the total electric and axial charge densities for the system at a selected temperature with both light and heavy masses. Our simulations are conducted using the exact statevector quantum simulator provided by Qiskit <cit.> and compared to the T = 0 limit obtained from exact diagonalization. Specifically, we perform our simulation using eqs. (<ref>) and (<ref>), following the quench protocol provided in Sec. <ref> at various chiral chemical potentials.


The real-time evolution is compared against the zero-temperature limit using exact diagonalization, serving as a benchmark for the results obtained from the quantum simulator at finite temperatures. We first note that the results at T=0 are consistent with the results in <cit.>, which performed the study under the periodic boundary condition at g=0 while we work with the open boundary condition at g≠0. We also observe that ⟨ Q⟩ =0 at all finite temperatures and it does not evolve over time, which is expected. The quantum simulation uses four qubits and applies the QITE algorithm for thermal state preparation. These methods allow for accurate modeling of thermal states and real-time evolution, offering a robust framework for understanding the underlying physics. The second order Suzuki-Trotter product formula <cit.> is used for the real-time evolution, with step sizes dt varying based on the masses: 0.1 for light masses and 0.01 for heavy masses. 



Fig. <ref> shows the real-time evolution of the thermal average of the axial charge ⟨ Q_5(t) ⟩_β for various temperatures, aT = 10.0,1.0,0.25. The mass takes values of ma=0.25, 1.0, 4.0 to examine how light and heavy masses affect the evolution of the vector and axial charge densities, showing how it evolves under different temperatures. The results highlight the role of a chiral chemical potential μ_5 and include both exact solutions and approximate results using quantum simulation. To confirm consistency with theoretical expectations, we show that the total vector charge is conserved and does not depend on time. 

In studying the chiral dynamics in the Schwinger model, our results share several important similarities with our previous results of nonlinear chiral magnetic waves (CMWs) <cit.>, particularly in observing how the oscillation of axial charges is influenced by mass. Here, chiral magnetic waves are collective excitations in a (1+1)-dimensional chiral medium under the influence of a magnetic field. These waves are a direct consequence of the interplay between axial (chiral) and vector (electric) currents in the presence of magnetic fields. Notably, the tendency that heavier mass results in faster oscillations of the axial charge is common <cit.>.
In both studies, it is observed that the axial charge density oscillates more rapidly with increasing mass. In the nonlinear CMWs, the nonlinearity arises because, in massive Schwinger model, the frequency of axial charge oscillations becomes significantly higher than that of the vector charge. This results in the axial charge oscillating rapidly while being nearly confined within static electric dipoles, a phenomenon referred to as the “thumper” solution  <cit.>. The interplay between the excited state (causing rapid oscillations) and the ground state (with almost static behavior) creates a non-linear dynamic in the system.




In Fig. <ref>, we present a comparison between our simulation results and exact diagonalization, observing good agreement when sufficiently small time steps are used. This figure illustrates how accurately our simulation can predict the system's behavior, validating our approach. Furthermore, the rapid oscillations observed in the simulation at a larger mass (ma = 4) also verified our expectation. These oscillations are attributed to the interplay between different energy eigenstates, and their amplitude increases as the temperature rises. The consistency between the simulation in Fig. <ref> underscores the accuracy of our model in capturing the dynamic properties of the system at varying conditions.

To study the dynamical interplay between finite temperature and real-time evolutions, we also present the density plots for ⟨ Q_5(t) ⟩_β in Fig. <ref>. The axial charge oscillates as a function of time t at different temperatures T, with increasing frequency for the heavier mass. Notably, the thermal state is first obtained using the QITE algorithm at finite β/a=1/(aT) values in the range of [0, 4], and then we obtain the density plot versus aT to the range of [∞, 0.25] with interpolation to T=0 temperature. The number of modes for the time oscillations is directly related to the mass in the Schwinger Hamiltonian. 



With the goal of extrapolating our results in the continuum, we perform the exact diagonalization with N=10 qubits in Fig. <ref>. Importantly, we have direct access to the finite temperature thermal state with exact matrix exponentiation using eq. (<ref>), and our exact results are in terms of aT and t. Here, we show the density plot of the thermal expectation value of ⟨ Q_5(t)⟩_T (with temperature in units of aT in the range of [0, 4]). Comparing to results simulated on the quantum simulator shown in Fig. <ref>, we observe the same behavior for both ma=1.0 and 4.0 cases, despite having many more qubits. Quantitatively, we can also observe good agreement between the quantum and classical simulations by counting the oscillation modes and the peak intensity values in the density plots.



§ CONCLUSION

This paper investigates the real-time dynamics of the chiral magnetic effect (CME) at finite temperature in the context of the massive Schwinger model, quantum electrodynamics (QED) in (1+1) dimensions. The study utilizes quantum simulation techniques, combining the quantum imaginary time evolution (QITE) algorithm for thermal state preparation and trotterization for real-time evolution, to examine the thermal properties of the system and the transitions induced by a chiral chemical potential via a quench process.

Our results illustrate the dynamic interplay between the fermion mass and the real-time dynamics of the axial charge. At zero temperature, the corresponding dynamics has been investigated in our previous studies   <cit.>. 
The new result in this paper is the temperature dependence of the real-time CME response. Specifically, at temperatures small compared to the fermion mass, the CME response is similar to the one found previously at zero temperature. However, at larger temperatures, the oscillations of the axial charge induced by the CME are dampened by thermal effects. This can be understood as a result of the contribution from excited fermion modes to the real-time CME response, that may depend on the temperature <cit.>. 

In addition, at large fermion mass the axial charge oscillations become very non-linear. The origin of this  non-linearity can be explained in the following way: in bosonized description of the massive Schwinger model, fermion mass induces non-linear interactions among the bosons, so the chiral magnetic waves become highly non-linear <cit.>.

Finally, our study of CME dynamics suggests that quantum simulations can be used to study other real-time phenomena at finite temperatures in gauge theories that form the basis of the Standard Model. 




§ ACKNOWLEDGEMENT

We thank Meijian Li for helpful comment on the manuscript. We are also grateful to Andrea Palermo and 
Ismail Zahed for useful discussions. This work was supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under Contract No.DE-SC0012704 (KI, DK, FZ), the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, Grants Nos. DE-FG88ER41450 (DK), DE-SC0012704 (DK) and DE-SC0011090 (FZ), and the  National Science Foundation under grant No. PHY-1945471 (ZK). WQ is supported by the European Research Council under project ERC-2018-ADG-835105 YoctoLHC; by the Spanish Research State Agency under project PID2020-119632GB-I00; by Xunta de Galicia (Centro singular de investigacion de Galicia accreditation 2019-2022), by European Union ERDF, and by the Marie Sklodowska-Curie Actions Postdoctoral Fellowships under Grant No. 101109293.


JHEP




 Paper Title: Optimally robust shortcuts to population inversion in cat-state qubits.


	
    
    
    Department of Physics, Fuzhou University, Fuzhou 350116, China
    
    Department of Physics, Fuzhou University, Fuzhou 350116, China
    
    Department of Physics, Fuzhou University, Fuzhou 350116, China
    yehong.chen@fzu.edu.cn
    Department of Physics, Fuzhou University, Fuzhou 350116, China
    Fujian Key Laboratory of Quantum Information and Quantum Optics, Fuzhou University, Fuzhou 350116, China
    Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
    
    Department of Physics, Fuzhou University, Fuzhou 350116, China
    Fujian Key Laboratory of Quantum Information and Quantum Optics, Fuzhou University, Fuzhou 350116, China

    § ABSTRACT 
        Cat-state qubits formed by photonic coherent states are a promising candidate for realizing fault-tolerant quantum computing. Such logic qubits have a biased noise channel that the bit-flip error dominates over all the other errors. 
        In this manuscript, we propose an optimally robust protocol using the control method of shortcuts to adiabaticity to 
        realize a nearly perfect population inversion in a cat-state qubit. 
        We construct a shortcut based on the Lewis-Riesenfeld invariant and examine the stability versus
        different types of perturbations for the fast and robust population inversion. 
        Numerical simulations demonstrate that the population inversion can be mostly insensitive to 
        systematic errors in our protocol. Even when the parameter imperfection rate for bit-flip control is 20%, 
        the final population of the target state can still reach ≥ 99%. 
        The optimally robust control provides a
        feasible method for fault-tolerant and scalable quantum computation.
    
    
    Optimally robust shortcuts to population inversion in cat-state qubits
    Yan Xia
    August 5, 2024
======================================================================



    

§ INTRODUCTION

    Quantum computers promise to drastically outperform classical computers on certain problems, such as factoring, (approximate) optimization, boson sampling, or
    unstructured database searching <cit.>. Building a large-scale quantum computer requires qubits that can  be protected from errors, i.e., utilizing quantum 
    error correction. During the past decades, many strategies using physical and logical qubits for quantum error correction have been developed. Noting that 
    quantum error correction with physical qubits usually requires huge physical resource overhead, this makes it difficult to scale up the number of 
    qubits for a large-scale quantum computer <cit.>. This is why in recent years, much attention has been paid to logic qubits formed by bosonic codes <cit.>, which 
    allow quantum error correction extending only the number of excitation instead of the number of qubits.
    
    A promising alternative with the potential to realize quantum error correction beyond the break-even point involves encoding logical qubits in continuous 
    variables <cit.>, such as coherent states. This gives rise to the cat-state codes, which are formed by even and odd coherent states of a single optical
    mode <cit.>. The cat-state qubits preserve the noise bias that experience only bit-flip noise, reducing the number of building blocks of layers for error 
    correction <cit.>. Moreover, the first experiment <cit.> realizing cat-state qubits showed a strong suppression of frequency fluctuations due to 1/f noise <cit.>. 
    All these make the cat-state qubits promising for hardware efficient universal quantum computing.
    
    In an implementation of quantum computation, high-fidelity single- and two-qubit quantum gates are essential elements of quantum computation because quantum 
    algorithms are usually designed as a sequence of such simple quantum gates <cit.>. Though several experiments have realized the control of cat-state qubits <cit.>, the robust control of a single cat-state qubit is still a problem to be 
    solved. In this manuscript, we propose an optimally robust shortcuts to adiabatic protocol for controlling a cat-state qubit. Shortcuts to adiabaticity <cit.> 
    are a series of protocols mimicking adiabatic dynamics beyond the adiabatic limit and have been widely applied for quantum state engineering. One of 
    the more prominent of these protocols is the method of “invariant-based reverse engineering” <cit.>, which can construct shortcuts only by redesigning 
    system parameters without destroying the initial form of the system Hamiltonian. This provides an alternative control method for the cat-state qubits with 
    large amplitudes because such qubits can be manipulated along only one direction on the Bloch sphere <cit.>. Moreover, the invariant-based reverse engineering 
    is compatible with various quantum optimized control techniques <cit.>. One can thus optimize the parameters to realize a high-fidelity population inversion 
    of a cat-state qubit.
    
    This manuscript is organized as follows. In Sec. <ref>, we present the model and the effective Hamiltonian
    for the protocol. The protocol of constructing shortcuts to adiabatic passage is given in Sec. <ref>.
    In Sec. <ref>, we analyze the systemic error sensitivity of the cat-state qubit. Then, in Sec. <ref> the optimal protocol to minimize the systemic error sensitivity is presented. Moreover, we discuss the influence of single-photon loss and pure dephasing on the protocol in Sec. <ref>. Finally,
    the conclusions are given in Sec. <ref>.
    
    

§ MODEL AND EFFECTIVE HAMILTONIAN

    Considering that a Kerr-nonlinear resonator with frequency ω_c is driven by a single-mode, two-photon excitation, where the driving frequency for the two-photon excitation is 
    twice the resonator frequency. In the rotating-wave approximation, the system Hamiltonian is given by (hereafter ħ=1)
    
    H_Kerr = -K a^† 2a^2 + P(a^†2 + a^2) .

    In the above expression, a and a^† are the annihilation and creation operators for photons, K is the strength of the 
    Kerr-nonlinearity, and P is the strength of the two-photon drive. 
    
    We can observe that Eq. (<ref>) is written in the rotating frame. In this frame, the 
    simplified Hamiltonian is described as having quasi-energy eigenstates with negative energies. Specifically, by applying the displacement transformation 
    D(±α) = exp [±α (a^† - a) ] to H_Kerr, the Hamiltonian in Eq. (<ref>) becomes 
    
    H^'   = D(±α) H_Kerr D^†(±α)    =-4K α^2a^†a - K a^† 2a^2∓ 2K α (a^† 2a + H.c.),

    where α = √(P/K). The vacuum |0⟩ is exactly an eigenstate of H^'. Therefore, the coherent states 
    D(±α)|0⟩=|±α⟩ or, equivalently, their superposition states 
    
    | 𝒞_±⟩ = 𝒩_± (| α⟩± | -α⟩),

    are the degenerate eigenstates of H_Kerr, where 𝒩_± = 1 / √(2(1 ± e^-2|α|^2)) are normalized coefficients. 
    In the limit of large α, one can obtain α^2 ≫α^1, α^0. Thus, Eq. (<ref>) is approximated by 
    H^'≃ -4K α^2a^†a, which is the Hamiltonian of a (inverted) harmonic oscillator. Therefore, in the original 
    frame, the eigenstates of H_Kerr are eigenstates of the parity operator (see Fig. <ref>). The first-excited states can be approximately 
    expressed as two orthogonal states |ψ_±^e,1⟩ = N_±^e,1 [D(α) ± D(-α)] |n=1⟩, where N_±^e,1 are normalized 
    coefficients and |n⟩ are Fock states. The energy gap between the cat states subspace 𝒞 and |ψ_±^e,1⟩ can be approximated as 
    ω_gap≃ 4K α^2.
        
    

    As shown in Fig. <ref>, the cat states subspace 𝒞 is separated from the rest of the Hilbert space 𝒞_⊥ by an energy gap of 
    approximately ω_gap≃ 4K α^2. In the limit of large α, the action of a can only flip the two cat states 
    | 𝒞_±⟩, i.e.,
    
    a | 𝒞_±⟩≃α | 𝒞_∓⟩ .

    Now, in the interaction picture, we add a control Hamiltonian <cit.>
    
    H_c(t)=   -E_J(t)/2{D[iφ_aexp(i ω_ct)]+ H.c.}   +ϵ(t) (a^†+a).

    A possible implementation of this control Hamiltonian is the superconducting circuits <cit.>
    by considering the Kerr-nonlinear 
    mode coupled capacitively to a
    Josephson junction and assuming that other modes (including the junction mode) are never excited.
    Accordingly, the time-dependent parameter E_J(t) is the effective Josephson energy and φ_a is the phase. 
    A single photon driving with time-dependent amplitude ϵ(t) is also applied to the system.
    For E_J(t),ϵ(t)≪ω_c, the effective Hamiltonian under the rotating-wave approximation becomes
    
    H'_c(t) = E_Je^-φ_a^2/2∑_m=0^∞L_m(φ_a^2)|n⟩⟨ n| + ϵ (a^†+a),

    where L_m(*)  is the Laguerre polynomial of order m.  Hereafter, for simplicity, we omit the explicit time
    dependence of parameters, e.g., E_j(t)→ E_J and ϵ(t)→ϵ.
    
 
       
    The total Hamiltonian now becomes
    H(t) = H_Kerr + H'_c(t). We can use the cat states | 𝒞_±⟩ to define the Pauli matrices, 
    
    σ_x = σ_+ + σ_-,   σ_y = i(σ_- - σ_+),   σ_z = σ_+σ_- - σ_-σ_+,

    where σ_+ = | 𝒞_+⟩⟨𝒞_- | is the raising operator and 
    σ_- = | 𝒞_-⟩⟨𝒞_+ | is the lowering operator. When
    
    E_J, ϵ≪ω_gap,

    the evolution of the system can be restricted to the cat-state subspace 𝒞, i.e., constructing a
   cat-state qubit as shown in Fig. <ref>. Projecting the system onto the cat-state subspace, the effective 
    Hamiltonian H'_c(t) can be represented as
    
    H_eff
                =E_Jexp[-(φ_a-2α)^2/2]/-2√(παφ_a)σ_z + ϵ (α^* + α) σ_x.

    We can choose φ_a=2α and rewrite the effective Hamiltonian in the matrix form as
    
    H_eff = 1/2[   Δ Ω_R; Ω_R  -Δ ],

    where the time-dependent parameters are Δ =-E_J/(α√(2π)) and Ω_R = 2 (α^* + α) ϵ.



    

    

§ NON-ADIABATIC EVOLUTION BASED ON THE LEWIS-RIESENFELD INVARIANTS

    Following the method of invariant-based reverse engineering <cit.>, we introduce a dynamical invariant I(t), which satisfies
    
    i∂/∂ tI(t)-[H_ eff(t),I(t)]=0.

    Then, the solution of the time-dependent Schrödinger equation
    
    i∂/∂ t|ψ(t)⟩=H_ eff(t)|ψ(t)⟩,

    can be expressed by a superposition of the eigenstates |ϕ_n(t)⟩ of I(t) as
    
    |ψ(t)⟩=∑_nc_nψ_n(t).

    Here, ψ_n(t)=e^iR_n(t)|ϕ_n(t)⟩ and c_n are time-independent amplitudes determined by the initial state, and R_n(t) are the 
    Lewis-Riesenfeld phases defined as
    
    R_n(t)=∫_0^t⟨ϕ_n(t')|i∂/∂ t'-H_ eff(t')|ϕ_n(t')⟩ dt'.




    Following the previous researches <cit.>,  for the Hamiltonian in Eq. (<ref>), the invariant I(t) can be given by 
    
    I(t)=1/2[       cosγ  sinγ e^iβ; sinγ e^-iβ      -cosγ ],

    where γ and β are two time-dependent dimensionless parameters to be determined later. The eigenstates of the Lewis-Riesenfeld invariant I(t) 
    can be thus derived as <cit.>
    
    | ϕ_+(t) ⟩   = cos(γ/2) e^i β | 𝒞_-⟩ + sin(γ/2) | 𝒞_+⟩, 
            | ϕ_-(t) ⟩   = sin(γ/2) | 𝒞_-⟩ - cos(γ/2) e^-i β | 𝒞_+⟩.

    According to Eq. (<ref>), we obtain the expressions of the time-dependent parameters Ω_R and Δ as 
    
    Ω_R    = γ̇ / sinβ,  Δ   = Ω_R γcosβ - β̇.

    To achieve the flipping of the cat states | 𝒞_±⟩, one needs to set the boundary conditions Ω_R(0) = Ω_R(t_f) = 0, and 
    
    γ(0) = π,    γ(t_f) = 0,    γ̇(0) = 0,    γ̇(t_f) = 0.

    We can arbitrarily choose the values of β(0) and β(t_f), according to Eq. (<ref>), when β approaches (n+1/2)π, the resulting of 
    |Ω_R| is minimized, imposing 
    
    [ β(0) = -π / 2, β(t_f/2 ) = -π / 2, β(t_f) = -π / 2,;           β̇(0) = π / (2t_f), β̇(t_f) = π / (2t_f). ]


    
  
    To satisfy the boundary conditions given in Eqs. (<ref>) and (<ref>), we can assume
    
    γ(t) = ∑_i=0^3 a_i t^i,       and     β(t) = ∑_i=0^4 b_i t^i,

    and thus determine their values as shown in Fig. <ref>(a). Accordingly, we can obtain E_J and ϵ [see Fig. <ref>(b) and 
    Fig. <ref>(c)]. Such parameters allow a population inversion from the even cat state
    |𝒞_+⟩ to the odd cat state |𝒞_-⟩ through a nonadiabatic passage. This is determined 
    by solving the Schrödinger equation i|ψ̇(t)⟩=H|ψ(t)⟩ of the total Hamiltonian 
    
    H=H_ Kerr+H_c(t).

    In Fig. <ref>(d), we display the dynamical evolution of the system when the initial state is |𝒞_+⟩.
    An almost perfect population inversion (P_-≃ 99.9% at t=t_f) is obtained as shown in the figure, where
    the populations of the states |𝒞_+⟩ and |𝒞_-⟩ are defined
    as 
    
    P_±(t)=|⟨ C_±|ψ(t)⟩|^2.


    


    
   

    

§ SYSTEMATIC ERROR SENSITIVITY

    Now, we consider the influence of systematic errors on the system dynamics. The ideal undisturbed Hamiltonian is H_eff. For systematic errors, the Hamiltonian in 
    actual experiments becomes H_01 = H_0 + μH_1, which also satisfies the Schrödinger equation
    
    i d/dt | ψ(t) ⟩ = ( H_0 + μH_1 ) | ψ(t) ⟩,

    where H_0 = H_eff, and H_1 is the disturbed Hamiltonian. 
   For simplicity, we assume that errors affect the pulse amplitude but not the
   detuning. The disturbed Hamiltonian H_1 under this assumption is
   
    H_1=ϵ(α^*+α)σ_x.

In the original control Hamiltonian H_c, this disturbed Hamiltonian corresponds to parameter deviations in
the single-photon drive ϵ(a+a^†).
   Then, we define the systematic error sensitivity as
    
    q_s = -1/2∂^2 P_-^2/∂μ^2|_μ=0 = -∂ P_-/∂( μ^2 )|_μ=0,

    where P_- is the population of the state | 𝒞_-⟩ at the final time t_f.



    Using perturbation theory up to O(μ^2), we obtain
    
    | ψ(t_f) ⟩ =    | ψ_0(t_f) ⟩ 
                                - iμ∫_0^t_f dt U_0(t_f,t) H_1(t) | ψ_0(t) ⟩ + ⋯,

    where | ψ_0(t_f) ⟩ is the solution without perturbation, and U_0(t_f,t) is the unperturbed time evolution operator. We assume that the protocol
    without errors (μ = 0) works perfectly, i.e. | ψ(t_f) ⟩ = |ϕ_+(t_f⟩)=e^i β(t_f) | 𝒞_- ⟩ with real β(t_f). Thus
    
    P_- ≈ 1 - μ^2 | ∫_0^t_f dt ⟨ϕ_-(t) |e^-iR_-H_1(t) e^iR_+| ϕ_+ (t) ⟩|^2.

    Substituting the above expression into the Eq. (<ref>), we can obtain the systematic error sensitivity

    q_s =   | ∫_0^t_f dt ⟨ϕ_-(t) |e^-iR_-H_1(t) e^iR_+| ϕ_+(t) ⟩|^2
            =   1/4| ∫_0^t_f e^2i R_+Ω_R (-cos^2 γ/2 e^2i β + sin^2 γ/2 ) dt |^2,

From Sec. <ref>, we know that when β approaches (n+1/2)π, we have e^2i β→ -1 and 
Ω_R(t) = γ̇ / sinβ≃ -γ̇. Therefore, q_s can be approximated as

    q_s ≃1/4| ∫_0^t_fγ̇ dt |^2 = π^2/4.

The relationship between the population P_- and the systematic error parameter μ is shown by the
green dashed-dotted curve in Fig. <ref>. A deviation rate of μ=±0.1 can lead to an infidelity 
    about 2.5%, which is small but 
    causes significant influence in quantum error correction.






    

§ OPTIMAL PROTOCOL

    Generally speaking, 
    the two-level Hamiltonian for an optimal control protocol <cit.> takes the following form:
    
    H_ opt = 1/2[                Δ Re[Ω] - i  Im[Ω]; Re[Ω] + i  Im[Ω]               -Δ ],

    where Re[*] and Im[*] denote the real and imaginary parts of the parameter *, respectively.
    The derivative of the Lewis-Riesenfeld phases can be obtained through computation
    
    Ṙ_± =±1/2 sinγ ( cosβ Re[Ω] - sinβ Im[Ω]).

    Using the derivations in Sec. <ref>, we can obtain the expressions for Re[Ω], Im[Ω], and Δ as
    
    Re[Ω] =   cosβsinγṘ_+ + sinβ Ṙ_+,  Im[Ω] =    -sinβsinγṘ_+ + cosβ Ṙ_+, Δ =   cosγṘ_+ - β̇.



    For the Hamiltonian H_opt in Eq. (<ref>), we can derive a new expression for the systematic error sensitivity
    
    q_s =   | ∫_0^t_f dt ⟨ψ_-(t) | H_1(t) | ψ_+(t) ⟩|^2 
                =   1/4| ∫_0^t_f dt [ e^2i R_+d/dt ( cosγsinγ) + e^2i R_+γ̇] |^2.

    Note that the boundary values γ(0) = π and γ(t_f) = 0, the expression can be further simplified to
    
    q_s = | ∫_0^t_f e^2i R_+γ̇sin^2 γ dt |^2.

    In the special case where R_+ does not vary with time (as in Sec. <ref> where R_+ is a constant), we obtain q_s = π^2/4.

 

    To make the systematic error sensitivity q_s = 0, we consider the case where R_+ varies with time, e.g.,
    
    R_+(t) = n/2 (2γ - sin2γ),       (n = 1, 2, 3, …),

    for R_+(t) in the above expression, we have 
    
    q_s = sin^2(n π)/4n^2,
 
     so, we have q_s = 0 when n≠ 0.  Note that in the limit of n → 0, we 
    obtain q_s →π^2/4, which is consistent with the previous statement below Eq. (<ref>). In this case, the expressions for Ω and Δ are as follows
    
    Re[Ω] =    (4n cosβsin^3γ + sinβ) γ̇,  Im[Ω] =    (-4n sinβsin^3γ + cosβ) γ̇, Δ =    4n γ̇cosγsin^2γ - β̇.

    For Ω in Eq. (<ref>) to be equivalent with Ω_R in Eq. (<ref>), we need to set Im[Ω]=0, resulting in
    
    β=4nsin^3γ.

    Then, taking this condition in Eq. (<ref>) and the boundary condition in Eq. (<ref>) and into account,
    we can redesign the parameters γ and β. 
    For instance, we can still use the polynomial expression in Eq. (<ref>) for γ, then we can obtain new β, as shown at Fig. <ref>(a).
    Accordingly, Ω and Δ can be calculated by Eq. (<ref>), so we can obtain E_J and ϵ, which are shown in Fig. <ref>(b) and Fig. <ref>(c), respectively. 
    
    Using these optimized parameters, our protocol becomes insensitive to systematic error 
    in the single photon drive (See the blue-solid, red-dashed, and purple-dotted curves in Fig. <ref>). 
    As can be seen in Fig. <ref>, the systematic error sensitivity can be
    significantly reduced by increase n. For n=5, a deviation with μ=± 0.3 in the parameter ϵ only
    leads to a decrease of 0.01% in the final population P_-(t_f), resulting
    in an optimally robust population inversion.
    Noting that the maximums of |ϵ| and |E_J| increase when n increases [see Figs. <ref>(b) and (c)],
    a longer operator time t_f is needed to satisfy |ϵ|,|E_J|≪ω_ gap for large n.
    Figure <ref>(d) shows the time-dependent population P_-(t). It is found that
    increasing n does not change the instantaneous population P_-(t). This because the instantaneous population P_-(t) is determined by the parameter γ, which keeps the same for different n.
   
   
  
    
The above discussion focuses on improving the robustness against parameter imperfections in ϵ. 
When parameter imperfections appear in E_J, our protocol can also achieve a robust population 
inversion as shown in Fig. <ref>. In this case, we consider an additional disturbed Hamiltonian 
    H_2=-νE_J(t){D[iφ_aexp(i ω_ct)]+ H.c.}/2,

where ν denotes the error rate.
The population of the target state |𝒞_-⟩ can still reach ≥ 99% when the error rate is ν=± 0.1
via our protocol with n=1 [see Fig. <ref>(a)]. Increasing the value of n can further
improve the robustness against systematic errors. However, to achieve such an optimal
robustness, an increase in the total evolution time is needed as discussed above. This becomes
a defect of the protocol when considering decoherence.
Therefore, for simplicity, we use the pulse with n=1 in the following numerical simulations.








    

§ DECOHERENCE

    For the resonator, we consider two types of noise: single-photon loss and pure dephasing. The system dynamics are described by the Lindblad master equation <cit.>
    
    ρ̇ = -i [H_ Kerr,ρ] + κ𝒟[a] ρ + κ^ϕ𝒟[a^†a] ρ,

    where 
    
    𝒟[o] ρ = o ρ o^† - 1/2 (o^†o ρ + ρ o^†o)

    is the standard Lindblad superoperator, κ is the single-photon dissipation rate, and κ^ϕ is the pure dephasing rate. 
    Projecting the whole system onto the cat-state subspace, we can obtain
    
    ρ̇_ eff≈    -i[H_ eff,ρ_ eff]   +κ|α|^2𝒟[A+A^-1/2σ_x+A-A^-1/2σ_y]ρ_ eff   +κ^ϕ|α|^4𝒟[A^2+A^-2/21-A^2-A^-2/2σ_z]ρ_ eff,

where A=tanh|α|^2 and 1=|𝒞_+⟩⟨𝒞_+|+|𝒞_-⟩⟨𝒞_-| is the unit matrix in the cat-state subspace. For large α, 
σ_y and σ_z terms are exponentially suppressed, resulting in

    ρ̇_ eff≈    -i[H_ eff,ρ_ eff]+κ|α|^2𝒟[σ_x]ρ_ eff,

i.e., leaving only the bit-flipping error. This is demonstrated in Fig. <ref>(a),
which shows that single-photon loss causes bit-flipping error but no leakage. The sum of populations

    P_S=P_++P_-,

in the cat-state subspace remains unchanged in the presence of single-photon loss.



According to Eq (<ref>), pure dephasing has no influence on the dynamics in the cat-state subspace.
However, pure dephasing can cause leakage out of the cat-state subspace [see Fig. <ref>(b)] because 

    a^†a|±α⟩=|α|^2|±α⟩±α D(±α)|1⟩.
 
This leakage probability is proportional to |κα/ω_ gap|^2 <cit.>.
The total population in the cat-state subspace reduces obviously as shown in the figure.
To further analyze the influence of pure dephasing on the dynamics in the cat-state subspace,
we define

    P_+^R=P_+/P_++P_-,      and     P_-^R=P_-/P_++P_-,

to be the normalized state populations in the cat-state subspace and display them in Fig. <ref>(c).
Obviously, a perfect population inversion is still possible in the presence of pure dephasing.

We also investigate the influence of total evolution time t_f on the protocol. As shown in Fig. <ref>,
an evolution time of t_f≃ 1/K is enough for our protocol to achieve 
the high-fidelity population inversion in the presence of decoherence. For instance, 
when t_f=1.1/K and κ=0.01K, the final population can reach P_-≃ 96%,
demonstrating the effectiveness of our optimized population inversion protocol.



 











§ DISCUSSION AND CONCLUSIONS


The proposed protocol is possible to be realized in superconducting
circuits <cit.>, especially the platforms with 3-dimension cavities <cit.>.
This is because that the 3-dimension cavities can provide
a relatively long coherent time (extending to microseconds) for photonic qubits <cit.>.
Cat-state qubits also belong to a larger family of bosonic qubits, most of which have
been realized with 3-dimension cavities <cit.>.
To be specific,
the Kerr nonlinearity and the two-photon drive can be respectively realized
by the Josephson junction (transmon) nonlinearity and four-wave mixing <cit.>.
The control Hamiltonian in Eq. (<ref>) is possible to realize
by considering the Kerr-nonlinear 
mode coupled capacitively to a
Josephson junction and assuming that other modes (including the junction mode) are never excited <cit.>. 
Following the first cat-state qubit experiment, we can consider the experimental 
parameters K/2π=6.7MHz, κ/2π≃ 0.01MHz, and κ^ϕ/2π≃0.045MHz.
With these parameters and t_f=1.1/K≃ 26 ns, the final population of the target state is P_-(t_f)≃ 95% 
in the presence of parameter imperfection with μ=ν=0.1 and decoherence.
When re-normalizing the state populations in the cat-state subspace using Eq. (<ref>),
the normalized state population of the odd cat state |𝒞_-⟩ becomes P_-^R≃ 98.6%. 

In conclusion, we have investigated a feasible control method 
to obtain the optimally robust shortcut to population inversion in cat-state qubits.
Focusing on the Kerr-cat qubit, which is realized by parametrically driving a Kerr-nonlinear resonator,
we have constructed shortcuts to adiabatic passages and minimized the systemic error
sensitivity based on the invariant-based reverse engineering.
Future work will involve extending our results to other logic qubits and the multi-qubit cases.
The existence of a set of optimal solutions for systematic errors also opens the
way to further optimization with respect to other error-correcting qubits.

    



    
    Y.-H.C. is supported by the National Natural Science Foundation of China
    under Grant No. 12304390 and the Project from Fuzhou University
    under Grant XRC-23051. Y.X. is supported
    by the National Natural Science Foundation of China
    under Grant No. 11575045, the Natural Science Funds
    for Distinguished Young Scholar of Fujian Province under
    Grant 2020J06011 and Project from Fuzhou University
    under Grant JG202001-2.



    
    
    




 Paper Title: No-Go Theorems for Universal Entanglement Purification.



APS/123-QED




These authors contributed equally.
Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL, USA


These authors contributed equally.
Department of Electrical and Computer Engineering, University of Illinois Urbana-Champaign, Urbana, IL, USA


Department of Electrical and Computer Engineering, University of Illinois Urbana-Champaign, Urbana, IL, USA


Microsoft Corporation, Redmond, WA, USA


Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL, USA



§ ABSTRACT 
    An entanglement purification protocol (EPP) aims to transform multiple noisy entangled states into a single entangled state with higher fidelity. In this work we consider EPPs that always yield an output fidelity no worse than each of the original noisy states, a property we call universality. We prove there is no n-to-1 EPP implementable by local operations and classical communication that is universal for all two-qubit entangled states, whereas such an EPP is possible using more general positive partial transpose-preserving (PPT) operations. We also show that universality is not possible by bilocal Clifford EPPs even when restricted to states with fidelities above an arbitrarily high threshold.


No-Go Theorems for Universal Entanglement Purification
    Tian Zhong
    August 5, 2024
======================================================




Introduction—

Entanglement is one of the most important resources for quantum information processing. Unfortunately, decoherence makes maintaining high entanglement fidelity difficult in practice. Even though quantum error correction can protect quantum information from environmental noises for both quantum computation <cit.> and quantum communication <cit.>, its low error threshold and high resource demands (quantum memory, gate counts, control requirements, etc.) are obstacles to its integration in current quantum architectures. Alternatively, entanglement purification protocols (EPPs) <cit.> offer a more resource-friendly method for distributing entangled states by transforming multiple noisy entangled states into a smaller number with higher entanglement fidelity using local operations and classical communication (LOCC). EPPs have been experimentally realized in different physical platforms <cit.>, and they provide a promising route for near-term implementation of measurement-based quantum computation <cit.>, distributed quantum computation <cit.>, distributed quantum sensing <cit.>, and the long-envisioned quantum internet <cit.>. 

The execution of ideal distributed quantum protocols, such as quantum teleportation <cit.>, typically requires pure entangled states in a specific form like the maximally entangled state |ϕ_2⟩=1/√(2)(|0⟩|0⟩+|1⟩|1⟩). One natural way to quantify the closeness of a given entangled state ρ to |ϕ_2⟩ is through its fidelity, F(ρ)=⟨ϕ_2|ρ|ϕ_2⟩. Most known EPPs were developed to work well for input states that are independent and identically distributed (i.i.d.). For instance, it is well known that 2-to-1 recurrence EPPs <cit.> can achieve arbitrarily high fidelity in the asymptotic regime via nested operations where at each level the input states are identical. However, the assumption of identical input states is rarely justified in practice. Due to the probabilistic nature of remote entanglement generation, entanglement pairs can be generated at different times,and need to be stored in quantum memories for varying amounts of time. This results in different amounts of memory decoherence, creating non-identical final states, even if the states are produced by the same source. In addition, performing a different number of entanglement purification cycles can result in a range of fidelity improvements, leading to non-identical input states for subsequent entanglement purification states. Entanglement pumping <cit.> is such an example with a well-known limitation on achievable fidelity.  Therefore, there is no easy way of obtaining identical states. Furthermore, obtaining accurate knowledge about the underlying error models for an ad hoc optimization via quantum benchmarking techniques <cit.> is also non-trivial due to the resource overhead, and the potential time dependence of error processes <cit.>. 

Motivated by these considerations, this work studies entanglement purification protocols that act on non-iid sequences of states, with each state being drawn from a given family of potential states.  In this setting, we consider whether it is possible to have EPPs that possess a type of universal monotonicity; namely, we demand that the EPP always outputs an entanglement fidelity no worse than the input fidelity of each state it acts on.  When this monotonicity breaks down, there are instances when the experimenter is better off aborting the EPP and just discarding all but one of the original states. 

In the following, we formally introduce the concept of universal EPPs, state theorems that clarify the fundamental limits of EPPs, and outline their proofs.  Mathematical details are provided in the Supplemental Material. To precisely define the concept of universal entanglement purification, we first recall that n-to-1 EPPs can be formally described as a two-branch quantum instrument <cit.> E={E_0,E_1}, where ℰ_0 and ℰ_1 are completely positive and trace non-increasing maps from the input space ⊗_i=1^n(ℋ_A_i⊗ℋ_B_i) to the output space ℋ_Â⊗ℋ_B̂, such that ℰ_0+ℰ_1 is trace-preserving. Here E_0 corresponds to success while E_1 corresponds to failure, such that the successfully purified output state is thus given by ρ'=ℰ_0(ρ)/tr(ℰ_0(ρ)). To appropriately compare the fidelities, we assume that each H_A_i and each H_B_i, together with ℋ_Â, and ℋ_B̂ are all two-dimensional. 

    Given a subset 𝒮 of two-qubit states, an n-to-1 EPP ℰ={ℰ_0,ℰ_1} is universal for 𝒮 if for n arbitrary states ρ_1,…,ρ_n∈𝒮 and ρ=⊗_i=1^nρ_i the successfully purified state ρ'=ℰ_0(ρ)/(ℰ_0(ρ)) has fidelity F(ρ')≥max{F(ρ_1),F(ρ_2),…,F(ρ_n)}.

Certain protocols are known to be universal for states with special structures. For instance, it can be shown that the 2-to-1 CNOT-based DEJMPS <cit.> protocol is universal for rank-three Bell diagonal states (BDS) with identical support. However, the input states are unlikely to belong to this restricted subset of entangled states in practice. Therefore, our first question is whether there exists an LOCC-implementable EPP that is universal for all entangled states. Unfortunately, we provide a negative answer to this question (Theorem <ref>). On the other hand, if we enlarge the class of EPPs beyond LOCC and consider slightly more general positive partial transpose-preserving protocols <cit.>, then we find that universality is possible for all states with fidelity greater than or equal to 1/2 (Proposition <ref>). This leads us to the question of whether there exists an LOCC-implementable EPP that is universal only for states with some sufficiently high fidelity.  Surprisingly, even this relaxed objective is difficult to achieve. Specifically, we prove that such protocols cannot be implemented by bilocal Clifford operations, which is a subset of LOCC that includes the original DEJMPS protocol (Theorem <ref>).


Universality for all entangled states—

Here we prove the no-go theorem that rules out the existence of a universal protocol for all entangled states. Deciding whether a protocol can be implemented by LOCC is notoriously difficult <cit.>. Therefore, we resort to two supersets of LOCC operations, namely the set of separable operations (denoted by SEP) and the set of operations that preserve the positive partial transpose (denoted by PPT). While SEP is a proper superset of LOCC, PPT is a proper superset of SEP. As discussed in Sec. II.A of the Supplemental Material <cit.>, without loss of generality, we only need to focus on the PPT and SEP properties of ℰ_0 to determine if the EPP ℰ={ℰ_0,ℰ_1} is in PPT or SEP.

Additionally, we observe the following necessary and sufficient condition of universality. Universality for two-qubit entangled states with fidelity greater than 1/2 is equivalent to universality for all entangled isotropic states {ρ:ρ=Fϕ_2+(1-F)I-ϕ_2/3,F>1/2} [We note that although there exist entangled states with fidelity below 1/2 <cit.>, they can always be transformed into states with fidelity above 1/2 through LOCC <cit.>. However, admittedly the filtering operations are state-dependent, and it assumes a priori knowledge of state. Nevertheless, the 1/2 fidelity threshold is well motivated by requirement of non-trivial quantum teleportation <cit.>.]. This is because if there exists an EPP ℰ that is universal for all entangled isotropic states, we can construct a protocol that is universal for all states with fidelity greater than 1/2. We first perform bilateral twirling τ(ρ)=∫_SU(2) (U⊗ U^*)ρ(U⊗ U^*)^† dU on each input state, which converts any state to an isotropic state with the same fidelity <cit.>, and then we perform ℰ. Moreover, as a consequence of the continuity of completely positive maps, we can always include the boundary of entangled states in the possible input set without affecting the result (for details see Sec. II.B of the Supplemental Material <cit.>). 
Finally, we can also twirl the output state without changing its fidelity. These observations significantly simplify our discussion, allowing us to derive the following results:

    For any n>1, there exists a PPT n-to-1 entanglement purification protocol that is universal for all two-qubit states with fidelity greater than 1/2. The success branch of the protocol is given by a PPT map E_0^(n)(ρ)=(ρϕ_2^⊗ n)ϕ+(ρ(ϕ_2^⊥)^⊗ n)ϕ_2^⊥/3. Moreover, any other n-to-1 PPT universal EPP {Ê_0^(n),Ê_1^(n)} satisfies pE_0^(n) = τ∘Ê_0^(n)∘τ^⊗ n for p∈(0,1].

The proof is in Sec. II.C of the Supplemental Material <cit.>. Now in order to determine if E_0^(n) are in SEP, we want to know whether the Choi operators are separable. While determining separability is an NP-hard problem <cit.>, a necessary condition of separability based on general Bloch representation of density matrices <cit.> is computable, allowing us to perform analytical studies. The criterion states that for a separable state ρ between two parties with dimensions N and M, we must have ‖ T‖_1=Tr(√(T^† T)) ≤√(MN(M-1)(N-1))/2, where T is the correlation matrix of ρ in a generalized Bloch decomposition. In Sec. II.D of the Supplemental Material <cit.>, we prove that the map violates this criterion whenever n is even.  This can be further used to prove non-separability for odd n as well.  Hence every E_0^(n) in Proposition <ref> is non-separable. Since all universal EPPs must have the form of E_0^(n) up to local twirling, which transforms every separable map into another separable map, we prove our main theorem.

    There is no LOCC-implementable n-to-1 qubit entanglement purification protocol that is universal for all two-qubit entangled states, ∀ n≥ 2.

Although Theorem <ref> is restricted to n-to-1 EPPs, it also has strong implications for n-to-k (k≤ n) scenarios. Let us define an n-to-k EPP as universal if among the 2k kept qubits (k held by Alice and k by Bob), there exists a remote pair of kept qubits (one held by Alice and the other by Bob) whose reduced state has fidelity no less than the highest input fidelity. For instance, the identity is a trivial universal n-to-n EPP mapping. However, for any universal n-to-k EPP, the output subsystem with the highest fidelity cannot be the same for every input. Otherwise, we could combine the EPP with a partial trace map and get a new n-to-1 EPP that violates Theorem <ref>.


Universality for restricted state sets—


We have demonstrated that an LOCC-implementable EPP which is universal for all entangled states cannot exist. Now we want to find protocols that are universal for restricted input states, especially those with sufficiently high fidelity. We turn to the n-to-1 bilocal Clifford entanglement purification (biCEP) protocols <cit.> (shown in Fig. <ref>), and study their universal properties. The n-to-1 biCEP protocols are defined as follows.

    In an n-to-1 biCEP protocol, Alice applies n-qubit Clifford C on her n qubits while Bob performs the entry-wise complex conjugate of C, i.e. C^*, on his n qubits. Then Alice and Bob measure in computational basis their last (n-1) qubits and communicate the measurement outcomes. The protocol is considered successful if all (n-1) pairs of measurement outcomes have the same parity (00 or 11), and unsuccessful otherwise.


The choice of biCEP has two motivations: (i) Clifford operations can be realized via finite layers of standard single- and two-qubit gates, e.g. the Hadamard gate, phase gate, CNOT gate and CZ gate <cit.>, which guarantees practicality; (ii) the n-to-1 biCEP family covers EPPs that can be constructed from all [[n,1,d]] stabilizer codes <cit.>, which demonstrates the connection between EPP and quantum error correction. As the successful output of biCEP protocols only depends on the diagonal elements of input states' density matrices in the Bell basis (see Sec. III.A.4 of the Supplemental Material <cit.>), we  assume BDS input without loss of generality, which is also practically justified by Pauli twirling <cit.>. We further specify that we will focus on “complete BDS sets” as input sets, which are complete in the sense that they contain the objective noiseless Bell state and cover all three possible Pauli errors. 

    A set of BDS, 𝒮, is a complete BDS set if (i) every ρ∈𝒮 is entangled, (ii) ϕ_2∈𝒮, and (iii) ∃ρ_1,ρ_2,ρ_3∈𝒮 s.t. |ψ^+⟩∈supp(ρ_1),|ψ^-⟩∈supp(ρ_2),|ϕ^-⟩∈supp(ρ_3), where supp(·) denotes operator support. Let ℭ be the collection of all such sets. 

We emphasize that we allow the lowest fidelity of a state in such sets to be arbitrarily close to 1. For example, a complete BDS set could be a family of high-fidelity isotropic states {ρ:ρ=Fϕ_2+(1-F)I-ϕ_23, F>1-ϵ} for any 1/2>ϵ>0.

The mechanism of biCEP protocols can be understood as follows (see more details in Sec. III.A.3 of the Supplemental Material <cit.>). For n BDS ρ_1,⋯,ρ_n, the total state ⊗_i=1^nρ_i equals the output of ϕ_2^⊗ n through a unilateral Pauli channel I⊗𝒩_P(ϕ_2^⊗ n), where 𝒩_P(ρ) = ∑_i=1^4^np_iP^(n)_iρ P^(n)_i with P^(n)_i being an n-qubit Pauli string, and p_i the corresponding probability of P_i^(n). Then we can see that the effect of the biCEP circuit is equivalent to transforming the unilateral Pauli channel into 𝒩_P(ρ) = ∑_i=1^4^np_iP^(n)_iρP^(n)_i, where each P_i^(n) is transformed to another P^(n)_i=CP^(n)_iC^† via the conjugate of Clifford C. 

It is clear that the error detection of biCEP is based on the Pauli string transformation capability of its Clifford C, and the transformed Pauli noise channel determines success or failure. 
Therefore, we can identify every biCEP by a single Clifford operator C. Our present goal is to identify the necessary and sufficient condition for biCEP to be universal for 𝒮∈ℭ as Lemma <ref>. We first define a class of Pauli strings that will simplify description.

    An n-qubit harmless string P_h^(n) satisfies one of the following two conditions: (i) its first (leftmost) component is the identity operator I and the remaining n-1 components are either the identity operator I or Pauli Z, e.g. I⊗ I⊗ Z, or (ii) there exist at least one Pauli X or Y operator among its n-1 components other than the first one, e.g. Y⊗ I⊗ X.

The definition is motivated as follows: If a type (i) harmless string is applied unilaterally to ϕ_2^⊗ n, the (n-1)-pair computational basis measurement outcomes will have identical parities as those of ϕ_2^⊗ n, indicating success, while the first slot is not affected by Pauli error resulting in a noiseless ϕ_2; if type (ii) harmless string is applied then the resulting state will indicate a failure.

    An n-to-1 biCEP protocol using n-qubit Clifford C is universal ∀𝒮∈ℭ if and only if C transforms all n-qubit Pauli strings which have at least one identity operator into harmless strings only.  

The proof is in Sec. III.B of the Supplemental Material <cit.>. For necessity we consider a special case where at least one input is noiseless, and then according to the definition of universality the successful output fidelity must be one. Moreover, as we have not assumed any knowledge of input states, the n input states have to be arbitrarily ordered, while the first slot is always unmeasured. Therefore, the unilateral noise Pauli channel could include all possible n-qubit Pauli strings which have at least one identity operator. If they are transformed to harmless strings, as long as the biCEP measurement outcomes indicate success, the unmeasured pair is guaranteed to be noiseless. The proof of sufficiency is based on a formal expression of successful biCEP outcome fidelity and some technical lower bounding.

Notice that the set of all entangled isotropic states is a complete BDS set, so Theorem <ref> together with Lemma <ref> implies the non-existence of an universal biCEP:

    For any 𝒮∈ℭ and n≥2, n-to-1 biCEP protocols cannot be universal for 𝒮.

We also provide an algebraic proof of this Proposition, which can be found in Sec. III.C of the Supplemental Material <cit.>.


No non-trivial universal biCEP with ordered fidelity—

So far we have made minimal assumptions about the input states, which strengthens the requirement of universality. In principle, we can obtain information about the input states, especially the ordering of their fidelities. In practice, we know when an entangled pair is successfully generated based on the heralding signal, and it is generally safe to assume that the latest generated state has the highest fidelity. We can take advantage of such ordered fidelity to relax the universality requirement. Specifically, we may choose to always keep the qubits of the highest-fidelity input unmeasured, which gives the following relaxed necessary condition for a biCEP to be universal with ordered fidelity:

    Given ordered fidelity, if a biCEP protocol using n-qubit Clifford C is universal for 𝒮∈ℭ, C transforms all n-qubit Pauli strings in the form of I⊗ P, where P is an arbitrary (n-1)-qubit Pauli string, into harmless strings only. 

The proof is omitted due to its similarity to the proof of Lemma <ref>, with the only change being that we fix the leftmost component of the Pauli strings to be the identity operator I. Here we emphasize that there always exists a trivial universal biCEP for complete BDS set given ordered fidelity: We can simply discard (n-1) input states except for the one with the highest fidelity, and then the “ successful output” will always have fidelity equal to the highest input fidelity. Given this relaxed necessary condition, finding a non-trivial universal biCEP might be possible, as we can sweep over Cliffords that satisfy this condition. However, we prove that these protocols are always trivial, i.e., the output fidelity is always equal to the highest input fidelity:

    Given ordered fidelity, every universal biCEP protocol for complete BDS sets does not perform better than the trivial protocol.

The proof can be found in Sec. III.D of the Supplemental Material <cit.>.


Entanglement-assisted biCEP—


We can further generalize our study of biCEP protocols to the scenario where we are supplied with m pure Bell states, which can be input to the entanglement purification circuit, but must remain unchanged at the end. Assuming the unilateral Pauli channel corresponding to the total state of certain n BDS is 𝒩_P, the total state of m pure Bell states together with the n BDS corresponds to another unilateral Pauli channel 𝒩_P' where the Kraus operators of 𝒩_P' are simply the Kraus operators of 𝒩_P tensored with m identity operators. The scenario of entanglement-assisted biCEP is visualized in Fig. <ref>. Note that multiplication relations and commutators between Pauli strings are unchanged if all relevant Pauli strings are tensored with with an equal number of identity operators. Therefore, the algebraic proofs of Proposition <ref> and Proposition <ref> still apply to these scenarios with pure Bell state assistance for biCEP. Combining all the above results on biCEP, we have the following theorem.

    There is no n-to-1 biCEP that is universal or non-trivially universal with ordered fidelity, when assisted by m pure Bell states, ∀𝒮∈ℭ, ∀ n≥ 2, ∀ m>0.



Discussion—

In summary, we have introduced a property of EPPs that is desirable in practice, universality; we found a unique PPT-preserving protocol that is universal for all entangled states with fidelity greater than 1/2, and we proved that it is not a separable operation, thus not LOCC-implementable. Subsequently, we relaxed the requirement by restricting ourselves to states with arbitrarily high fidelity, but it turned out that universal protocols are still practically challenging since they cannot be realized by bilocal Clifford circuits. Moreover, we showed that if fidelity ordering of input states is provided, any biCEP that is universal is necessarily trivial, meaning that the purification output always has the same fidelity as the state with the highest fidelity. The results for biCEP hold even if assisted by an arbitrary number of pure Bell states.

In the context of distributed quantum information processing, our results imply that 1G quantum repeaters <cit.> which distribute high-fidelity entanglement over long distance are difficult to construct without using large amount of physical resources and taking advantage of additional information such as underlying error patterns (see more detailed discussion in Sec. I of the Supplemental Material <cit.>). Universality of the DEJMPS protocol for rank-3 BDS implies that biased (Pauli) noise is favorable; such biased noise can be obtained in various physical platforms, e.g. neutral atoms <cit.>[We note that Ref. <cit.> considers error conversion for Rydberg state, while Rydberg state is in general not directly used for quantum memory.], and stabilized cat qubits encoded in bosonic systems <cit.>. From a theoretical perspective, our work advanced the beyond-i.i.d. literature, by contributing to the fundamental understanding of entanglement purification with non-i.i.d. input <cit.>. We derived new fundamental limits on the capability of LOCC and provided a new example of operational gap between PPT and SEP (LOCC) in low dimensions. We also deepened the understanding of Clifford gates by revealing interesting structures of n-qubit Pauli group under Clifford conjugate. 

Our introduction of universality opens up various new research directions. For example, it would be interesting to go beyond EPPs for two-qubit states and study universality of qudit EPPs, and multi-partite EPPs, e.g. for graph states <cit.>. The idea of universality can also be extended to magic state distillation <cit.> where magic states will become non-identical if they undergo different number of distillation cycles. Furthermore, the presented no-go results have not ruled out all possibilities of LOCC-implementable EPPs that are universal for two-qubit states with arbitrary error patterns, because we could in principle further increase the allowed lowest fidelity of input states and consider other EPPs beyond biCEP. There should then exist a larger parameter space of PPT universal entanglement purification protocols, and it is possible that some of these PPT entanglement purification protocols are LOCC-implementable. The search for separable universal entanglement purification protocol with higher fidelity threshold is an important next step that may require discovery of new techniques to determine separability for quantum states with symmetry <cit.>. 


Acknowledgments—

We acknowledge support from the NSF Quantum Leap Challenge Institute for Hybrid Quantum Architectures and Networks (NSF Award 2016136). T.Z. would like to acknowledge the support from the Marshall and Arlene Bennett Family Research Program. X.C. and E.C. are supported by the U.S. Department of Energy Office of Science National Quantum Information Science Research Centers.









    






§ MOTIVATION FOR INTRODUCING UNIVERSALITY


Analysis of entanglement purification protocols (EPPs) connects beyond-i.i.d. quantum information theory, entanglement theory, and other advanced mathematical structures. As a property of both the purification protocol and the input state set, universality is an interesting theoretical topic on its own. Here we elaborate on the motivation for introducing universality with special attention to practical considerations. Although we consider entanglement distribution quantum networks as the main application of this work, our discussion also applies to general quantum information processing architectures which use imperfect quantum memories to store quantum states and run probabilistic operations, with purification/distillation protocols as examples.



 §.§ Requirements for benchmarking, optimization and scheduling in quantum networks

Once built, quantum networks <cit.> will serve as the infrastructure necessary for distributed quantum information processing (QIP). Quantum networks will be used to distribute entanglement that will be in turn consumed by network applications. Most currently envisioned quantum applications consume specific forms of entangled states. For example, quantum state teleportation <cit.> and gate teleportation <cit.>, operations required for distributed quantum computing <cit.>, both consume EPR pairs, and quantum sensing <cit.> with beyond standard quantum limit performance consumes multi-partite entangled states such as spin-squeezed states or GHZ states. 

Distributing entanglement over long-distances will require long duration of operations, leading to significant idling decoherence. This poses strict requirements on counteracting its effects. Counteracting decoherence can be achieved by (i) quantum error correction (QEC) <cit.> which protects encoded quantum information by continuous monitoring and active correction, and (ii) quantum error detection, with various examples of quantum state purification/distillation protocols <cit.>. Due to the requirement of active correction, QEC is more challenging in practice. Therefore, we do not anticipate that QEC can be practically incorporated in near-term QIP architectures, especially quantum networks, and this justifies our focus on entanglement purification protocols in this work.

In entanglement distribution quantum networks, the probabilistic nature of operations leads to the non-identicality of entangled states. Long distance communication requires using photons as flying qubits to mediate remote entanglement generation. Errors occur due to losses during photon transmission and the fundamental limit on success probability of linear optical Bell state measurements <cit.>, which do not seem solvable in the near future. Probabilistic entanglement generation requires storing successfully generated entangled states in quantum memories with two uses: (i) quantum networks resort to divide-and-conquer strategy to avoid exponentially decaying communication success probability with distance <cit.>; (ii) entanglement purification needs multiple entangled states as inputs. As a result, the earlier generated states stored in quantum memories accumulate errors due to idling decoherence, leading to non-identicality of the entangled states. The non-identicality is exacerbated by the states spending different amounts of idling time in quantum memories. The non-identicality occurs even if the originally generated states were identical. In addition, entanglement purification is also probabilistic. Therefore, entangled states could have undergone different levels of purification, which naturally could be non-identical.

It has been shown that EPPs can be optimized if the input states are known <cit.>, and the optimal solution may depend on the input states and the operation error models. However, knowledge of the input states is non-trivial. The states are determined by their form during successful entanglement generation, and the idling error accumulated between the successful state generation and the start of entanglement purification. As a result, to determine the form of the input states, we need to know both the initial state and the idling error model. Although information about the quantum states and channels can be learned through measurements, we cannot directly measure the states prior to EPP, as this would corrupt them. Therefore, the information must be obtained earlier, either through state and process tomography, or through specialized quantum benchmarking/learning protocols <cit.>. However, there is no guarantee that the information about the initial states and the error processes will coincide with the current input states. In other words, even if we can learn information about the errors, we can only obtain optimal solution for the specific problem corresponding to a scenario in the past, while error processes vary over time <cit.>. We could argue that we do not need an accurate knowledge of the quantum states to obtain close-to-optimal protocols. Such solutions could be an interesting subject of future work, but but finding them may be difficult because the transition between optimal solutions can be abruptly different when error processes change <cit.>. Even if the quantum processes are stationary, additional resources such as repeated measurements are required in order to learn the details from the processes.

Coordinating the actions of the large number of components in quantum networks requires scheduling optimization, including optimization of entanglement swapping and purification. Scheduling policy optimization is difficult due to the large state space, and optimal policy is only known for the simplest scenarios that do not include entanglement purification <cit.>. For instance, in quantum repeater networks with entanglement generation buffer time <cit.> or continuous (pre-) entanglement generation <cit.>, additional idling time is intentionally introduced to optimize entanglement distribution rate and latency, and entanglement purification can be performed within the time windows before entanglement is requested. This enlarges the parameter space for optimization. We consider the scenario where we schedule a fixed EPP. Although there is no known EPP that is universal for arbitrary Bell diagonal states with a certain fidelity threshold, it is still possible that existing EPPs can offer positive fidelity gain with non-identical input states conditioned on success, but the requirement on small difference between input states could be stringent. For example, we consider the well-known 2-to-1 DEJMPS protocol <cit.> and isotropic states as inputs. If one input has fidelity 0.9 and the other input has fidelity 0.85, the successful output fidelity will be F'(0.9,0.85)≈ 0.9055>0.9; meanwhile, if we fix the 0.9 fidelity input and change the other input to have 0.83 fidelity, the successful output fidelity will be F'(0.9,0.83)≈ 0.8967<0.9. Note that when the input fidelities increase, the difference will be even smaller. Therefore, in order to ensure non-trivial fidelity increase, we must wisely choose from all available entangled pairs. To decide, we must calculate the output fidelities for every possible combination of input states, which introduces additional computational overhead, and requires the knowledge of entangled states. 

In summary, there are two approaches to distributing high-fidelity entanglement with entanglement purification: (i) optimizing EPPs, and (ii) optimizing the scheduling for fixed EPPs. However, both approaches have a significant optimization overhead, and also require knowledge of quantum states which results in a quantum benchmarking overhead.



 §.§ Relaxed requirements for universal entanglement purification protocols

In the above, we discussed the operational requirements in quantum networks from an engineering and an architectural perspective. It is clear that when the adopted EPP is not universal for arbitrary error models, the required operational overhead is very costly. In contrast, a universal EPP is a “one-cure-for-all” solution, and the requirements can be relaxed, as explained in the following.

First, assume there exists a universal EPP for arbitrary error models. There is no need to know the underlying error models of quantum memories and entanglement generation. It suffices to know that the input states have a fidelity above the fidelity threshold of the universal EPP. This can be done e.g. by lower bounding the fidelity based on results of rough quantum benchmarking, which significantly reduces the resource overhead. 

Second, although a fixed universal EPP is not necessarily the optimal protocol for some specific input states, it is guaranteed that the successful output will always be non-trivial, i.e. the successful output fidelity will always at least as high as the highest input fidelity. This is operationally desirable, and therefore unless a better performance is required, using a universal EPP can avoid the need to perform additional purification circuit optimizations. 

Finally, a universal EPP strongly relaxes the scheduling requirements. A universal EPP only requires that every input state has a fidelity above the fidelity threshold, which can be verified individually, and cross-referencing other entangled pairs' information is not needed. For instance, we can employ a memory-wise cut-off policy. Approximate information about initial state and error models allows lower bounding entanglement fidelity. We only need to re-initialize the quantum memories at a certain cut-off time when we can no longer guarantee that the fidelity is above the threshold. The states can be used for EPP at any time before the cut-off.




§ GENERAL ENTANGLEMENT PURIFICATION PROTOCOLS


In this section, we offer elaborated discussion on general entanglement purification protocols, i.e. we simply view EPPs as two-branch quantum instruments while ignoring their explicit implementation. In Sec. <ref>, we briefly review the needed background information on state-operation duality and the conditions for quantum operation to be PPT and SEP, respectively. We also present a self-contained overview of a computable separability criterion, i.e. the Bloch representation criterion, which proves to be useful for studying separability of Choi states which are of our interest. Then in Sec. <ref> we prove technical results which allow us to include the boundary of entangled state set when studying universality. Subsequently, we prove the unique form of Choi state for PPT entanglement purification protocols that are universal for all entangled states with fidelity above 1/2 (Proposition 2 in the main text) in Sec. <ref>. Finally in Sec. <ref>, based on the Bloch representation criterion and the explicit form of Choi state, we prove that the Choi states are non-SEP for arbitrary even n≥ 2, and moreover, the Choi states for odd n>2 are also non-SEP. We also include a brief, interesting spin-off discussion on the elementary function representation of a special case of hypergeometric function, which is closely related to the combinatorial proof of non-separability of the Choi states for even n≥ 2.



 §.§ Preliminaries

Recall that we define general EPP as a two-branch quantum instrument ℰ={ℰ_0,ℰ_1}, where CPTNI maps ℰ_0 and ℰ_1 correspond to success and failure, respectively.



  §.§.§ State-operation duality

The effect of the operation ℰ_0 (the successful branch of n-to-1 EPP ℰ) can be described by the Choi operator (matrix, state) J=(Id^A_1B_1 ⋯ A_nB_n⊗ℰ_0^A_1'B_1' ⋯ A_n'B_n')(|Ω⟩⟨Ω|), where |Ω⟩=∑_i|i⟩_A_1B_1 ⋯ A_nB_n|i⟩_A_1'B_1' ⋯ A_n'B_n' and |i⟩_A_1B_1 ⋯ A_nB_n are basis vectors of the 2n-qubit (EPP input) Hilbert space. As an instance of state-operation duality (isomorphism), the PPT and SEP properties of quantum operations are equivalent to the respective properties of their Choi state:

    A quantum operation ℰ:ℒ(ℋ_A⊗ℋ_B)→ℒ(ℋ_Â⊗ℋ_B̂) is PPT if and only if its Choi matrix J∈ℒ(ℋ_A⊗ℋ_B⊗ℋ_Â⊗ℋ_B̂) is PPT with respect to the AÂ-BB̂ partition.


    A quantum operation ℰ:ℒ(ℋ_A⊗ℋ_B)→ℒ(ℋ_Â⊗ℋ_B̂) is separable if and only if its Choi matrix J∈ℒ(ℋ_A⊗ℋ_B⊗ℋ_Â⊗ℋ_B̂) is separable with respect to AÂ-BB̂ partition.

An EPP ℰ={ℰ_0,ℰ_1} is in SEP if both the success and the failure branches are SEP operations, i.e. if the Choi operators of ℰ_0 and ℰ_1 are both separable <cit.>, while it is in PPT if both branches are PPT operations, i.e. if both Choi operators have a positive partial transpose <cit.>. Note that whenever the Choi operator of ℰ_0 is separable, we can always replace the unwanted state ℰ_1(ρ)/tr(ℰ_1(ρ)) with the maximally mixed state, making the EPP separable. The same argument applies if the Choi operator of ℰ_0 is PPT. 



  §.§.§ Bloch representation separability criteria

As mentioned in the main text, for bipartite states on the M× N-dimensional Hilbert space (one party has dimension M and the other N), they have the following general Bloch decomposition

    ρ = 1/MN(I_M⊗ I_N + ∑_i=1^M^2-1r_iλ_i⊗ I_N + ∑_j=1^N^2-1s_jI_M⊗λ_j + ∑_i=1^M^2-1∑_j=1^N^2-1t_ijλ_i⊗λ_j )

where λ_i are SU(M) generators and λ_j are SU(N) generators. Based on this representation, we have the following necessary and sufficient condition for bipartite separable states, and a derived necessary condition for bipartite separability <cit.> which is used in this work. We provide simple proofs for both.

    A bipartite state ρ_AB on ℋ_A⊗ℋ_B is separable if and only if there exist triples (0≤ p_i≤ 1,𝐮_i,𝐯_i), ∑_ip_i=1 such that its Bloch representation satisfies T=∑_ip_i𝐮_i𝐯_i^T, 𝐫=∑_ip_i𝐮_i, 𝐬=∑_ip_i𝐯_i.


    First we note that a bipartite state ρ_AB is a pure separable state ρ=|ψ⟩⟨ψ|_A⊗|φ⟩⟨φ|_B if and only if its Bloch representation (Eqn. <ref>) satisfies T=𝐫𝐬^T with ‖𝐫‖_2=√(M(M-1)/2) and ‖𝐬‖_2=√(N(N-1)/2), where 𝐫,𝐬 are the column vectors with each component being r_i, s_j, respectively. The matrix equation guarantees that ρ=ρ_A⊗ρ_B where ρ_A(B)=Tr_B(A)(ρ), and the 2-norm (Euclidean norm) condition guarantees that ρ_A(B) are pure. 

    Now observe that any bipartite state ρ_AB is separable if and only if it can be decomposed into a weighted sum of pure separable states. We complete the proof using the above condition for a pure separable state.


    If a bipartite state ρ_AB on ℋ_A⊗ℋ_B is separable, then its coefficient matrix T_ij=t_ij must satisfy 
    
    ‖ T‖_1 = Tr(√(T^† T)) ≤√(MN(M-1)(N-1))/2,

    where M=ℋ_A, N=ℋ_B, and ‖·‖_1 denotes 1-norm which equals the sum of the singular values, i.e. ‖ T‖_1 = Tr(√(T^† T)).


    According to the inequality of singular values (triangle inequality of 1-norm), i.e. ‖ A+B‖_1≤‖ A‖_1 + ‖ B‖_1, we have that for the T matrix of a separable bipartite state
    
    ‖ T‖_1 =   ‖∑_ip_i𝐮_i𝐯_i^T‖_1≤∑_ip_i‖𝐮_i𝐯_i^T‖_1
    
            =   √(MN(M-1)(N-1))/2∑_ip_i‖𝐞_𝐮_i𝐞_𝐯_i^T‖_1 = √(MN(M-1)(N-1))/2

    where 𝐞_𝐮_i(𝐯_i) is the unit vector in the same direction as 𝐮_i(𝐯_i), and for the last equality we use the fact that for the outer product of two unit vectors, in singular value decomposition we can always find unitaries U,V (in this specific case we may only need real orthogonal matrices as 𝐮_i,𝐯_i are real) such that U𝐞_𝐮_i𝐞_𝐯_i^TV^† = diag(1,0,0,…,0), and thus the sum of the singular values is guaranteed to be 1.




 §.§ Continuity of completely positive maps and boundaries of entangled states



  §.§.§ Results on trace distance

It is well known that the trace distance d(X,Y)=1/2X-Y_1 is monotonic under completely positive and trace-preserving maps <cit.>. Here we establish monotonicity under general completely positive trace-non-increasing (CPTNI) maps with a similar proof:

    For any completely positive and trace-non-increasing map ℰ, we have d(ℰ(ρ),ℰ(σ)) ≤ d(ρ,σ).


    First we prove that for two positive operators X and Y that are not necessarily unit trace, there exists an operator 0 ≤ P≤𝕀 such that
    
    d(X,Y)=(P(X-Y)).
 
    To show this, let us write the spectral decomposition of X-Y as A-B, where A and B are both positive operators and have orthogonal support. Then, d(X,Y)=1/2|X-Y|=1/2|A-B|=1/2( A+ B). Suppose without loss of generality A ≥ B. Let P' be the projector onto the support of A and define P= A+ B/2 AP'. Then, 
    
    (P(X-Y))=(P(A-B))= A+ B/2 A(P'(A-B)P')=1/2( A+ B) = d(X,Y).


    Now we apply this result to ℰ(ρ) and ℰ(σ). Let 0≤ Q≤𝕀 be an operator such that d(ℰ(ρ),ℰ(σ))=(Qℰ(ρ)-Qℰ(σ)). Suppose ρ-σ has the spectral decomposition C-D where C and D are both positive operators and have orthogonal support. Since ρ-σ=0, we have C= D. Now,
    
    d(ρ,σ)    = 1/2|ρ-σ| = 1/2|C-D| = 1/2( C +  D) =  C ≥ℰ(C) ≥(Qℰ(C)) ≥(Qℰ(C)-Qℰ(D)) 
       = (Qℰ(ρ)-Qℰ(σ)) = d(ℰ(ρ),ℰ(σ)),

    where the first inequality follows from the fact that ℰ is trace-non-increasing, the second inequality follows from Q≤𝕀, and the third inequality follows from the positivity of Q and ℰ(D).


    d(⊗_iρ_i,⊗_iσ_i) ≤√(2∑_i d(ρ_i,σ_i)).


    The proof starts with the Fuchs–van de Graaf inequality, and follows with a straightforward derivation:
    
    d(⊗_iρ_i,⊗_iσ_i)    ≤√(1-F(⊗_iρ_i,⊗_iσ_i))
       = √(1-∏_i F(ρ_i,σ_i))
       ≤√(1-∏_i[1-d(ρ_i,σ_i)]^2)
       ≤√(1-(1-2∑_i d(ρ_i,σ_i)))
       = √(2∑_i d(ρ_i,σ_i)),

    where the first equality follows from the fact that fidelity is multiplicative under tensor products; the second inequality is from another application of Fuchs–van de Graaf inequality; the third inequality is obvious by recalling 0≤ d(ρ_i,σ_i)≤ 1.




  §.§.§ The boundary of an input state set

Using the above results, it is easy to prove the following statement:

    A purification protocol is universal for a set of states S if and only if it is universal for S∪∂ S, where ∂ S is the boundary of S.


    The if direction is clear. To show the converse, suppose a purification protocol ℰ=(ℰ_0,ℰ_1) is not universal for S∪∂ S. This means that there exist states ρ_1,⋯,ρ_n∈ S∪∂ S such that 
    
    F(ℰ_0(ρ)/ℰ_0(ρ)) < max_i F(ρ_i),

    where ρ=⊗_i=1^nρ_i, and we have omitted the reference state which is the maximally entangled state. If all of ρ_i∈ S, then this means ℰ cannot be universal for S. Therefore, we focus on the case where at least one of ρ_i∈∂ S∖ S. According to the definition of the boundary, for any point x∈∂ S, there exists in every neighborhood of x a point that is in S. If we replace each such ρ_i by a state ρ̃_̃ĩ∈ S that is ϵ close to ρ_i in trace distance and define ρ̃=(⊗_ρ_i∈∂ S∖ Sρ̃_̃ĩ) ⊗(⊗_ρ_j∈ Sρ_j), we deduce
    
    d(ρ̃,ρ) ≤√(2nϵ).

    Thus by monotonicity of trace distance under CPTNI map, we get
    
    d(ℰ_0(ρ̃),ℰ_0(ρ)) ≤√(2nϵ).

    By continuity of trace distance and fidelity, this means that |F(ℰ_0(ρ̃)/ℰ_0(ρ̃)) - F(ℰ_0(ρ)/ℰ_0(ρ))| can be made arbitrarily small by choosing arbitrarily small ϵ. Therefore, we can always find ρ̃ as a tensor product of a state from set S which violates the universality condition:
    
    F(ℰ_0(ρ̃)/ℰ_0(ρ̃)) < max_i F(ρ_i),

    which means that the protocol cannot be universal for S.

Based on this result, we can safely include F=1/2 isotropic state in the set in the following when we consider possibility of universality for all qubit entangled isotropic states.



 §.§ Choi matrix of -to-1 universal EPP with fidelity threshold 1/2

Here we prove the Proposition 2 in the main text, while we rephrase the statement a bit by explicitly demonstrating the Choi operators.
 
    There exist PPT n-to-1 entanglement purification protocols that are universal for all two-qubit states with fidelity greater than 1/2, given by the Choi operator J_n = (⊗_i=1^n ϕ_2^A_iB_i)⊗ϕ_2^ÂB̂ + (⊗_i=1^n (ϕ_2^⊥)^A_iB_i)⊗(ϕ_2^⊥)^ÂB̂/3, where ϕ_2^⊥ = I-ϕ_2. The protocols are unique up to twirling the input and output systems and a multiplicative constant C∈(0,1] on the Choi operator which determines the probability of success.


    Let us first show that this entanglement purification protocol is PPT and universal. It is obvious that given the input states as n isotropic states with fidelities F_1,F_2,…,F_n the output fidelity of this operation is 
    
    F'(F_1,F_2,…,F_n) = ∏_i=1^nF_i/∏_i=1^nF_i + ∏_i=1^n(1-F_i).

    Now consider the difference F'(F_1,F_2,…,F_n)-F_j as follows
    
    F'(F_1,F_2,…,F_n)-F_j =   (∏_i≠ jF_i/∏_i=1^nF_i + ∏_i=1^n(1-F_i) - 1)F_j
    
            =   ∏_i≠ jF_i - ∏_i=1^nF_i - ∏_i=1^n(1-F_i)/∏_i=1^nF_i + ∏_i=1^n(1-F_i)F_j
    
            =   (1-F_j)(∏_i≠ jF_i - ∏_i≠ j(1-F_i))/∏_i=1^nF_i + ∏_i=1^n(1-F_i)F_j ≥ 0,

    for F_1,F_2,…,F_n≥ 1/2. This shows that the operation is universal for n isotropic states with fidelity above 1/2, thus also universal for arbitrary n states with fidelity above 1/2. The partial transpose of J_n w.r.t. A-B bipartition is
    
    J_n^T_B = 1/2^n+1[(P_s-P_a)^⊗(n+1) + 1/3 (P_s+3P_a)^⊗(n+1)],

    since ϕ_2^T_B = 𝔽/2 = (P_s-P_a)/2 and (ϕ_2^⊥)^T_B = I-𝔽/2 = (P_s+3P_a)/2, where P_s and P_a are projectors onto a 2-qubit symmetric subspace and an antisymmetric subspace, respectively. We observe all projectors' coefficients from the second term are positive, while only the projectors that contain odd number of P_a have negative coefficients from the first term, but their corresponding coefficients from the second term are guaranteed to have larger absolute value due to 3^# P_a/3≥ 1 for odd # P_a, where # P_a denotes the number of # P_a in a tensor product of (n+1) projectors. Therefore, the sum of the two terms gives a convex combination of projectors, which is positive, and thus J_n is PPT.

    For uniqueness, note that universal entanglement purification protocols necessarily output ϕ_2 whenever one of the input states is ϕ_2. Therefore, the Choi operator of any universal entanglement purification protocol must have the form (up to twirling)
    
    J = ∑_s⃗∈{0,1}^× n a_s⃗{⊗_i=1^n [ϕ_2^s_i(ϕ_2^⊥)^1-s_i]^A_iB_i}⊗ϕ_2^ÂB̂ + a [⊗_i=1^n (ϕ_2^⊥)^A_iB_i] ⊗(ϕ_2^⊥)^ÂB̂/3

    where a≥0 and a_s⃗≥0  ∀ s⃗. Now we have that 
    
    J^T_B∝   ∑_s⃗∈{0,1}^× n a_s⃗{⊗_i=1^n [(P_s-P_a)^s_i(P_s+3P_a)^1-s_i]^A_iB_i}⊗(P_s-P_a)^ÂB̂
         + a [⊗_i=1^n (P_s+3P_a)^A_iB_i] ⊗(P_s+3P_a)^ÂB̂/3

    J^T_B≥0 implies in particular that the coefficient in front of P_s^⊗ n⊗ P_a is non-negative. This means that 
    
    a ≥∑_s⃗ a_s⃗.

    Now the fidelity of the output state conditioned on success is given by
    
    F(F_1,⋯,F_n) = ∑_s⃗ a_s⃗∏_i=1^n F_i^s_i(1-F_i)^1-s_i/∑_s⃗ a_s⃗∏_i=1^n F_i^s_i(1-F_i)^1-s_i + a∏_i=1^n(1-F_i).

    In particular, when F_2=F_3=⋯=F_n=1/2, we have
    
    F(F_1,1/2,⋯,1/2) = ∑_s⃗:s_1=1 a_s⃗ F_1 + ∑_s⃗':s'_1=0 a_s⃗' (1-F_1)/∑_s⃗:s_1=1 a_s⃗ F_1 + ∑_s⃗':s'_1=0 a_s⃗' (1-F_1) + a(1-F_1).

    If F_1∈(1/2,1) and at least one of {a_s⃗:s_1=0} is positive, we have the strict inequality
    
    F(F_1,1/2,⋯,1/2)    < ∑_s⃗:s_1=1 a_s⃗ F_1 + ∑_s⃗':s'_1=0 a_s⃗' F_1/∑_s⃗:s_1=1 a_s⃗ F_1 + ∑_s⃗':s'_1=0 a_s⃗' F_1 + a(1-F_1)≤aF_1/aF_1+a(1-F_1) = F_1.

    Thus for the operation to be universal, it must be that a_s⃗=0 whenever s_1=0. By the same logic, we find that a_s⃗ must vanish whenever any s_i=0. Moreover, if a>∑_s⃗ a_s⃗, then the second inequality above also becomes strict inequality. Therefore, it must be that a=a_(1,⋯,1), and all other a_s⃗ are zero.




 §.§ No LOCC (even SEP) -to-1 universal EPP with fidelity threshold 1/2

Now we show the analytical approach to calculate the 1-norm of the T matrix in Bloch representation for Choi matrix of PPT n-to-1 universal entanglement purification protocols. Recall:

    J_n ∝(⊗_i=1^n ϕ_2^A_iB_i)⊗ϕ_2^ÂB̂ + 1/3(⊗_i=1^n (ϕ_2^⊥)^A_iB_i)⊗(ϕ_2^⊥)^ÂB̂,

and the Pauli string decomposition of Bell state density matrix ϕ_2^AB and its complementary projector (ϕ_2^⊥)^AB:

    ϕ_2^AB =   1/4(I_A⊗ I_B + X_A⊗ X_B - Y_A⊗ Y_B + Z_A⊗ Z_B),
    (ϕ_2^⊥)^AB =    I-ϕ_AB = 1/4(3I_A⊗ I_B - X_A⊗ X_B + Y_A⊗ Y_B - Z_A⊗ Z_B).

The symmetric nature of ϕ^AB and (ϕ^⊥)^AB in terms of Pauli string expansion guarantees that the T matrix in Bloch representation for J_n is diagonal, and therefore its 1-norm which equals sum of the singular values can be calculated as the sum of the absolute values of the diagonal elements. The key point here is that the diagonal elements are proportional to the coefficients of λ_iA⊗λ_iB terms in J_n, where λ_i are generators of SU(2^n+1), which can be chosen proportional to the (n+1)-qubit Pauli string P_i, for the n-to-1 case. Note that the 1-norm is unitary invariant, and therefore the calculation result does not depend on the choice of SU(2^n+1) generators, which allows us to take Pauli strings as the natural choice. Then regarding the diagonal elements of T matrix we have the following result:

    In the Bloch representation of the Choi matrix J_n,
    
    T_ii = 2^n/3^n+1coeff(G_n+G_n^⊥, P_iA⊗ P_iB),

    where coeff(G,x) denotes the coefficient of the term x in expression G. To make the expressions more concise we have defined G_n=(I_A⊗ I_B + X_A⊗ X_B - Y_A⊗ Y_B + Z_A⊗ Z_B)^⊗(n+1) and G_n^⊥=(3I_A⊗ I_B - X_A⊗ X_B + Y_A⊗ Y_B - Z_A⊗ Z_B)^⊗(n+1)/3.    


    According to the Bloch representation we have
    
    T_ii = ℋ_Aℋ_B/4Tr(ρλ_iA⊗λ_iB),

    where ρ is a unit-trace (normalized) density matrix. In our case both parties A and B of the Choi matrix have n+1 qubits, so ℋ_A=ℋ_B=2^n+1. The Choi matrix corresponds to a trace-non-increasing quantum operation, and is thus not necessarily normalized. It is straightforward to calculate the normalization factor 𝒩 for ρ=𝒩J_n as
    
    𝒩 = 1/Tr(J_n) = 1/1 + 1/3× 3^n+1 = 1/3^n+1.

    Suppose we use (n+1)-qubit Pauli string P_i to construct SU(2^n+1) generator λ_i=rP_i. To satisfy the orthonormality of generators we must have
    
    Tr(λ_iλ_j) = r^2Tr(P_iP_j) = r^2Tr(I_2^n+1)δ_ij = r^22^n+1 = 2δ_ij⇔ r = 2^-n/2.

    Combining the above results we have
    
    T_ii =   2^n+1× 2^n+1/41/3^n+1coeff(J_n, P_iA⊗ P_iB)Tr[(P_iA⊗ P_iB)(λ_iA⊗λ_iB)]
    
            =   4^n2^n/3^n+1coeff(J_n, P_iA⊗ P_iB)Tr[(λ_iA⊗λ_iB)(λ_iA⊗λ_iB)]
    
            =   4^n+12^n/3^n+1coeff(J_n, P_iA⊗ P_iB) =  2^n/3^n+1coeff(G_n+G_n^⊥, P_iA⊗ P_iB),

    where for the last equality we have used the 1/4 prefactor in Eqn. <ref>.

For the 1-norm of diagonal T we have ‖ T‖_1=∑_i|T_ii|, and therefore we need to evaluate the absolute values of the coefficients corresponding to each P_iA⊗ P_iB, which leads to the following theorem that gives a closed-form expression of ‖ T‖_1 for arbitrary n.

    The 1-norm of the T matrix in the Bloch representation of an n-to-1 PPT universal entanglement purification protocols' Choi matrix J_n is:
    
    ‖ T‖_1(even n) = 2^2n+1(3^n-1)+2^n(3^n+1-1)/3^n+1, ‖ T‖_1(odd n) = 2^n(2^n+1-1).



    The main argument of this proof lies in number counting of generator tensor products whose coefficients in G_n and G_n^⊥ have different signs, i.e. we want to find the index set
    
    ℐ={i|coeff(G_n,P_iA⊗ P_iB)coeff(G_n^⊥,P_iA⊗ P_iB)<0}.

    The definition of this index set is motivated by the expansion of ‖ T‖_1 according to Lemma <ref>:
    
    ‖ T‖_1 =   2^n/3^n+1∑_i|coeff(G_n+G_n^⊥, P_iA⊗ P_iB)|
    
            =   2^n/3^n+1(∑_i|coeff(G_n, P_iA⊗ P_iB)| + ∑_i|coeff(G_n^⊥, P_iA⊗ P_iB)|.
                    - .2∑_j∈ℐmin[|coeff(G_n, P_jA⊗ P_jB)|,|coeff(G_n^⊥, P_jA⊗ P_jB)|]),

    where for the second equality we have used the fact |a+b|=|a|+|b|-2min[|a|,|b|], ab<0. The evaluation of the first two terms in the bracket of the second equality is straightforward as follows
    
    ∑_i|coeff(G_n, P_iA⊗ P_iB)| + ∑_i|coeff(G_n^⊥, P_iA⊗ P_iB)| = 4^n+1-1 + 6^n+1-3^n+1/3.

    In the following we will discuss odd and even n separately. Before that we consider what kind of Pauli strings can have coefficients with opposite signs in G_n and G_n^⊥ by enumerating the following 4 cases that cover all possibilities:
    
        
  * P with odd #Y and even (#X+#Z): coeff(G_n, P)<0, coeff(G_n^⊥, P)>0,
        
  * P with odd #Y and odd (#X+#Z): coeff(G_n, P)<0, coeff(G_n^⊥, P)<0,
        
  * P with even #Y and even (#X+#Z): coeff(G_n, P)>0, coeff(G_n^⊥, P)>0,
        
  * P with even #Y and odd (#X+#Z): coeff(G_n, P)>0, coeff(G_n^⊥, P)<0.
    
    where for further simplicity we replace P_iA⊗ P_iB with P, and #· means the number of certain Pauli operator in a Pauli string. Then it is clear that only the 1st and 4th case, i.e. #Y and (#X+#Z) with a different parity, are included in ℐ. Meanwhile, we note that if #I=0 then 1/3=coeff(G_n^⊥, P)<coeff(G_n, P), and coeff(G_n^⊥, P)≥coeff(G_n, P)=1 if #I>0.

    For odd n. Because n+1 is even and #Y and (#X+#Z) have opposite parity, #I is necessarily positive. Therefore, we are sure that coeff(G_n^⊥, P)≥coeff(G_n, P)=1 and thus
    
    ∑_j∈ℐmin[|coeff(G_n, P_jA⊗ P_jB)|,|coeff(G_n^⊥, P_jA⊗ P_jB)|] = |ℐ|.

    Now we start to count the number of Pauli strings in ℐ. For case (1), we have the number of such Pauli strings
    
    N_odd n^(1) =   ∑_i=0^(n-1)/2n+12i+1∑_j=0^(n-2i-1)/2n-2i2j∑_k=0^2j2jk
    
            =   ∑_i=0^(n-1)/2n+12i+1∑_j=0^(n-2i-1)/2n-2i2j2^2j
    
            =   1/2∑_i=0^(n-1)/2n+12i+1(3^n-2i-1)
    
            =   1/23^n+11/2(4^n+1/3^n+1 - 2^n+1/3^n+1) - 1/21/2[(1+1)^n+1-(1-1)^n+1] = 4^n-2^n,

    where we have repeatedly used the binomial theorem, e.g. ∑_i=0^nni=(1+1)^n=2^n, and for the summations with only odd terms in the binomial expansion, we use the following fact
    
    (a+b)^n - (a-b)^n = ∑_i=0^nnka^n-kb^k - ∑_i=0^nnka^n-kb^k(-1)^k = 2∑_odd knka^n-kb^k.

    Similarly, we have the number of Pauli strings in case (4)
    
    N_odd n^(4) =   ∑_i=0^(n-1)/2n+12i∑_j=0^(n-2i-1)/2n-2i+12j+1∑_k=0^2j+12j+1k
    
            =   ∑_i=0^(n-1)/2n+12i∑_j=0^(n-2i-1)/2n-2i+12j+12^2j+1
    
            =   1/2∑_i=0^(n-1)/2n+12i(3^n-2i+1-1)
    
            =   1/2(2^2n+1 + 2^n - 1) + 1/2(1-2^n) = 4^n.

    Combining the above two expressions, we can then calculate ‖ T‖_1(n) for odd n:
    
    ‖ T‖_1(odd n) =   2^n/3^n+1[4^n+1-1 + 6^n+1-3^n+1/3 - 2(4^n - 2^n + 4^n)]
    
            =    2^n(2^n+1-1).


    For even n. Now n+1 is odd, so it is possible that #I=0, for which we have 1/3=coeff(G_n^⊥, P)<coeff(G_n, P). Therefore, we separate the #I=0 and #I>0 cases. Firstly, for case (1) with #I=0 we have
    
    N_even n^(1),#I=0 = ∑_i=0^n/2n+12i+1∑_j=0^n-2in-2ij = ∑_i=0^n/2n+12i+12^n-2i = 1/2(3^n+1-1).

    For case (4) with #I=0 we have
    
    N_even n^(4),#I=0 = ∑_i=0^n/2n+12i∑_j=0^n-2i+1n-2i+1j = ∑_i=0^n/2n+12i2^n-2i+1 = 1/2(3^n+1+1).

    Then, we move on to case (1) with #I>0
    
    N_even n^(1),#I>0 =   ∑_i=0^(n-2)/2n+12i+1∑_j=0^(n-2i-2)/2n-2i2j∑_k=0^2j2jk
    
            =   ∑_i=0^(n-2)/2n+12i+1∑_j=0^(n-2i-2)/2n-2i2j2^2j
    
            =   1/2∑_i=0^(n-2)/2n+12i+1(3^n-2i-2^n-2i+1+1)
    
            =   1/4(4^n+1-2^n+1-2-2× 3^n+1+6+2^n+1-2) = 4^n - 3^n+1/2 + 1/2.

    For case (4) with #I>0 we have
    
    N_even n^(4),#I>0 =   ∑_i=0^(n-2)/2n+12i∑_j=0^(n-2i-2)/2n-2i+12j+1∑_k=0^2j+12j+1k
    
            =   ∑_i=0^(n-2)/2n+12i∑_j=0^(n-2i-2)/2n-2i+12j+12^2j+1
    
            =   1/2∑_i=0^(n-2)/2n+12i(3^n-2i+1-2^n-2i+2+1)
    
            =   1/4(2^n+1-2n-2-2× 3^n+1+8n+6+4^n+1+2^n+1-6n-6) = 4^n -3^n+1/2 + 2^n - 1/2.

    Combining the above 4 scenarios, we calculate ‖ T‖_1(n) for even n:
    
    ‖ T‖_1(even n) =   2^n/3^n+1[4^n+1-1 + 6^n+1-3^n+1/3 - 2(4^n -3^n+1/2 + 2^n - 1/2 + 4^n - 3^n+1/2 + 1/2).
                  - .2/3(1/2(3^n+1-1) + 1/2(3^n+1+1))]
    
            =   2^2n+1(3^n-1)+2^n(3^n+1-1)/3^n+1.


With the above theorem, we have the following result about separability of the unique PPT universal entanglement purification protocol.

    There does not exist any n-to-1 universal qubit entanglement purification protocol with fidelity 1/2 that can be implemented by LOCC for even n. The PPT n-to-1 universal qubit entanglement purification protocol with fidelity threshold 1/2 exactly saturates the necessary condition of separability for odd n.


    Recall Lemma <ref> that the upper bound of the (normalized) separable Choi matrix is 
    
    √(ℋ_Aℋ_B(ℋ_A-1)(ℋ_B-1))/2 = 1/22^n+1(2^n+1-1) = 2^n(2^n+1-1).

    Then the second statement about odd n is obvious. For even n we compare the 1-norm and the upper bound by taking their difference
    
    2^2n+1(3^n-1)+2^n(3^n+1-1)/3^n+1 - 2^n(2^n+1-1) =    2^n3^n+1+3^n-2^n+2/3^n+1 = 2^n+2(3^n-2^n)/3^n+1>0.

    And thus the first statement about even n is also proved.


    It has been proven <cit.> that for a bipartite system with equal subsystem dimensions, the computable cross norm <cit.> or realignment <cit.> (CCNR) criterion is stronger than criterion <ref> from Bloch representation, and both criteria are equivalent when both subsystems are maximally mixed, which is exactly the case for our Choi matrices. 


The following lemma shows that there is no n-to-1 universal protocol for odd n either:

    ∀ n>1, if there exists an SEP n-to-1 EPP that is universal for all entangled isotropic states, there also exists an SEP (n-1)-to-1 EPP that is universal for the same state set.


If there exists an SEP n-to-1 EPP ℰ^(n) that is universal for all entangled isotropic states, then we can construct an (n-1)-to-1 protocol ℰ^(n-1) by fixing the n-th input of ℰ^(n) to be 1/2ϕ_2+1/6(I-ϕ_2), which is a separable state and can be prepared by LOCC:

    ℰ^(n-1)(ρ) = ℰ^(n)(ρ⊗[1/2ϕ_2+1/6(I-ϕ_2)]).

By hypothesis, ℰ^(n) is universal and SEP, meaning ℰ^(n-1) must be universal and SEP as well.




  §.§.§ Spin-off: elementary representation of a hypergeometric function

We have extensively used binomial sums to study the separability of PPT universal entanglement purification protocols. Meanwhile, it is know that there is connection between combinatorial sums and the hypergeometric function (HGF) <cit.>. During our study, we have rediscovered an elementary representation of HGF in a special case. The (Gaussian/ordinary) HGF is defined as follows:

    _2F_1(a,b,c;z)=∑_n=0^∞(a)_n(b)_n/(c)_nz^n/n!,

where (x)_n is the so-called rising factorial, i.e. (x)_n=∏_k=0^n-1(x+k) and (x)_0=1. Using the definition, we see that the following binomial summation, which is used in the proof of Proposition <ref>, can be expressed with HGF <cit.>:

    2^2n+1 - 2^n/3^n = ∑_i=0^(n-1)/2n+12i+13^-2i = (1+n) _2F_1(1/2-n/2,-n/2,3/2;1/9),

considering that n is positive, odd integer. This can be seen as follows. Firstly the HGF can be expanded

    _2F_1(1/2-n/2,-n/2,3/2;1/9) =   ∑_i=0^∞(1/2-n/2)_i(-n/2)_i/(3/2)_i(1/9)^i/i!
    
        =    1 + ∑_i=1^∞∏_k=0^i-1(1/2-n/2+k)(-n/2+k)/(3/2+k)(1/9)^i/i!
    
        =    1 + ∑_i=1^(n-1)/2∏_k=0^i-1(1/2-n/2+k)(2k-n)/(2k-3)(1/9)^i/i! = 1 + ∑_i=1^(n-1)/2P_i(1/9)^i/i!,

where for the third equality we use the fact that the repeated multiplication will become zero for i>(n-1)/2 because (1/2-n/2+k)=0 when k=i-1=(n-1)/2. We also write out the binomial sum

    ∑_i=0^(n-1)/2n+12i+13^-2i = 1+n + ∑_i=1^(n-1)/2(n+1)!i!/(n-2i)!(2i+1)!(1/9)^i/i! = 1+n + ∑_i=1^(n-1)/2Q_i(1/9)^i/i!.

It is easily checked that P_1(1+n)=Q_1, and P_i+1=(n-2i-1)(n-2i)/[2(2i+3)]P_i, and Q_i+1=(n-2i-1)(n-2i)/[2(2i+3)]Q_i. Therefore, according to mathematical induction we have P_i(1+n)=Q_i,∀ i=1,2,…, (n-1)/2. Then it is obvious that Eqn. <ref> holds.




§ BILOCAL CLIFFORD ENTANGLEMENT PURIFICATION PROTOCOLS


In this section, we discuss the universality of bilocal Clifford entanglement purification (biCEP) protocols in more details. In contrast to the previous section on general EPPs, biCEP protocols have explicit implementation based on Clifford circuits, and are thus practical for experimental realization. At the beginning in Sec. <ref>, we review background information that is needed for a self-contained presentation, including the Pauli and Clifford groups, Bell states and their properties, the mechanism of biCEP protocols, the effect of off-diagonal density matrix element on the output fidelity, definitions of some types of Pauli strings which are useful in the following discussion, and the correspondence between biCEP protocols and stabilizer codes. Then in Sec. <ref> we prove the necessary and sufficient conditions for biCEP to be universal for complete BDS sets as defined in the main text (Lemma 7 in the main text). Finally, Sec. <ref> is devoted to the algebraic proof of the non-existence of biCEP protocol that is universal for complete BDS sets (Proposition 8 in the main text), while Sec. <ref> proves that any biCEP protocol that is universal with ordered fidelity is necessarily trivial (Proposition 10 in the main text).



 §.§ Preliminaries and the biCEP mechanism

Here we offer additional background information on bilocal Clifford entanglement purification (biCEP).



  §.§.§ Pauli and Clifford groups

First recall the 4 single-qubit Pauli operators with their common matrix representations

    I = σ_0 = 
        [ 1 0; 0 1 ], 
        X = σ_1 = 
        [ 0 1; 1 0 ], 
        Y = σ_2 =  
        [  0 -i;  i  0 ], 
        Z = σ_3 =  
        [  1  0;  0 -1 ].

Tensor products of n single-qubit Pauli operators, e.g. I_1⊗ X_2⊗…⊗ Z_n, are usually called n-qubit Pauli strings. A Pauli group on n qubits is defined as the set of all n-qubit Pauli strings with phase factors, i.e. 𝒫_n={e^iθπ/2P_1⊗ P_2⊗…⊗ P_n}, θ=0,1,2,3, P_i=I,X,Y,Z. In some cases, the phase factors correspond to global phases of quantum states and can be ignored, which means we can equivalently consider the Pauli group without phase factors, or, the Pauli group modulo a subgroup {e^iθπ/2I^⊗ n}, θ=0,1,2,3, i.e. 𝒫_n={P_1⊗ P_2⊗…⊗ P_n}, P_i=I,X,Y,Z. 
 
Another very relevant group is the Clifford group (the group that consist of all Clifford gates on n qubits) 𝒞_n={C∈U(2^n)| C𝒫_nC^†=𝒫_n}. By definition, each Clifford group element C induces a group automorphism from Pauli group 𝒫_n to itself through conjugation, i.e. ∀ P^(n)∈𝒫_n, P^(n)↦ CP^(n)C^†∈𝒫_n, and it is obviously bijective while preserving group structure as (CP_1^(n)C^†)(CP_2^(n)C^†) = CP_1^(n)P_2^(n)C^†. As a direct corollary, commutators of Pauli strings are also preserved by Clifford conjugate as C[P_1,P_2]C^† = C(P_1P_2-P_2P_1)C^† = CP_1P_2C^† - CP_2P_1C^† = CP_1C^† CP_2C^† - CP_2C^† CP_1C^† = [CP_1C^†,CP_2C^†]. 



  §.§.§ Bell states and useful properties

First let's recall the transpose trick:

    For a maximally entangled state between two parties with a d-dimensional Hilbert space |Φ⟩_AB=(1/√(d))∑_i=0^d-1|i⟩_A|i⟩_B, the following property holds:
    
    (O_A⊗ I_B)|Φ⟩_AB = (I_A⊗ O_B^T)|Φ⟩_AB

    for any d× d matrix O.

The 3 Bell states |ψ^+⟩=(|01⟩ + |10⟩)/√(2), |ψ^-⟩=(|01⟩ - |10⟩)/√(2), |ϕ^-⟩=(|00⟩ - |11⟩)/√(2) can be considered as the result of an unilateral Pauli error applied to |ϕ_2⟩=(|00⟩ + |11⟩)/√(2):

    |ψ^+⟩∝ (I⊗ X)|ϕ_2⟩∝ (X⊗ I)|ϕ_2⟩, |ψ^-⟩∝ (I⊗ Y)|ϕ_2⟩∝ (Y⊗ I)|ϕ_2⟩, |ϕ^-⟩∝ (I⊗ Z)|ϕ_2⟩∝ (Z⊗ I)|ϕ_2⟩.

This suggests that when dealing with n noisy Bell states distributed to two parties we can treat all noise processes as happening on only one of the two parties, i.e. 

    ⊗_i=1^nρ^(i)_AB = ⊗_i=1^n(Id_A⊗𝒩^(i)_B)(|ϕ_2⟩⟨ϕ_2|) = (Id_A⊗𝒩_B)(|ϕ_2⟩⟨ϕ_2|^⊗ n),

where 𝒩^(i)(ρ) = ∑_j=0^3p^(i)_jσ_jρσ_j^†, and thus 𝒩 can be expressed in Kraus representation with Kraus operators being n-qubit Pauli strings. Here we explicitly denote two parties with subscripts A and B. In fact the above properties of Bell states are also results of the transpose trick.

It is noteworthy that n pure Bell pairs distributed to two parties as a whole are equivalent to a bipartite maximally entangled state where the dimension of a single party's Hilbert space is d=2^n, i.e. the n-qubit space:

    |ϕ_2⟩⟨ϕ_2|_AB^⊗ n =   1/2^n/2⊗_i=1^n(|0⟩^(i)_A|0⟩^(i)_B + |1⟩^(i)_A|1⟩^(i)_B)
    
        =   1/2^n/2(|00… 0⟩_A|00… 0⟩_B + |00… 1⟩_A|00… 1⟩_B + … + |11… 1⟩_A|11… 1⟩_B).

This can be used in combination with the transpose trick to further explain the mechanism of biCEP as follows.



  §.§.§ The biCEP mechanism

Recall the definition of biCEP protocols:

    An n-to-1 biCEP protocol intakes n independent noisy Bell pairs between Alice and Bob. Alice applies n-qubit Clifford C on her n qubits while Bob performs the entry-wise complex conjugate of C, i.e. C^*, on his n qubits. Then Alice and Bob measure in computational basis their (n-1) qubits from (n-1) input Bell pairs, respectively, and perform a single round of classical communication of the measurement outcomes. The protocol is considered successful if all (n-1) pairs of measurement outcomes have the same parity (00 or 11), and unsuccessful otherwise.

We focus on the circuit of biCEP before measurement to explicitly demonstrate its entanglement purification mechanism as:

    (C_A⊗C^*_B)(⊗_i=1^nρ^(i)_AB) =   (C_A⊗C^*_B)∘(Id_A⊗𝒩_B)(|ϕ_2⟩⟨ϕ_2|^⊗ n)
    
        =   (Id_A⊗C^*_B)∘(C_A⊗Id_B)∘(Id_A⊗𝒩_B)(|ϕ_2⟩⟨ϕ_2|^⊗ n)
    
        =   (Id_A⊗C^*_B)∘(Id_A⊗𝒩_B)∘(C_A⊗Id_B)(|ϕ_2⟩⟨ϕ_2|^⊗ n)
    
        =   (Id_A⊗C^*_B)∘(Id_A⊗𝒩_B)∘(Id_A⊗C^T_B)(|ϕ_2⟩⟨ϕ_2|^⊗ n)
    
        =   (Id_A⊗C^*_B∘𝒩_B∘C^T_B)(|ϕ_2⟩⟨ϕ_2|^⊗ n)
    
        =   (Id_A⊗C_B∘𝒩^*_B∘C^†_B)^*(|ϕ_2⟩⟨ϕ_2|^⊗ n)
    ≃   (Id_A⊗C_B∘𝒩_B∘C^†_B)(|ϕ_2⟩⟨ϕ_2|^⊗ n)

where C(ρ)=Cρ C^†, ∀ C∈𝒞_n, and from line 3 to line 4 we have used Lemma <ref>. On the second to last line, the complex conjugate will only add a π phase to Pauli Y and is still equivalent to Y itself as we work with Pauli group without a phase factor. Therefore, in the last line we see that the effect of the biCEP circuit is to transform Pauli error string on the n Bell pairs according to the conjugate by the chosen Clifford gate, and the resulting noise channel remains a mixture of Pauli strings, as mentioned in the main text. 



  §.§.§ Pauli twirling does not change the successful biCEP output fidelity

Having understood the mechanism of biCEP protocols, here we explicitly demonstrate that we can safely restrict our discussion to the BDS input without loss of generality, because for arbitrary input states the biCEP output fidelity will be identical to the the successful output fidelity when all input states are Pauli twirled into BDS:

    For arbitrary n input states ρ_i,i=1,2,…,n to a certain biCEP protocol, the successful output fidelity is always equal to the successful output fidelity when each ρ_i is Pauli twirled into BDS ρ'_i and input to the same biCEP protocol.


    The total input state density matrix, i.e. ⊗_i=1^nρ_i, is a linear combinations of the following tensor products:
    
    |ψ^(i_1)⟩⟨ψ^(j_1)|⊗|ψ^(i_2)⟩⟨ψ^(j_2)|⊗…⊗|ψ^(i_n)⟩⟨ψ^(j_n)|,

    where i_l,j_m=0,1,2,3, and for simplicity we defined:
    
    |ψ^(0)⟩ = |ϕ_2⟩, |ψ^(1)⟩ = |ψ^+⟩, |ψ^(2)⟩ = |ψ^-⟩, |ψ^(3)⟩ = |ϕ^-⟩.

    It is then obvious that when input states are restricted to BDS, we always have i_l=j_l, but when we allow input states to be non-diagonal in Bell basis, we can have i_l≠ j_l. Now we consider the effect of an arbitrary n-to-1 biCEP protocol on the input state ⊗_i=1^nρ_i. Notice that the above tensor product can be re-written as:
    
    (|ψ^(i_1)⟩|ψ^(i_2)⟩…|ψ^(i_n)⟩) (⟨ψ^(j_1)|⟨ψ^(j_2)|…⟨ψ^(j_n)|) = (I_n⊗ P_n^i_1,i_2,…,i_n)|Φ_n⟩⟨Φ_n|(I_n⊗ P_n^j_1,j_2,…,j_n),

    where |Φ_n⟩ = 2^-n/2∑_k=0^2^n-1|kk⟩ is the maximally entangled state corresponding to n noiseless 2-qubit maximally entangled state |ψ^0⟩=|ϕ_2⟩, and I_n⊗ P_n denotes the unilateral n-qubit Pauli string. According to the biCEP mechanism, after the bilocal Clifford circuit, every above tensor product will be transformed into:
    
    (I_n⊗P_n^i_1,i_2,…,i_n)|Φ_n⟩⟨Φ_n|(I_n⊗P_n^j_1,j_2,…,j_n),

    where P_n is the result of the Clifford conjugate of P_n, corresponding to transformed i,j. Now recall the success criterion of biCEP. We need all (n-1) measured qubit pairs to have equal parity, i.e. the success probability will be determined by the POVM: Π_succ = ⊗_k=2^n(|00⟩⟨ 00| + |11⟩⟨ 11|). Then the success probability for the above tensor product as input to the biCEP protocol is
    
    p_succ =   tr[(⊗_k=2^n|ψ^(i_k)⟩⟨ψ^(j_k)|)(⊗_k=2^n(|00⟩⟨ 00| + |11⟩⟨ 11|)) ]
    
            =   ∏_k=2^ntr[|ψ^(i_k)⟩⟨ψ^(j_k)|(|00⟩⟨ 00| + |11⟩⟨ 11|)],

    which is obviously zero as long as there exists k∈{2,…,n} s.t. i_k≠j_k. Now let us explicitly consider the input tensor products that include off-diagonal terms in the Bell basis. It is clear that P_n^i_1,i_2,…,i_n≠P_n^j_1,j_2,…,j_n in this case, and then the difference is either among k∈{2,…,n} or within k=1. If the difference is among k∈{2,…,n} the tensor product will not contribute to the success output; if the difference is within k=1 then even in the case of success the tensor product does not contribute to the fidelity.




  §.§.§ Types of relevant Pauli strings

When the input states are BDS, their total tensor product can be decomposed into a sum of tensor products of pure Bell states corresponding to different patterns of single Pauli errors on |ϕ_2⟩^⊗ n. As we know, the Pauli error string transformation by an n-qubit Clifford gate determines the properties of the corresponding n-to-1 biCEP protocol. Moreover, for simplicity we can also represent the tensor product of Bell pairs with their corresponding Pauli strings, which we also call error pattern strings. Therefore, we define different types of Pauli strings that are of interest. As in the main text, we use a superscript to denote the number of qubits the Pauli strings act on, and a subscript to denote their type.

    A single noiseless string P_s^(n) is an n-qubit Pauli string in which there exists at least one identity operator I.

We have encountered them in the main text, and for the sake of conciseness we give them a specific name in the Supplemental Material. These Pauli strings correspond to the tensor product of n Bell states where at least one of them is the objective |ϕ_2⟩ (with no error), which gives them the name “single noiseless”. Suppose the input states to biCEP include a noiseless |ϕ_2⟩, the Kraus operators of their corresponding unilateral Pauli channel are certain to be single noiseless strings. There are (4^n-3^n) such Pauli strings for n Bell pairs.


    A harmless string P_h^(n) is an n-qubit Pauli string which satisfies one of the following two conditions: (i) its first (leftmost) component is always the identity operator I when the rest n-1 components are either the identity operator I or Pauli Z, or (ii) there exist Pauli X or Y operator(s) among its n-1 components other than the first one.

If any tensor product of n Bell states has an error pattern string that is transformed to a harmless string by the biCEP circuit, there is zero probability that the output Bell state is not |ϕ_2⟩ when we consider the purification successful. This suggests unit output fidelity conditioned on success. On the other hand, if the measurement outcomes on n-1 Bell pairs suggest a failure, it does not contribute to the output fidelity conditioned on success. There are [2^n-1+4×(4^n-1-2^n-1)] = (4^n-3× 2^n-1) such Pauli strings for n Bell pairs.


    A correct string P_c^(n) is an n-qubit Pauli string whose first (leftmost) component is the identity operator I, while the rest n-1 components are either the identity operator I or the Pauli Z.

By definition, correct strings are also harmless strings, and they will contribute positively to the output fidelity conditioned on success because they give rise to n-1 pairs of measurement outcomes all with an even parity on both parties while the unmeasured pair remains intact. They satisfy the first condition in the definition of harmless strings. There are 2^n-1 such Pauli strings for n Bell pairs.


    An incorrect string P_ic^(n) is an n-qubit harmless string that is not a correct string.

These strings will not give rise to successful measurement outcomes because there will be at least a pair of measurement outcomes with odd parity. Thus they do not contribute to the successful output fidelity. They satisfy the second condition in the definition of harmless strings. There are (4^n-2^n+1) such Pauli strings for n Bell pairs.


    An undetectable string P_u^(n) is an n-qubit Pauli string whose first (leftmost) component is Pauli X, Y or Z, while the rest n-1 components are either the identity operator I or the Pauli Z.

If any n Bell states' error pattern string is transformed to an undetectable string through the biCEP circuit, the biCEP protocol will be considered successful because a single Pauli Z error on a Bell state does not vary the parity of measurement outcomes in computational basis, while the output Bell state is not |ϕ_2⟩ and this contributes to infidelity of the purification output. There are 3× 2^n-1 such Pauli strings for n Bell pairs.



  §.§.§ Correspondence between QEC and EPP

In this section, we demonstrate that any [[n,1,d]] stabilizer code can be used to construct an n-to-1 biCEP by running the error correction code in an error detection mode <cit.>, with only encoding and decoding without intermediate syndrome measurement. The correspondence between QEC and EPP is demonstrated in Fig. <ref>. In fact, such correspondence goes beyond k=1 to general stabilizer codes <cit.>. We note that this is mainly for conceptual understanding the mechanism of EPP, but not necessarily a desirable way of implementing a QEC in practice <cit.>. 


We first consider one way to implement quantum error correction corresponding to (a): We have one physical quantum state |ψ⟩ to protect, and four ancilla qubits are prepared in a certain computational basis state; then an encoding circuit U_enc is applied on all five qubits to create an encoded logical quantum state; at the end the decoding circuit, i.e. the inverse of encoding circuit U_dec = U_enc^-1 = U_enc^† (such that when there is no error the initial state is exactly unchanged), is applied to the encoded quantum state, and four ancilla qubits are measured in computational basis; combined with the knowledge of the ancilla qubits' initial state and measurement outcomes, we should be able to infer whether there is an error happening in between the encoding and the decoding and can in principle perform error correction on the physical data qubit. 

The above mechanism of QEC can be understood in an equivalent approach based on a shared entangled state corresponding to (b): At the beginning several (five in the figure) pure Bell states |ϕ_2⟩ are prepared by Alice (left) and half of each pair will be sent to Bob (right). Alice measures four out of her five qubits in the computational basis after applying the transpose of encoding circuit in (a) which is also the complex conjugate of the decoding circuit, i.e. U_enc^T = U_dec^*; Bob will apply the same decoding circuit as in (a) and measure four qubits initially entangled with the four qubits measured by Alice; combining the measurement outcomes from Alice and his own, Bob can in principle infer whether certain error has happened during transmission of the five qubits he receives and perform the corresponding error correction operation; the transmission of a physical state |ψ⟩ from Alice to Bob can be achieved a posteriori, supposing Alice measures her remaining qubit in an arbitrary basis which will project Bob's remaining qubit to an arbitrary state |ψ⟩. Note that in (b) if we only look at the operations on Bob's side, it is obvious that they are identical to (a), and if Alice does not measure her remaining qubit the entire bipartite process is exactly a 5-to-1 entanglement purification. 

Furthermore, we note that the encoding and decoding circuits are Clifford for any stabilizer QEC codes, and thus any [[n,1,d]] stabilizer code corresponds to an n-to-1 biCEP. To see this, consider the constructive algorithm of stabilizer encoding <cit.>. While the technical detail of the algorithm is beyond the scope of this work, it is worth mentioning that the encoding circuit can be constructed by controlled Pauli strings and single-qubit Clifford gates (single-qubit Pauli and Hadamard) only. Therefore, to argue that the encoding circuit is Clifford we only need to show that the controlled Pauli strings are Clifford gates:

    Any controlled Pauli string applied on (n+1) qubits (n≥ 1) is Clifford.


    Consider an arbitrary controlled Pauli string (|0⟩⟨ 0|⊗ I + |1⟩⟨ 1|⊗ P) with an n-qubit Pauli string P. We then consider an arbitrary (n+1)-qubit Pauli string P^(1)⊗ P^(n), where P^(1) is an arbitrary single-qubit Pauli and P^(n) is an arbitrary n-qubit Pauli string, and P^(1) corresponds to the control qubit. Then we can evaluate the controlled Pauli string conjugate of the arbitrary Pauli string
    
    (|0⟩⟨ 0|⊗ I + |1⟩⟨ 1|⊗ P)P^(1)⊗ P^(n)(|0⟩⟨ 0|⊗ I + |1⟩⟨ 1|⊗ P)
    
            =    |0⟩⟨ 0|⊗ I (P^(1)⊗ P^(n)) |0⟩⟨ 0|⊗ I + |0⟩⟨ 0|⊗ I (P^(1)⊗ P^(n)) |1⟩⟨ 1|⊗ P
       + |1⟩⟨ 1|⊗ P (P^(1)⊗ P^(n)) |0⟩⟨ 0|⊗ I + |1⟩⟨ 1|⊗ P (P^(1)⊗ P^(n)) |1⟩⟨ 1|⊗ P
    
            =   ⟨ 0|P^(1)|0⟩|0⟩⟨ 0|⊗ P^(n) + ⟨ 0|P^(1)|1⟩|0⟩⟨ 1|⊗ P^(n)P
       + ⟨ 1|P^(1)|0⟩|1⟩⟨ 0|⊗ PP^(n) + ⟨ 1|P^(1)|1⟩|1⟩⟨ 1|⊗ PP^(n)P

    Recall the representations of P^(1)=I,X,Y,Z. We see from the last line that the result will always be a Pauli string up to a global phase, which completes the proof.




 §.§ Necessary and sufficient condition of universal biCEP for complete BDS sets

Here we prove the universality condition for biCEP (Lemma 7 in the main text), and we separate the proof into two parts for necessity and sufficiency, respectively.



  §.§.§ Necessity

If an n-to-1 biCEP protocol is universal for a certain class of input states, by definition of universality the output fidelity conditioned on success is one when the input states include at least one noiseless Bell state. Hence, the necessary condition for biCEP to be universal for complete BDS sets can be obtained using the notion of error pattern strings:

    If an n-to-1 biCEP protocol using n-qubit Clifford gate C is universal for a complete BDS set, then all single noiseless strings of length n will be transformed by C into harmless strings only.
    


    Recall that any complete BDS set 𝒮 includes ϕ_2, and satisfies that each of Φ^-,Ψ^+,Ψ^- is in the support of at least a BDS in 𝒮, where we have rewritten Φ^±=|ϕ^±⟩⟨ϕ^±|, Ψ^±=|ψ^±⟩⟨ψ^±| for simplicity. Then consider arbitrary combination of n BDS from 𝒮 including at least a ϕ_2, and we know that the tensor product of such n BDS equals to (ϕ_2)^⊗ n undergoing a unilateral Pauli channel. Obviously the Kraus operators of the unilateral Pauli channel for such input states are single noiseless strings. Moreover, all single noiseless strings are possible when we exhaust all possible input state combinations. 
    
    Then suppose an n-to-1 biCEP is universal. The output fidelity for such input states should be unity conditioned on success. The output states before measurement are statistical mixtures of Bell state tensor products corresponding to transformation results of single noiseless strings. Unit fidelity of the successful output state requires that single noiseless strings can only be transformed into harmless strings by definition.




  §.§.§ Sufficiency

Before proving the sufficient condition, we first provide a formal expression of output fidelity from a biCEP which satisfies Lemma <ref> (i.e. candidate of universal biCEP) given the input state:

    ρ_in = ⊗_i=1^n[F_iϕ_2 + (1-F_i)(x_iΨ^+ + y_iΨ^- + z_iΦ^-)]

where F_i is the fidelity of the i-th input BDS, and x_i+y_i+z_i=1 to satisfy normalization of the error probability. Because the tensor product of input Bell diagonal states can be decomposed as a sum of a weighted tensor product of pure Bell states, which correspond to different error pattern strings, here we define the weight function:

    W:{ρ_in}×𝒫_n→ [0,1] is the weight function that output the weight of a tensor product of pure Bell states corresponding to a Pauli error string P∈𝒫_n in the decomposition of input state ρ_in.


    The weight function satisfies the following property of normalization: ∑_P∈𝒫_nW(ρ_in,P)=1, ∀ρ_in. Without loss of clarity, in the following we will omit the input state in weight function.

We would like to express the biCEP output according to the underlying transformation of Pauli strings by the corresponding Clifford. To make description concise, we define the following types of Pauli strings:

    A Type 1 (T_1) Pauli string has the form of I⊗ P, P∈𝒫_n-1.


    A Type 2 (T_2) Pauli string has the form of {X,Y,Z}⊗ P, P∈𝒫_n-1.

Then we have the output fidelity and success probability from an universal biCEP candidate:

    If a biCEP protocol satisfies the necessary condition of universal for general BDS input, then the success probability and output fidelity conditioned on success are given as follows
    
    p_suss =   ∑_s∈ T_1,s→ s'∈CorrectW(s) + ∑_t∈ T_2,t→ t'∈CorrectW(t) + ∑_r∈ T_2,r→ r'∈UndetectableW(r), 
    
            F_out =   ∑_s∈ T_1,s→ s'∈CorrectW(s) + ∑_t∈ T_2,t→ t'∈CorrectW(t)/∑_s∈ T_1,s→ s'∈CorrectW(s) + ∑_t∈ T_2,t→ t'∈CorrectW(t) + ∑_r∈ T_2,r→ r'∈UndetectableW(r)
    
            =   1/1 + ∑_r∈ T_2,r→ r'∈UndetectableW(r)/∑_s∈ T_1,s→ s'∈CorrectW(s) + ∑_t∈ T_2,t→ t'∈CorrectW(t).



    According to the assumption that the biCEP protocol is an universal candidate for a complete BDS set, T_1 Pauli strings are all single noiseless strings and thus will only be transformed to harmless strings following Lemma <ref>; T_2 Pauli strings can be transformed to either harmless or undetectable strings. Then the expressions follow naturally from the definition of success criterion of biCEP protocols.

Equipped with the above preparation, we prove the sufficient condition of universality as follows:

    If an n-qubit Clifford gate C transforms all single noiseless strings of length n into harmless strings only, then the n-to-1 biCEP protocol using C is universal for all complete BDS sets.
    


    In this proof we first find the lower bound of successful output fidelity, and then show that this lower bound is always not less than the highest input fidelity as long as all input fidelities are greater than 1/2, which is one requirement in the definition of complete BDS set. 
    
    It is obvious from Lemma <ref> that:
    
    F_out≥1/1 + max_{F_m}{∑_r∈ T_2,r→ r'∈UndetectableW(r)}/min_{F_m}{∑_s∈ T_1,s→ s'∈CorrectW(s)} + min_{F_m}{∑_t∈ T_2,t→ t'∈CorrectW(t)}

    where max(min)_{F_m}{·} means the highest (lowest) possible value of a certain object given fixed input state fidelities {F_m}. The highest and lowest possible values involved can be found by considering “worst case scenarios”:
    
        
  * ∑_s∈ T_1,s→ s'∈CorrectW(s)≥∏_m=1^nF_m, where the right hand side equals the probability of the tensor product of n noiseless ϕ_2's which corresponds to the identity string. The identity will always be transformed back to identity which is a correct string, and the worst case scenario is when there is no other type 1 string transformed to correct string.
        
  * ∑_t∈ T_2,t→ t'∈CorrectW(t)≥ 0. There is no guarantee that there are type 2 strings which will be transformed to correct strings, thus the worst case scenario is that no type 2 string is transformed to correct strings which gives the value 0.
        
  * ∑_r∈ T_2,r→ r'∈UndetectableW(r)≤∏_m=1^n(1-F_m). Suppose the rank of m-th input BDS is r_m. We have in total ∏_m=1^n(r_m-1) type 2 strings which are not single noiseless strings. These strings are possible to be transformed to undetectable strings. As there are 3× 2^n-1 undetectable strings, as long as ∏_m=1^n(r_m-1)≤ 3× 2^n-1 the worst case scenario would be that all these ∏_m=1^n(r_m-1) type 2 strings are transformed to undetectable strings.
    
    With above bounds, we have the explicit lower bound of output fidelity for fixed input state fidelities {F_m}
    
    F_out≥∏_m=1^nF_m/∏_m=1^nF_m + ∏_m=1^n(1-F_m) = F_out^min

    which is independent of the previously chosen i-th input Bell state, as expected from the arbitrariness of the choice. 

    Now we move on to prove that this lower bound is in fact guaranteed to be higher than the highest input fidelity F_max=max[{F_m}] if F_m≥ 1/2. We rewrite the lower bound expression as
    
    F_out^min =    F_max∏_m≠ m_maxF_m/F_max∏_m≠ m_maxF_m + (1-F_max)∏_m≠ m_max(1-F_m)
    
            =    F_maxF_max∏_m≠ m_maxF_m + (1-F_max)∏_m≠ m_maxF_m/F_max∏_m≠ m_maxF_m + (1-F_max)∏_m≠ m_max(1-F_m)≥ F_max

    where m_max refers to the index of the input Bell state with highest fidelity F_max, and the inequality holds because 1-F_m≤ F_m if F_m≥ 1/2.
    In this way, the statement is proved.




  §.§.§ Example usage of the necessary and sufficient condition

Here we demonstrate the effectiveness of the universal condition by proving the universal property of DEJMPS (a 2-to-1 biCEP with C=CNOT) protocol through Pauli string checking only, which is proved elsewhere <cit.> analytically.

    DEJMPS protocol is universal for bit-flipped Bell states (in the form of ρ_X = Fϕ_2 + (1-F)Ψ^+) with fidelity higher than 1/2 as input, but not universal for all entangled isotropic states.


    Notice that the set of bit-flipped Bell states is not a complete BDS set, the universality condition also needs to be modified a bit from the conditions proved above, and we only need to require that all Pauli strings with at least one identity operator while all remaining Pauli operators being X can be transformed by the Clifford gate to harmless strings. The proof of this modification is very similar to the above proofs, so is omitted. Now we only need to check the transformation of the following 2-qubit Pauli strings by CNOT:
    
    I⊗ I→ I⊗ I, X⊗ I→ X⊗ X, I⊗ X→ I⊗ X,

    and the transformed Pauli strings are obviously all harmless strings so DEJMPS is indeed universal for bit-flipped Bell states. Then notice that the set of all entangled isotropic states is a complete BDS set. We need to evaluate the transformation of some additional Pauli strings:
    
    I⊗ Y→ Z⊗ Y, I⊗ Z→ Z⊗ Z,
       Y⊗ I→ Y⊗ X, Z⊗ I→ Z⊗ I.

    Notice that Z⊗ I which is unchanged is not a harmless string. This means that DEJMPS cannot stay universal for all entangled isotropic states.




 §.§ Proof of no universal biCEP protocol

Here we prove the Proposition 8 in the main text.


    For any 𝒮∈ℭ and n≥2, n-to-1 biCEP protocols cannot be universal for 𝒮.


    We utilize the fact that Pauli string transformation via Clifford conjugate is an automorphism of Pauli group, which means that any group structure must be preserved, specifically group multiplication. Thus we examine group structures within single noiseless strings and harmless strings.

    Firstly, we can make the following observations about single noiseless strings:
    
        
  * They include n Pauli subgroups which are isomorphic to (n-1)-qubit Pauli group 𝒫_n-1, in form of I_i⊗(⊗_j≠ iP_j), P_j∈𝒫_1 where the Pauli operator on the i-th qubit is fixed to be identity.
        
  * These subgroups are necessarily overlapping, and any 2≤ m≤ n of these subgroups have 4^n-m overlapping terms which are isomorphic to (n-m)-qubit Pauli group 𝒫_n-m. This can be verified by calculating the number of total Pauli strings according to the overlapping structure of these n subgroups:
        
    ∑_k=1^n(-1)^k+1nk4^n-k = 4^n - 3^n

        according to inclusion-exclusion principle. It is clear that this number equals the number of single noiseless strings. Meanwhile, all Pauli strings in such subgroups are single noiseless strings according to definition, thus we have shown that single noiseless strings indeed possess the structure with these subgroups.
    

    We can see that P_h^(n) at least contains one such set of Pauli strings, i.e. S_0≡{I⊗ P, P∈𝒫_n-1}. We thus prove the theorem by contradiction through the attempt of constructing other such sets via replacing strings in S_0 with other harmless strings not in S_0. 
        
    Suppose we can construct a different set satisfying same properties as S_0. The replacing strings from outside S_0 can only have one type or three types of Pauli operators on their first site, because supposing there are only two types of Pauli operators on the first site, product between non-S_0 strings can get the third type of Pauli operator on the first site, which is obviously outside the set thus contradicting the assumption. Furthermore, if the replacing strings have only one type of non-identity Pauli on the first site, then there are necessarily 4^n-1/2 original S_0 strings in the newly constructed set with 4^n-1 elements in total. This can be seen as follows: Suppose that there are m S_0 strings including the identity string remaining; then as the m strings are different, multiplication of any non-S_0 string and remaining S_0 strings will produce m-1 non-S_0 strings different from the one involved in multiplication. This means that there should be at least m added non-S_0 strings. Meanwhile, closed-ness under multiplication requires that there are at most m added non-S_0 strings, because for l different non-S_0 strings with same type of Pauli on the first site there are at least l different S_0 strings, including identity string, which will be generated from multiplication between the l non-S_0 strings. Therefore, from the requirement on the size of newly constructed set we have that m+m=4^n-1⇔ m=4^n-1/2. Similarly, suppose there are three different first-site Pauli operators among added non-S_0 strings. the multiplication between added strings and remaining S_0 strings, together with multiplication between added strings with same type of first-site Pauli requires that the number of strings with any first-site Pauli to be equal, i.e. 4n=4^n-1⇔ m=4^n-2.

    Suppose we can construct another subset of harmless strings that share same properties with S_0. The added string with non-identity first-site Pauli should not have same last (n-1)-qubit Pauli string as that of any remaining S_0 string, because otherwise the multiplication of the non-S_0 string and S_0 string can get outside harmless string. This means that we can divide the constructed set into two non-intersecting subsets, of which one is S_0 subgroup I⊗{P^(n-1)} and the other is {X,Y,Z}⊗ p_X,Y,Z{P^(n-1)}, where p_X,Y,Z∉{P^(n-1)} and p_X,Y,Z{P^(n-1)} are cosets of 𝒫_n-1 subgroup {P^(n-1)}. Then the 4^n-1 strings in the constructed set actually have all 4^n-1 different (n-1)-qubit Pauli strings on their last n-1 sites, i.e. other than the leftmost Pauli. Therefore, the first part I⊗{P^(n-1)} should contain all 2^n-1 = 4^(n-1)/2 correct strings. Now we come across an obvious contradiction that any 2≤ m≤ n of these constructed subgroups have at least have 4^(n-1)/2 overlapping terms which are the correct strings as mentioned above, while they should have 4^n-m overlapping terms which are isomorphic to (n-m)-qubit Pauli group 𝒫_n-m. 
    
    In this way, we know that such construction is impossible, i.e. we cannot find a way to construct n different but overlapping harmless string subsets to satisfy the properties of single noiseless strings. This contradicts with the assumption that there exists a universal biCEP, which concludes that there does not exist universal biCEP for any integer n>1.




 §.§ Bilocal Clifford protocols that are universal with ordered fidelity are necessarily trivial

Our definition of universality (Definition 1 in the main text) only requires non-decreasing output fidelity in comparison with highest input fidelity, it is thus possible that the output fidelity is always equal to the highest input fidelity. Obviously such result can be achieved by Clifford which can be decomposed into the form I⊗ C_n-1, C_n-1∈𝒞_n-1 supposing we have ordered fidelity and always put the input state with highest fidelity on the first site and do not measure it. We call such universal biCEP protocols trivially universal. In the following we prove the Proposition 10 in the main text, that given ordered fidelity, all universal biCEP must be trivial. Here we rephrase the proposition in terms of Pauli string transformation.


    If the underlying Clifford gate of a biCEP maps Pauli strings in the form of I⊗ P where P is arbitrary (n-1)-qubit Pauli string to harmless strings only, then for input states ρ_1,…,ρ_n the output fidelity equals the fidelity of ρ_1, where we keep the qubits of ρ_1 unmeasured.


    Consider Pauli strings X⊗ I_n-1, Y⊗ I_n-1, and Z⊗ I_n-1. Each of these three strings commutes with all of strings of the form I⊗ P, where P∈𝒫_n-1. We have shown earlier in the proof of no universal biCEP that if the set {I⊗ P: P∈𝒫_n-1} is mapped to harmless strings, then it must be mapped to a set that contains all strings of the form I⊗{I,Z}^⊗ n-1, i.e. correct strings. 
    
    Since Clifford conjugation preserves commutation relations, we must have that all three of X⊗ I_n-1, Y⊗ I_n-1, and Z⊗ I_n-1 are mapped to undetectable strings. This can be shown as follows: Suppose that there exist one of them mapped to harmless string, then it can only be mapped to incorrect string (all correct strings are mapped to from I⊗ P); incorrect strings have at least one X or Y among their last n-1 Pauli operators, while none correct strings has X or Y among their last n-1 Pauli operators; this means that any incorrect string cannot commute with all correct strings; however, X⊗ I_n-1, Y⊗ I_n-1, and Z⊗ I_n-1 commute with any I⊗ P, therefore even after Clifford conjugation they should still commute, and this can never be achieved if X⊗ I_n-1, Y⊗ I_n-1, and Z⊗ I_n-1 are mapped to incorrect strings, leading to contradiction.
    
    Furthermore, since Clifford conjugation preserves multiplication, we find that for any P∈𝒫_n-1, if I⊗ P is mapped to a correct string, X⊗ P, Y⊗ P, and Z⊗ P are mapped to undetectable strings. Conversely, if I⊗ P is mapped to an incorrect string, then X⊗ P, Y⊗ P, and Z⊗ P are mapped to incorrect strings. Now we can calculate the output fidelity of a universal biCEP via Lemma <ref>, where
    
    ∑_r∈ T_2,r→ r'∈UndetectableW(r)    = ∑_I⊗ P →{P_c^(n)}[ W(X⊗ P) + W(Y⊗ P) + W(Z⊗ P) ], 
    ∑_s∈ T_1,s→ s'∈CorrectW(s)    = ∑_I⊗ P →{P_c^(n)} W(I⊗ P), 
    ∑_t∈ T_2,t→ t'∈CorrectW(t)    = 0.

    Notice that for any P we always have W(X⊗ P) + W(Y⊗ P) + W(Z⊗ P) = (1-F_1)W(I⊗ P). It is then easy to see that F_out = F_1.






 Paper Title: Perfect quantum strategies with small input cardinality.










strandafir@us.es
Departamento de Física Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain


Department of Physics, Stockholm University, 10691 Stockholm, Sweden


adan@us.es
Departamento de Física Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain
Instituto Carlos I de Física Teórica y Computacional, Universidad de Sevilla, E-41012 Sevilla, Spain

Perfect quantum strategies with small input cardinality
    Adán Cabello
    August 5, 2024
=======================================================





§ ABSTRACT 
A perfect strategy is one that allows the mutually in-communicated players of a nonlocal game to win every trial of the game. Perfect strategies are basic tools for some fundamental results in quantum computation and crucial resources for some applications in quantum information. 
Here, we address the problem of producing qudit-qudit perfect quantum strategies with a small number of settings. For that, we exploit a recent result showing that any perfect quantum strategy induces a Kochen-Specker set. We identify a family of KS sets in even dimension d ≥ 6 that, for many dimensions, require the smallest number of orthogonal bases known: d+1. 
This family was only defined for some d. We first extend the family to infinitely many more dimensions.
Then, we show the optimal way to use each of these sets to produce a bipartite perfect strategy with minimum input cardinality. As a result, we present a family of perfect quantum strategies in any (2,d-1,d) Bell scenario, with d = 2^kp^m for p prime, m ≥ k ≥ 0 (excluding m=k=0), d = 8p for p ≥ 19, d=kp for p > ((k-2)2^k-2)^2 whenever there exists a Hadamard matrix of order k, other sporadic examples, as well as a recursive construction that produces perfect quantum strategies for infinitely many dimensions d from any dimension d' with a perfect quantum strategy. We identify their associated Bell inequalities and prove that they are not tight, which provides a second counterexample to a conjecture of 2007.






§ INTRODUCTION






 §.§ Motivation




The qubit is the basic unit of quantum information <cit.>. Bell nonlocality <cit.> is arguably the most powerful resource of quantum theory. Interestingly, “maximal” bipartite Bell nonlocality cannot be achieved with qubit-qubit correlations, but requires qudit-qudit correlations with d ≥ 3.

There are several ways to quantify nonlocality. However, it has recently been shown <cit.> that nonlocality is maximal at the same time in, at least, four of them: (I) When nonlocality is quantified through the nonlocal fraction <cit.>. In this case, maximum nonlocality corresponds to nonlocal fraction 1. Then, the correlation is said to be fully nonlocal (FN) <cit.>. (II) When nonlocality is quantified by the minimum distance to a face of the nonsingnaling polytope without local points <cit.>. In this case, maximum nonlocality occurs when the correlation is in a face of the nonsingnaling polytope without local points. Then, the correlation is said to be face nonsignaling (FNS) <cit.>. (III) When nonlocality is quantified based on whether or not it allows for a GHZ-like proof <cit.> of Bell theorem. In this case, maximum nonlocality occurs when the correlation allows for such a proof. Then, the correlation is said to allow an “all-versus-nothing” (AVN) <cit.> proof of nonlocality. (IV) When nonlocality is quantified by the wining probability of a nonlocal game whose classical winning probability is smaller. In this case, maximum nonlocality occurs when the correlation allows the players to always win the game. Then, the correlation is said to allow for a perfect quantum strategy (PQS) <cit.> or “quantum pseudo-telepathy” <cit.>. For precise definitions of FN, FNS, and AVN, we refer the reader to Appendix <ref>. A detailed definition of PQS is presented below. In this paper, we will focus and use the language of PQSs. However, our results can also be read in terms of FN correlations, FNS correlations, and AVN proofs. 

In the bipartite case, a nonlocal game <cit.> G = (X × Y, A × B, π, W) is characterized by: (i) Two input sets: X for the first player, Alice, and Y for the second player, Bob. (ii) Two output sets: A for Alice and B for Bob. (iii) A distribution of probability for the inputs: π(X × Y). (iv) The winning condition, i.e., the condition that inputs and outputs must satisfy for winning the game: W(X × Y × A × B) ∈{0,1}, where W=1 means winning and W=0 means losing. Therefore, the winning probability of the game is

    ω(G) = ∑_x,y,a,bπ(x,y) p(a,b|x,y) W(a,b,x,y).
 
The game admits a PQS if there is a quantum correlation p(a,b|x,y), where a ∈ A, b∈ B, x ∈ X, and y ∈ Y, such that ω(G)=1. 

PQSs have a special status in foundations of quantum computation and play a crucial role in the proofs of some fundamental results such as the quantum computational advantage for shallow circuits <cit.>, MIP^∗=RE <cit.>, and the impossibility of classically simulating quantum correlations with arbitrary relaxations of measurement and parameter independence <cit.>.

For n ≥ 3, n-partite PQSs can be achieved with qubits <cit.> and there are systematic methods to produce them for any number of parties <cit.>. However, for n=2, PQSs require pairs of qudits with d ≥3 <cit.>. 
Bipartite PQSs are therefore a fundamental motivation for exploring high-dimensional quantum correlations (other motivations are faster data rates, improved communication security <cit.>, higher resistance to noise <cit.>, and lower detection efficiencies to attain the loophole-free regime <cit.>).

On the other hand, bipartite PQSs require more than two measurements per party <cit.> and more than two outcomes per measurement (at least for some measurements) <cit.>. 

Three types of bipartite PQSs are known:

    
  * Bipartite PQSs based on qudit-qudit maximally entangled states and Kochen-Specker (KS) <cit.> sets of rays in dimension d ≥ 3 <cit.>. KS sets are defined in Sec. <ref>.
    
  * Bipartite PQSs based on magic sets (as defined in <cit.>) of Pauli observables <cit.> or parallel repetitions of them <cit.>. Magic sets are related to KS sets <cit.>.
    
  * Bipartite PQSs based on state-independent contextuality sets of vectors <cit.> whose corresponding orthogonality graph has fractional packing number equal to the Lovász number and larger than the independence number <cit.>. In light of the one-to-one connection between PQSs and KS sets <cit.> (see Appendix <ref>), these sets must be KS sets.


The simplest known example of a bipartite PQS, and probably the simplest allowed by quantum theory
<cit.>, occurs in the (2,3,4), Bell scenario, where there are 2 parties and each of them has 3 settings with 4 outcomes. This PQS requires Alice and Bob to share a ququart-ququart maximally entangled state. It was introduced in <cit.> and makes use of the Peres-Mermin two-qubit proof of the KS theorem <cit.>. This PQS is usually called the “magic square” correlation <cit.> and has been experimentally tested using photonic hyperentanglement: either polarization-path hyperentanglement <cit.> or polarization-orbital angular momentum hyperentanglement <cit.>. To our knowledge, no other bipartite PQS has been experimentally tested.

The aim of this work is to show that, for infinitely many (finite) even dimensions 
d ≥ 4, there are qudit-qudit PQSs using only d-1 local settings per party. The case d = 4, the magic square, is well-known, but the new ones, which are not directly connected to the case d = 4, form an especially beautiful and seemingly fundamental family of PQSs.





 §.§ Structure




In Sec. <ref>, we introduce a family of bipartite nonlocal games, the “colored odd-pointed star games”. What makes this family special is explained in Secs. <ref> and <ref>.

In Sec. <ref>, we show that that, infinitely many of the “stars” used in these games can be associated to a KS set in dimension d. The interest of these KS sets is triple: they are very symmetric (their graph of orthogonality is vertex transitive), the impossibility of a KS assignment is easy to check (as they allow for a “parity proof” KS set <cit.>), and they have a small number of bases (d+1, which implies that they are the KS sets with the smallest number of bases known in each dimension d). As we will see, this last property is crucial for our purpose of identifying bipartite perfect quantum strategies with small input cardinality.
However, this extended family of KS sets is interesting in its own right, as is the fact that there is a connection between some of its new members and the smallest KS set in quantum theory <cit.>, which happens to be in d=4. Moreover, this family seems to be the “natural” (although nontrivial) extension to any even d of the smallest KS. For all these reasons, these results are presented in a section, Sec. <ref>, that can be read independently from the rest of the article.

In Sec. <ref>, we prove that colored odd-pointed star games correspond to the optimal ways of distributing the KS sets of Sec. <ref> between two players, while achieving a PQS with minimal input cardinality. 

In Sec. <ref>, we discuss the Bell inequalities corresponding to the PQSs introduced in Secs. <ref> and <ref> and prove that, unlike the Bell inequalities corresponding to the magic square <cit.> and GHZ <cit.> games, the Bell inequalities corresponding to the new PQSs are not tight. This provides a second counterexample <cit.> to a conjecture formulated in <cit.>.

In Sec. <ref>, we discuss some advantages of the new PQSs with respect to existing PQSs. We also compute the necessary visibility to achieve quantum advantage and compare it the visibility needed for different variants of the colored odd-pointed star games.

Finally, in Sec. <ref>
we summarize the results and list some open problems.





§ THE COLORED ODD-POINTED STAR GAMES






 §.§ The game




Here, we introduce a class of bipartite nonlocal games that, in infinitely many cases, allow for a PQS with quantum advantage. We call them colored odd-pointed star games.



Let N be an odd number. The referee gives Alice a number x chosen from

    X := {1,2, …, N-2}

and gives Bob a number y chosen from

    Y := {3, 4, …, N}.

Alice outputs a number a ∈ A and Bob outputs a number b ∈ B, where

    A    :=   {1, 2, …, N}∖{x}, 
    
        B    :=   {1, 2, …, N}∖{y}.

See Fig. <ref>. Alice and Bob win when the sets {x,a} and {y,b} are either the same or disjoint. That is, whenever

    |{x,a}∩{y,b}| ∈{0,2}.

Therefore, the winning condition is

    W(x,y,a,b) ≡ 1 + |{x,a}∩{y,b}|  2.










This game has no perfect classical strategy
(see Corollary <ref> in Sec. <ref>). However, as we will show in Sec. <ref>, there are infinitely many N for which there is a PQS. 

The colored odd-pointed star game can be viewed as being played on a configuration, which is a set of points P and lines L such that each point is incident (i.e., lies on) to the same number of lines, and each line is incident to the same number of points. The configuration that our game is played on is the J(N,2)-configuration with odd N (or N-pointed star). The name follows from the fact that, as we shall see in Sec. <ref>, it arises from the Johnson graph J(N,2). The J(N,2)-configuration has N lines labeled {1,2,…,N} with N-1 points each. Each pair Line i and Line j meet at exactly one point which we label by {i,j} (or ij for short).  

Alice's (Bob's) inputs are red (blue) lines of the J(N,2)-configuration. Their outputs correspond to points on their respective lines. For example, if Alice is given (red) Line 5, Alice's “output 6” corresponds to point {5,6} (i.e., the point in the intersection between Line 5 and Line 6). The cases N=7 and N=9 are illustrated in Figs. <ref> and <ref>, respectively.

Alice and Bob win either if they output the same point or if they output points in different lines. That is, they lose if they output different points in the same line.

The reason for the J(N,2)-configuration with odd N is that, as we will see in Sec. <ref>, for infinitely many values on N, it can be realized with a KS set in dimension d=N-1 and having only N bases. The reason why some lines have only one color while other lines have the two colors is that, as we will see in Sec. <ref>, this minimizes the input cardinality of the PQS. 











 §.§ Example: N=7
 


 
N=7 is the smallest instance of the colored odd-pointed star game where there is a PQS. The best classical strategy yields a winning probability of 24/25. For example, an optimal classical strategy is the following: Alice gives the output corresponding to the point 12 when the input corresponds to Line 1 or to Line 2, outputs 34 for inputs 3 or 4, and outputs 56 for input 5. Bob outputs 34 for inputs 3 or 4, outputs 56 for inputs 5 or 6, and outputs 67 for input 7. The strategy is summarized in Table <ref>. This strategy leads to a single failure: when Alice is given Line 5 and Bob is given Line 7.









A PQS is the following. Alice and Bob share the following qudit-qudit maximally entangled state:

    |ψ⟩ = 1/√(6)∑_k=0^5|k⟩⊗|k⟩.

If Alice receives Line j as input, she measures on her qudit the orthogonal basis j, defined as those vectors in Table <ref> having j as superindex. This measurement projects Alice's qudit state into one state v^i,j (or v^j,i). Then Alice outputs the number i. Similarly, if Bob receives Line k, he measures on his qudit the conjugate orthogonal basis k, defined as the conjugates of the vectors in Table <ref> having k as superindex. This projects Bob's qudit state into a state v^i',k, and then Bob outputs the number i'.









That a set of 21 6-dimensional vectors exists such that each point in the J(7,2) configuration can be assigned to a different vector and such that vectors in the same line are mutually orthogonal 
was presented in <cit.>. We will refer to this set of vectors as a quantum realization of J(7,2). Interestingly, this set is a KS set.


 
A Kochen-Specker (KS) set is a finite set of vectors 𝒱 in a Hilbert space of finite dimension d ≥ 3, which does not admit an assignment f: 𝒱→{0,1} satisfying: (I) Two mutually orthogonal vectors v_i and v_j cannot have f(v_i)=f(v_j)=1. (II) For every orthogonal basis (set
of d mutually orthogonal vectors) {v_i}_i=1^d, ∑_i f(v_i)=1.


It is sometimes useful to refer to a KS set as a pair (𝒱, ℬ), where 𝒱 is the finite set of vectors and ℬ is a specific subset of the orthogonal bases formed by elements of 𝒱. 
The reason is that, in some cases, the no existence of f : 𝒱→{0,1} satisfying (I) and (II) follows from the no existence of f satisfying (II) for the orthogonal bases in ℬ.

The way to prove that f does not exist for any quantum realization of J(7,2) in dimension 6 is by noticing that there is an odd number (7) of lines (orthogonal bases). Therefore, there are 7 equations of the type ∑_i f(v_i)=1. If we add them all, then the right-hand side is an odd number (7). However, this is impossible to achieve since each vector is in an even number (two) of lines. Sets that can be proven to be KS sets with a simple parity argument of this type are said to admit a parity proof.

Interestingly, the KS set in Table <ref> is the KS set with the smallest number of vectors (and, more interestingly for us, bases) known in d=6. Moreover, it has been proven that it is the KS set with the smallest number of vectors in d=6 that admits a symmetric parity proof <cit.>. The set is also symmetric in another sense: the graph J(7,2) is vertex transitive (i.e., its automorphism group acts transitively on its vertices).





§ AN INFINITE FAMILY OF SYMMETRIC KS SETS WITH A SMALL NUMBER OF BASES
 



In this section we prove that, for infinitely many N, there are N-1-dimensional vectors such that each point in the J(N,2)-configuration can be assigned to a different vector such that vectors in the same line are mutually orthogonal. This implies that, for these values of N, there is a KS set in dimension N-1 and N bases (and admitting a symmetric parity proof, and associated to a vertex-transitive graph). As we will see in Sec. <ref>, this also implies that, for these values of N, there is a bipartite PQS with quantum advantage with N-2 inputs per party and N-1 outputs.





 §.§ Construction of the KS sets




The problem of for which odd N there is a quantum realization in dimension N-1 for the J(N,2)-configuration is a special case of a well-studied problem in graph theory - that of computing orthogonal representations. Given a graph G=(V,E), where V is the vertex set and E is the edge set, an orthogonal representation (OR) in ℂ^d of G is a labeling of the vertices of G with non-zero vectors in ℂ^d such that adjacent vertices are labeled by orthogonal vectors. Note that some papers use a different definition in which non-adjacent vertices are labeled by orthogonal vectors. The dimension of the orthogonal representation is d, and the orthogonal rank is the minimum dimension d for which there is an orthogonal representation of G in ℂ^d. The orthogonal rank is denoted by ξ(G). For general graphs, computing orthogonal representations of minimal dimension is very difficult and little is known. In fact, there is not even an efficient algorithm to compute ξ(G) (for example, deciding whether ξ(G) ≤ t for some positive integer t ≥ 3 is NP-hard in the real case - i.e., substituting ℂ for ℝ - see <cit.>). Bounds exist for ξ(G): 

    ω(G) ≤ξ(G) ≤χ(G),

where ω(G) is the clique number (maximum number of mutually adjacent vertices), and χ(G) is the chromatic number (fewest colors needed in order to color each vertex so that adjacent vertices are assigned different colors). 
 
The Johnson graph J(N,m) is the graph whose vertices are the m-subsets of {1,…, N}, where an edge exists between vertices S_1 and S_2 if and only if S_1 and S_2 are not disjoint. Orthogonal representations of J(N,2) in dimension N-1 are KS sets. Here, the bounds given in (<ref>) are useful: for J(N,2) with N odd, N ≥ 5, it is known that ω(G) = N-1 and χ(G) = N, so that the minimum dimension d of any orthogonal representation must be one of N-1 or N. Note that in the N=3 case we have ω(G) = N, so that there is no orthogonal representation in dimension N-1 (and thus no associated KS set). The previous discussion can be summarized as follows:

For odd N, ξ(J(N,2)) ∈ {N-1, N}.
    If ξ(J(N,2)) = N-1, then there is a KS set in dimension N-1.

Showing that ξ(J(N,2)) = N-1 amounts to proving that a KS exists. However, one still needs to explicitly construct the orthogonal representation in order to obtain the KS set.

In the next subsection we will provide a construction that works for infinitely many dimensions. In the subsection following that, we present a novel recursive construction that allows one to cover infinitely many dimensions from any known dimensions.





 §.§ The Jungnickel-Lisoněk construction
 



For each dimension d=2^kp^m, with m ≥ k ≥ 1, it is possible to explicitly construct an orthogonal representation of J(d+1,2) in ℂ^d.
The construction uses a result of Jungnickel <cit.> in order to build a generalized Hadamard matrix GH(2p, 2) and then produce the orthogonal representation via a pair of results of Lisoněk <cit.>.

A generalized Hadamard matrix GH(g, λ) over a group G of order g is a gλ× gλ matrix whose entries are elements of G and for which each element of G occurs exactly λ times in the difference of any pair of rows. 
For example, the matrix

    D^(3) :=
    [ 0 0 0 0 0 0; 1 2 0 2 0 1; 1 0 2 2 1 0; 0 2 2 0 1 1; 2 2 0 1 1 0; 2 0 2 1 0 1 ]

is a generalized Hadamard matrix GH(3,2) over ℤ_3. The reader may notice a resemblance between the matrix D^(3) and the KS set given in Table <ref>. As we shall see in this section, the KS set is constructed from the matrix.

By EA(q) we denote the group of order q obtained by the direct product of groups of prime order (for example, if q = 12, then EA(12) = ℤ_2 ⊕ℤ_2 ⊕ℤ_3). The following theorem describes Jungnickel's construction of generalized Hadamard matrices.

 <cit.> 
    Let q be an odd prime power and let c be any non-square element of GF(q). Fix an ordering σ on the elements of GF(q). Define matrices A_i = (a^i_σ(x), σ(y)) for i=1, …, 4 via
    
    a^1_σ(x),σ(y)   = xy + (x^2/4), 
     a^2_σ(x), σ(y)   = xy + (cx^2/4), 
    
            a^3_σ(x), σ(y)   = xy - y^2 - (x^2/4), 
     a^4_σ(x), σ(y)   = [xy - y^2 - (x^2/4)]/c

    and set 
    
    D := [ A_1 A_2; A_3 A_4 ].

    Then D is a generalized Hadamard matrix GH(q, 2) over EA(q).


In the case that q is prime, EA(q) ≃ℤ_q, so we obtain a generalized Hadamard matrix over ℤ_q. For example the GH(3,2) matrix H given in Eq. (<ref>) was obtained from Jungnickel's construction with q=3. The Kronecker product of generalized Hadamard matrices is also a generalized Hadamard matrix. Exploiting this observation and Jungnickel's construction, one immediately obtains generalized Hadamard matrices GH(p, λ) over ℤ_p for each n := pλ = 2^kp^m where m ≥ k ≥ 1 (see <cit.>).

The Jungnickel-Lisoněk construction for KS sets does not work directly from generalized Hadamard matrices, but from S-Hadamard matrices, which we now define. A matrix H = (h_i,j) ∈ℂ^n × n is an S-Hadamard matrix of order n if

    
  * HH^* = nI ,
    
  * |h_i,j| = 1 for each 1 ≤ i,j ≤ n, 
    
  * ∑_j=1^n h^2_k,jh^2_ℓ, j = 0 for each 1 ≤ k,ℓ≤ n with k ≠ℓ.


The next proposition allows one to construct S-Hadamard matrices from generalized Hadamard matrices over ℤ_g (for g ≥ 3). 

 <cit.>
    Let g ≥ 3, and let H = (h_i,j) be a generalized Hadamard matrix GH(g, λ) over ℤ_g. Denote by ζ_g a primitive gth root of unity. Then the matrix S = (ζ_g^h_i,j) is an S-Hadamard matrix.


In the case that N := gλ is even, we obtain a KS set with at most N+12 vectors, and N+1 orthogonal bases via the construction described in <cit.>. We describe this construction now, following the notation of <cit.>.

Let H = (h_i,j)_i,j=1^n be an S-Hadamard matrix of order n. By h_i we denote the ith row of H. If h_1 is not the all-ones vector, we normalize H by multiplying each of its entries h_i,j by h_1,j^-1.
For vectors u = (u_1, …, u_n),v = (v_1, …, v_n) ∈ℂ^n, we write u ∘ v to denote the vector (u_1v_1, …, u_nv_n). 

From H, we construct the KS set (𝒱, ℬ) where 𝒱 is the set of vectors and ℬ is the set of orthogonal bases. We index each vector by a pair of indices i,j each between 1 and N+1 - so the elements of 𝒱 are each written in the form v^i,j. It should be understood that v^i,j and v^j,i denote the same vector.

The vectors are:

    
  * v^1,s := h_s-1 for each s = 2, …, n+1. 
    
  * v^2,s := h_s-1∘ h_s-1 for each s=3, …, n+1.
    
  * v^r,s := h_r-1∘ h_s-1 for each pair r=3,…, n+1, s=r+1,… n+1.

One can check that 

    B_r = {v^r,i : i=1,…, n, i ≠ r}

is an orthogonal basis for each r=1, …, n+1.  Finally, setting ℬ = {B_1, …, B_n+1}, the previously described parity argument shows that pair (𝒱, ℬ) does indeed form a KS set (there are an odd number of orthogonal bases, and an even number of vectors in each orthogonal basis). Note that the vectors v^{i,j} generated by this construction may not be pairwise linearly independent. 

We say that an orthogonal representation is faithful if non-adjacent vertices are mapped to by non-orthogonal vectors. Whenever we may produce a faithful orthogonal representation of a graph G via a KS set 𝒦, we shall say that 𝒦 induces a faithful orthogonal representation of G.

Using the procedure that we have described with the matrix D^(3), we obtain the KS set given in Table <ref>. This induces a faithful orthogonal representation of the Johnson graph J(7,2). By construction the co-ordinates of each v^{i,j} are of the form ζ_3^m for some m ∈{0,1,2}. In order to simplify the presentation, we replace each ζ_3^m by its exponent m in the vector so that for example v^{1,4} is denoted by (1, 0, 2, 2, 1, 0) but is really (ζ_3, 1, ζ_3^2, ζ_3^2, ζ_3, 1). The vectors are written in this form in Table <ref>. 

We emphasize that the graph J(7,2) is simply the graph obtained from the J(7,2)-configuration (see Fig. <ref>) by viewing each point of the J(7,2)-configuration as a vertex, with a pair of vertices being adjacent whenever the corresponding points are colinear in the J(7,2)-configuration.

We also give the KS set obtained from the Jungnickel-Lisoněk construction defining an orthogonal representation of J(11,2) in Table <ref>. The orthogonal representation in this case is not faithful. For example, the vectors v^{1,2} and v^{3,4} are orthogonal but are not in a common maximum clique. The components of each vector are all of the form ζ_5^m, where ζ_5 is a primitive 5th root of unity and m ∈{0, 1, 2, 3, 4}. As before, we replace the power of root of unity with its exponent (i.e., ζ_5^m with m) in the presentation of the vectors for simplicity.

To summarize, whenever there is a generalized Hadamard matrix GH(g,λ) over ℤ_g for some positive integer λ satisfying gλ = N-1, we obtain a PQS for the colored odd-pointed star game played on the J(N,2)-configuration. Therefore, the Jungnickel-Lisoněk construction ensures that there is a perfect quantum strategy for infinitely many N (those of the form 2^kp^m+1 with m ≥ k ≥ 1, m,k integers). There are also generalized Hadamard matrices over ℤ_g so that gλ = N-1 for different values of N than those we have mentioned; our intention here was to illustrate that there is an infinite class, for which our choice is enough. There are several other constructions of infinite families of generalized Hadamard matrices given in <cit.>. The reader should take care to note that these constructions are given over EA(g) for general prime powers g, so one must restrict g to be prime. Additionally, <cit.> provide recursive constructions for generalized Hadamard matrices. For smaller cases, generalized Hadamard matrices over ℤ_g are known to exist for each order k := gλ =1, …, 100, with the exception of 2, 4, 8, 32, 40, 42, 60, 64, 66, 70, 78, 84, 88 <cit.> (note that, in some of these cases, it is still open whether such a generalized Hadamard matrix exists). Therefore, the smallest Johnson graph for which we may not produce a labeling via this method is J(9,2). We refer the reader to the Sec. 5.2 of 
Ref. <cit.> for a more complete list of the possible choices of N. 

Finally, let us remark that Lisoněk's construction of a KS set does not need a generalized Hadamard matrix, but only an S-Hadamard matrix. Note that here we refer to the “Lisoněk construction” as opposed to the “Jungnickel-Lisoněk construction” since we are not restricting the generalized Hadamard matrix to the ones built from Jungnickel's construction. 
It may be possible that for certain choices of N suitable S-Hadamard matrices exist even when generalized Hadamard matrices do not.

















 §.§ A novel construction for dimensions not covered by the Jungnickel-Lisoněk construction




The N=9 case is the smallest example of where there is no suitable generalized Hadamard matrix to produce a KS set for J(N,2). Here, we show that, in fact, there is a KS set for J(9,2). Additionally, we provide a novel construction that allows one to produce a KS set for J(N^k,2) for each non-negative integer k whenever one has a KS set for J(N,2) with N a prime power. This allows us to extend the Jungnickel-Lisoněk construction in infinitely many dimensions. As an example, we are able to produce a KS set for J(121,2). The direct constructions illustrated in <cit.> do not produce a generalized Hadamard matrix in dimension 120 for the Jungnickel-Lisoněk construction. In fact, the recursive construction we present produces a KS set for J(N',2) for each N' ≡ N  N(N-1) whenever N' is sufficiently large.

Given a graph G = (V,E), the line graph of G, denoted L(G) = (V',E') is the graph whose vertices are the edges of G (i.e V' = E) and for which two vertices w,x ∈ V' are adjacent if and only if w and x are incident to a common vertex of G. The Johnson graph J(N,2) is the line graph of the complete graph K_N. 

A subgraph H of G is a spanning graph of G if V(H) = V(G). A k-regular (each vertex has degree k) spanning graph of G is called a k-factor of G, and a k-factorization of G is a set of k-factors of G, {H_1, …, H_ℓ} whose edge sets form a partition of the edge set of G.

 
    Let G be a graph with a k-factorization H_1, …, H_ℓ such that L(H_i) has an orthogonal representation in ℂ^k for each i=1,…,ℓ. Then, there is an orthogonal representation for L(G) in ℂ^kℓ.



    Let |E(G)| = n', so n := n'/k = |E(H_i)| for each i=1,…,ℓ.  Further, let
    
    {v_j^i : j =1, …, n}

    be the k-dimensional orthogonal representation of L(H_i) for each i=1,…,ℓ. Each of these vectors is associated to a vertex of L(H_i), and thus to an edge of H_i, and thus an edge of G. Now define vectors 
    
    v_j^i := (0, …, 0_i-1, v_j^i, 0, …, 0_ℓ - i),

    where 0 denotes the k-dimensional zero vector and v_j^i is the ith block of size k. As each of these vectors are associated to an edge of G, they are associated to vertices of L(G). It is straightforward to check that a pair of such vectors is orthogonal whenever the corresponding vertices of L(G) are adjacent. Additionally, by construction, the dimension of each each of these vectors is kℓ. Therefore, we see that the vectors v_j^i do indeed form a kℓ-dimensional orthogonal representation of G as required.


Consider the 4-factorization of K_9 into two isomorphic copies of the Paley graph of order 9 (henceforth P(9)) as shown in Fig. <ref>. Paley graphs are important objects in graph theory - among other important properties, they are self-complementary and strongly regular. 
The Paley graph of order 9 encodes the commutativity relationships of the Peres-Mermin “magic” square <cit.>, and its line graph encodes all the 54 “essential” orthogonalities of the 18 vector KS set of <cit.> (the 18 vector KS set actually has 63 orthogonalities, but one may ignore 9 of them and still find that no assignment f may be produced). Indeed, the 18 vector KS set provides a 4-dimensional orthogonal representation of P(9).
Applying Proposition <ref>, we immediately obtain the 8-dimensional orthogonal representation given in Table <ref> which is a KS set for J(9,2). Fig. <ref> illustrates explicitly the two copies [subfigures (a) and (b)] of the 18-vector KS set <cit.> that appear in the J(9,2)-configuration. Here, the 18-vector KS set appears in the same format as in <cit.> and <cit.>, where the 9 orthogonal bases are given by the 6 straight lines and 3 ellipses. The two copies are further emphasized in our depiction of the J(9,2)-configuration of Fig. <ref> (c) by displaying the vertices corresponding to each copy in the same colors (black and purple) as in Fig. <ref>. 









We now introduce a general construction for building KS sets for Johnson configurations from smaller ones. This construction is based on balanced incomplete block designs (BIBDs). 


    Given positive integers v,b,r,k,λ, a balanced incomplete block design is a pair (S,T) where:
    
        
  * S is a set of size v,
        
  * T is a collection of b subsets of size k called blocks,
        
  * each element of S is in exactly r blocks,
        
  * each pair of elements of S is in exactly λ blocks.
    
    If there is a partition (t_1, …, t_m) of the blocks of T so that each set of blocks t_i is a partition of S, then we say that the BIBD is resolvable, and write RBIBD for short. Finally, we call the partition (t_1, …, t_m) a resolution.


Note that the parameters v,k,λ determine the others, so in case we want to specify the parameters, we write (v,k,λ)-RBIBD. 

The key point is that an (N,k,1)-RBIBD yields a (k-1)-factorization of K_N, via the following recipe.

    
  * Let S be the set of vertices of K_N.
    
  * Consider some t_i in the resolution. It is a partition of the vertices into sets of size k. Associate to each of these the complete graph K_k, and associate to t_i the (vertex-disjoint) union of all of these complete graphs. Then t_i corresponds to a (k-1)-regular spanning subgraph of K_N. Call this graph H_i.
    
  * Since λ = 1, the graphs H_1, …, H_m are edge-disjoint and the union of their edge sets is the edge set of K_N. Therefore H_1, …, H_m is a (k-1)-factorization of K_N.


Notice further that since each H_i is a disjoint union consisting of isomorphic copies of the graph K_k, a (k-1)-dimensional orthogonal representation of L(K_k) provides a (k-1)-dimensional orthgonal representation of L(H_i) – one can simply use the same vectors for each copy of K_k. By applying Proposition <ref>, we obtain the following result.


    If there exists a (N,k,1)-RBIBD for some odd k, and a (k-1)-dimensional orthogonal representation of J(k,2), then there is an (N-1)-dimensional orthogonal representation of J(N,2).

Such RBIBDs exist for each choice of k that is a prime power, since the affine geometry AG(2, k) is a (k^2,k,1)-RBIBD (see for example <cit.>). As a consequence, we immediately obtain a 120-dimensional orthogonal representation of J(121) which notably does not seem to be attainable via the generalized Hadamard constructions that we have seen. 

In fact, one can generalize this to all powers of the odd prime power k: there is a (k^m, k, 1)-RBIBD for each positive integer m <cit.>.

Combining this with the previous construction, we obtain the following:


    Let n be an odd prime power such that there is a generalized Hadamard matrix over ℤ_p for some p | (n-1). Then, there is a (n^m - 1)-dimensional orthogonal representation of the Johnson graph J(n^m) for each positive integer m, and consequently a KS set associated to each of these Johnson graphs.


In fact, for any choice of k, and N ≡ k  k(k-1) sufficiently large, there exists an (N,k,1)-RBIBD <cit.>. Therefore, we obtain orthogonal representations for J(N,2) in dimension N-1 for infinitely many dimensions for each choice of k for which we have an orthogonal representation in dimension k-1. To emphasize, using the k=7 representation we have given in Table <ref>, one immediately obtains KS sets for J(N,2) for each N ≡ 7  42 with N sufficiently large.

The best known lower bound for “sufficiently large” is currently exp(exp(k^12k^2)) <cit.>. This bound can likely be improved – indeed in the k=7 case the largest possible N for which there may be no (N,k,1)-RBIBD is 294427 <cit.>. We also remark (although no suitable orthogonal representation for J(k,2) exists in these cases) that in the case of k=5 there is a (N, 5, 1)-RBIBD for each N > 645 satisfying the congruence condition and in the case of k=3,4 the congruence condition is not just necessary but also sufficient. Such explicit bounds do not seem to have been obtained for other k (see <cit.>).









Aside from the J(9,2) case, the smallest case for which the there is no suitable generalized Hadamard matrix is J(33,2) (in dimension 32). 
Future research is needed to check if a suitable S-Hadamard matrix exists in this case, or if one may leverage a k-factorization of K_33 as we have done in this section.





§ PERFECT QUANTUM STRATEGIES WITH MINIMAL INPUT CARDINALITY
 



Let us now go back to the bipartite (colored odd-pointed star) games. We will finally justify our choice of input sets for Alice and Bob. In this section, b will be used to denote an orthogonal basis of a KS set.


Let us assume that we have an orthogonal representation of J(N,2) in dimension N-1. Denote the vectors by 𝒱 = {v^i,j : 1 ≤ i < j ≤ N} with each v^i,j associated to vertex {i,j} of J(N,2). Then, it is straightforward to see that, for each i=1,…,N, the set b_i = {v^i,j : j =1, …, N, j ≠ i} forms an orthogonal basis. Let ℬ := {b_1, …, b_N}. From Sec. <ref>, we know that (𝒱, ℬ) forms a KS set. The natural generalization of the PQS outlined in Sec, <ref> yields a perfect quantum strategy for the colored odd-pointed star game played on the J(N,2)-configuration. 

Consider sets S_A, S_B ⊆ ℬ. A classical assignment consists of a pair  

    f_A:𝒱× S_A →{0, 1}
       f_B: 𝒱× S_B →{0,1}

satisfying

    
  * for b_i ∈ S_A, b_k ∈ S_B, and vectors v^i,j∈ b_i, v^k,ℓ∈ b_k, if |{i,j}∩{k,ℓ}| = 1, then we cannot have f_A(v^i,j, b_i) = f_B(v^k,ℓ, b_k) = 1,
    
  * 
        
  * for each orthogonal basis b_i ∈ S_A, ∑_j ≠ i f_A(v^i,j, b_i) = 1,
        
  * for each orthogonal basis b_k ∈ S_B, ∑_ℓ≠ k f_B(v^k, ℓ, b_k) = 1.
 

Note that we tacitly assume that f_A(v,b) = 0 for each b ∈ S_A if v ∉ b, and f_B(v,b) = 0 for each b ∈ S_B if v ∉ b. 

Given a classical assignment f_A, f_B, we say that an orthogonal basis b_i is labeled for S_A if we have specified the values f_A(v^i,j,b_i) for each of the vectors v^i,j∈ b_i (i=1, …, N, i ≠ j). In order to do this, it is sufficient to specifiy the vector v^* ∈ b_i with f_A(v^*,b_i) = 1. Similarly, we say that b_i is unlabeled for S_A if we have not yet designated the values f_A(v^i,j,b_i). Of course, the same holds replacing A with B. 



     The pair (S_A, S_B) is a bipartite KS set (B-KS) if no classical assignment exists for (S_A, S_B).
 

 It is clear that (ℬ, ℬ) is always a B-KS since (𝒱, ℬ) is a KS set,. Additionally, if S_B ⊆ S_B' for some S_B' ⊆ℬ, then (S_A, S_B') is also a B-KS.
A B-KS set (S_A, S_B) is an optimal bipartite KS set if |S_A||S_B| is minimum over all choices of S_A, S_B ⊆ℬ.

The relationship between B-KS sets and the colored odd-pointed star game is as follows:

    
  * B-KS sets correspond exactly to the input sets for Alice and Bob in the colored odd-pointed star game for which there is no perfect classical strategy,
    
  * optimal B-KS sets correspond exactly to such input sets of minimum input cardinality.


Throughout the proofs, we take advantage of the symmetries of the graph J(N,2). Namely, any permutation σ∈𝔖_N of {1,2,…,N} induces an automorphism of J(N,2) by mapping each vertex ij to σ(i)σ(j). It therefore also maps the maximum clique C_i := {ij: j=1,…,N, j ≠ i} to the maximum clique C_σ(i) := {σ(i)j: j=1,…,N, j ≠σ(i)}. This induces an automorphism of the KS set mapping each v^i,j to v^σ(i), σ(j), and consequently each b_i to b_σ(i).

 
    The optimal B-KS sets are exactly the pairs (S_A, S_B) satisfying:
    
    S_A    = {b_m : m ∉{i,j}}
    
            S_B    = {b_n : n ∉{k,ℓ}}

    for any choice of distinct i,j,k,ℓ∈{1,2,…,N}. 
 


We first show that min(|S_A|, |S_B|)≥ N-2. Let us proceed by contradiction, assuming without loss of generality that |S_A| ≤ N - 3, and showing that a classical assignment exists in such a case. We may assume that S_B = ℬ.
Additionally, by exploiting the symmetries of J(N,2), we may assume without loss of generality that b_1, b_2, b_3 ∈ℬ∖ S_A.  Then set 

    f_B(v^1,2, b_1), f_B(v^1,2, b_2), f_B(v^1,3, b_3) := 1.

Since N is odd, this leaves an even number of orthogonal bases to label for S_B. By exploiting the fact that the intersection of each pair of orthogonal bases is a single vector, we are able to greedily label the remaining orthogonal bases of S_B – we simply pick an unlabeled pair of orthogonal bases b_i, b_j and set:

    f_B(v^i,j,b_i), f_B(v^i,j,b_j) := 1.

We continue this process until all orthogonal bases of S_B have been labeled. 
Then, we label the orthogonal bases of S_A (which are each in S_B) exactly the same as we have labeled the bases of S_B, so:

    f_A(v,b) := f_B(v,b)

for each b ∈ S_A ⊆ S_B, v∈ b.
This constitutes a classical assignment, and so we see that such a pair (S_A, S_B) cannot be a B-KS set. Therefore, min(|S_A|, |S_B|) ≥ N-2.

Next, we show that the choices of (S_A, S_B) described in the theorem statement are indeed bipartite KS sets. Let i,j,k,ℓ∈{1,2,…,N} be distinct, and let S_A, S_B be defined as in Eqs. (<ref>)–(<ref>). By symmetry, we may assume without loss of generality that i = 1,j = 2, k = 3, ℓ=4, so that S_A = {b_3, b_4, …, b_N} and S_B = {b_1, b_2, b_4, …, b_N}.

We prove that (S_A, S_B) is a B-KS set by showing that the existence of a classical assignment (f_A, f_B) of (S_A, S_B) implies the existence of a classical assignment (f'_A, f'_B) of (ℬ, ℬ). This would contradict the fact that (ℬ, ℬ) is a bipartite KS set, thus that (𝒱, ℬ) is a KS set, and finally that 𝒱 is an orthogonal representation of J(N,2) in dimension N-1. 

Since (f_A, f_B) is a classical assignment, we must have that for each b ∈ S_A ∩ S_B, 

    f_A(v,b) = f_B(v',b)  v = v'

otherwise Condition 1 of the classical assignment definition is violated. Consider extending (f_A, f_B) to an assignment of (ℬ, ℬ) by labeling the missing bases of S_A with the choices made in f_B (i.e setting f_A(v,b_1) := f_B(v,b_1) for each v ∈ b_1, f_A(v',b_2) := f_B(v',b_2) for each v' ∈ b_2) and vice-versa. This either forms a classical assignment of (ℬ, ℬ), or Condition 1 is violated in the process. In the case that a violation occurs, it must be between b_1 and b_2 or b_3 and b_4. This is because every other pair of bases has one point labeled by f_A and the other by f_B. Therefore, we may assume without loss of generality that

    f_B(v^1,2, b_1) = f_B(v^2,m, b_2) = 1

for some m ≥ 3 so that |{1,2}∩{2,m}| = 1. However such a scenario is impossible since there is a violation of Condition 1 for any choice of labeling of b_m for f_A: setting f_A(v^m,m',b_m) = 1 for any m' ≠ m, we find that either |{m,m'}∩{1,2}| = 1 or |{m,m'}∩{2,m}| = 1. Thus, the classical assignment (f_A, f_B) of (S_A, S_B) may indeed be extended to an assignment (f'_A, f'_B) of (ℬ, ℬ).

As stated earlier, this would imply that 𝒱 does not define an orthogonal representation of J(N,2) in dimension N-1, which is a contradiction. Therefore, (S_A, S_B) must indeed be a B-KS set, and since min(|S_A|, |S_B|) ≥ |ℬ| - 2, this is optimal. 

Any other optimal B-KS set (S'_A, S'_B) must also have |S'_A| = |S'_B| = |ℬ| - 2. If it is not of the type described above, then the B-KS set must correspond to two (possibly equal) vectors in a common orthogonal basis, say b_N without loss of generality (again each vector corresponds exactly to the pair of orthogonal bases missing from S'_A or S'_B). In this case, there is a simple classical assignment: setting f_A(r_i,i+1, b_i) = f_B(r_i,i+1, b_i) = 1 for each valid choice of even i, and f_A(r_j,j-1, b_j) = f_B(r_j,j-1, b_j) = 1 for each valid choice of odd j. By “valid choice” we mean that one should simply ignore assigning f_A for the b_i or b_j that are not in S_A, and do the same replacing A with B.

Therefore, we see that the optimal B-KS sets are exactly those sets (S_A, S_B) in the statement of the theorem.


In the context of the colored odd-pointed star games, we have shown the following. For any input choice of N-2 lines for Alice and N-2 lines for Bob with no line missing for both Alice and Bob (i.e each of the possible N lines from the J(N,2)-configuration is given to either Alice or Bob or both), there is no perfect classical strategy. Moreover, these are exactly the choices with minimal input size. We record this below.

 
    Giving each of Alice and Bob N-2 lines so that none of the N lines are missing for both Alice and Bob yields the minimum input cardinality |A||B| for the colored odd-pointed star games. With this choice of input sets A,B there is a PQS but no perfect classical strategy. Any choice of input sets not of this form either has larger input cardinality |A||B| or a perfect classical strategy.






§ BELL INEQUALITIES
 



Although the main interest of PQSs is that they are perfect rather than merely better than classical, a legitimate question is how much better than classical they are. This is equivalent to asking which are the Bell inequalities associated to the colored odd-pointed star games. Related to this is also the question <cit.> of whether 
Bell inequalities associated to PQSs are tight (i.e., facets of the local polytope) or not. Recall that, for the simplest PQSs (the GHZ-Mermin <cit.> and magic square <cit.> correlations), the corresponding Bell inequalities (<cit.>, respectively) are tight (as proven in <cit.>, respectively). However, recently, it has been shown <cit.> that there are PQSs whose associated Bell inequalities are not tight.

In this section we present the general form of the Bell inequalities associated to the colored odd-pointed star games and prove that neither of them is tight.





 §.§ Bell inequalities associated to the colored odd-pointed star games




Here we introduce the linear combination of probabilities, i.e., Bell inequality, that is associated to the game described in Sec. <ref>. We first write the Bell inequality associated to the colored odd-pointed star game with N=7, and then we generalize the analysis for an arbitrary N. It is convenient for this purpose to use the configuration view of the colored odd-pointed star game.

In order to write the Bell inequality associated to the J(7,2)-configuration, we are going to momentarily ignore the line's colors and exploit the symmetries that appear when considering that Alice and Bob could receive any of the seven lines of the J(7,2)-configuration as input.

As before, Alice and Bob have six possible outputs, corresponding to the points in their respective lines. For convenience, we order the points of each Line i in lexicographical order: so {i,j} comes before {i,j'} if and only if j < j'. For example, for Line 5 of the N=7 case, this produces the order 15, 25, 35, 45, 56, 57.
This allows us to represent each of the outputs of Alice and Bob in a manner that does not depend on the input line. 
 
More precisely, we set Alice's outputs to be a=1, …, 6, where choosing a=n corresponds to choosing the n^th point on the input line in the order we have just described. To emphasize, if Alice outputs a=6 for Line 5 described above, this corresponds to Alice choosing point 57 (and not point 56 as in the notation used in previous sections). We emphasize this further in a remark to avoid any confusion.


    In this section only, the choice of output n ∈{1,2,…,6} (and more generally {1,2,…,N-1}) for either Alice and Bob corresponds to choosing the n^th point on their given input line. 


With these preparations, we can define the coefficients that represent the winning conditions of the game as stated in Sec. <ref>:

    c(a,b|x,y) = {[      δ(a=b),      for x=y; c'(a,b|x,y),      for x<y;  c(b,a|y,x),     for x>y, ].

where δ(z)=1 (0) when z=True (False) and 

    c'(a,b|x,y)    = δ(a ≠ b)
       + δ(a=b)δ(x≤ b ≤ y-1)
       - δ(a=b-1)δ(x<b<y)
       - δ(a=y-1)δ(b≠ x) 
       - δ(a ≠ y-1)δ(b=x),
    
where a,b=1,…,6 and x,y=1,…,7. 

Once we have the coefficients c(a,b|x,y) defined, we can include the lines' colors and write the Bell inequality associated to the J(7,2)-configuration

    ∑_x = red∑_y = blue∑_a,b c(a,b|x,y) p(a,b|x,y) ≤ 24,

where the summation over red (blue) denotes that Alice (Bob) is going to only be inquired for the red (blue) lines. In the colored odd-pointed star games, with N=7, these summations represent x=1,2,3,4,5 and y=3,4,5,6,7.

To generalize this Bell inequality for the colored odd-pointed star game we define the possible inputs as before to be any choice from {1,2,…,N} (where Alice will be given input from {1,2,…,N-2} and Bob will be given input from {3,4,…,N}), and the output set to be {1,2,…,N-1}.

Then, we can write the colored odd-pointed star game as the following Bell inequality

    ∑_x = red∑_y = blue∑_a,b c(a,b|x,y) p(a,b|x,y) ≤ (N-2)^2-1,

with the coefficients c(a,b|x,y) defined in Eq. (<ref>) for a,b=1,…, N-1 and x,y=1,…,N.





 §.§ Proof that the colored odd-pointed star Bell inequalities are not tight




Here we give an upper bound to the maximum dimension of the faces defined by Eq. (<ref>) and we generalize this method for the case of the colored odd-pointed star game Eq. (<ref>). As a consequence of this upper bound, we conclude that these hyperplanes do not define facets of the classical polytope, i.e., they are not tight Bell inequalities.

For our argument it is enough to focus on the lines that are common to Alice and Bob. For the J(7,2)-configuration these are x',y'={3,4,5}. It is helpful to arrange these combinations in a table, see Table <ref>, and write the coefficients c(a,b|x,y) = M_x,y explicitly as matrices

    M_3,4 = [ 0 1 0 1 1 1; 1 0 0 1 1 1; 0 0 1 0 0 0; 1 1 0 0 1 1; 1 1 0 1 0 1; 1 1 0 1 1 0;   ],


    M_3,5 = [ 0 1 0 1 1 1; 1 0 0 1 1 1; 1 1 0 0 1 1; 0 0 1 0 0 0; 1 1 0 1 0 1; 1 1 0 1 1 0;   ],


    M_4,5 = [ 0 1 1 0 1 1; 1 0 1 0 1 1; 1 1 0 0 1 1; 0 0 0 1 0 0; 1 1 1 0 0 1; 1 1 1 0 1 0;   ].

Moreover, due to Eq. (<ref>) we have that M_j,i = M_i,j^t. Where we denote by A^t the transpose of matrix A.
 

Our argument is based on the next observation. To saturate the local bound of Eq. (<ref>), i.e., to reach 24, a local strategy must only lose in one combination of inputs (x,y). We can show that a local strategy that loses for (x=3,y=4) cannot reach the local bound of 24. In order to see this, first, let us consider the deterministic strategies δ_a,f(x) and δ_b,g(y) used by Alice and Bob, respectively. Where f(x) and g(y) are deterministic assignments of outputs for each input. Suppose that a local strategy p^L(a,b|x,y)=δ_a,f(x)δ_b,g(y) loses only for the inputs (x=3,y=4). This means that ∑_a,b c(a,b|3,4)δ_a,f(3)δ_b,g(4) = 0, or equivalently c(f(3),g(4)|3,4)=0. Moreover, since to win in (x=3,y=3) and (x=4,y=4) both deterministic strategies have to be the same, due to c(a,b|z,z)=δ_a,b, we have that f(3)=g(3) and f(4)=g(4). This implies that this strategy also loses for (x=4,y=3) due to

    ∑_a,b c(a,b|4,3)δ_a,f(4)δ_b,g(3)   = c(f(4),g(3)|4,3)
       = c(g(4),f(3)|4,3)
       = 0,

where in the last step we have used the symmetry c(a,b|x,y) = c(b,a|y,x). This proves that any local strategy that loses for (x,y) = (3,4) will also lose for (x,y) = (4,3) reaching at most a value of 23. The same argument can be applied for any combination of inputs x' ≠ y', with x' and y' being the common lines of Alice and Bob. 

The above argument shows that among all the local strategies that reach 24, none of them will lose in (x,y) = (3,4),(4,3),(3,5),(5,3),(4,5),(5,4). As a consequence a linear combination of all the local points that reach 24 would create a vector that has entries equal to zero whenever c(a,b|x' ≠ y')=0. This is already an indication that the saturating points are contained in a subspace of relatively low dimension. Next we show that this subspace is of a lower dimension than the necessary to be a facet. 

For a Bell inequality to define a facet of the local polytope, the set of points that saturate its local bound should span an affine subspace of dimension dim(NS)-1. dim(NS) is the dimension of the no-signaling space and it is given by dim(NS) = |X||Y|(|A|-1)(|B|-1) + |X|(|A|-1) + |Y|(|B|-1). For our case, (|X|=|Y|=5, |A|=|B|=6), we have dim(NS) = 675. As it can be noted, so far we have parameterized the correlations p={ p(a,b|x,y)}_abxy using a space of dimension |X||Y||A||B|, which is larger than dim(NS). In order to represent the correlations in a space of dimension dim(NS) we can use the parametrization introduced in <cit.>, also known as Collins-Gisin (CG) notation. In this notation the correlations p are parameterized using dim(NS) probabilities such that p = {p(a,b|x,y), p^A(a|x), p^B(b|y)}, where p^A (p^B) denotes the marginal probability of Alice (Bob). And, we also have that a = 1,…, |A|-1 and x = 1,…, |X|, while b = 1,…, |B|-1 and y = 1,…, |Y|. The CG notation is useful here because once we have removed the redundancy on the dimension of the correlation space we can directly translate the number of zeros of a linear combination of vectors with the dimension of its orthogonal complement.

Now the joint part of the correlations p(a,b|x,y) will be described by a 5 × 5 matrix, instead of 6 × 6, for every combination of inputs (x,y). Therefore we should only consider the first 5 × 5 entries in Eqs.(<ref>-<ref>), or in other words ignore the last row and the last column of these matrices. Then, a linear combination of all the saturating points will have: 12 zeros coming from M_3,4, 12 zeros from M_3,5, and 12 zeros from M_4,5, which including the coefficients in the transposition M^t_x,y reaches a total of 72 zeros. Therefore in the most favorable case the maximum dimension of the hyperplane defined by Eq. (<ref>) would be dim(NS)-72. This upper bound on the dimension of the hyperplane defined by Eq. (<ref>) shows that it is not a facet.

Now we extend this proof to the family of colored odd-pointed star games. Clearly for a colored odd-pointed star game the number of common lines is larger. But, since we only aim at proving that the dimension of the hyperplane is lower than dim(NS)-1, we can focus only on two of the analogous lines we have already analyzed in the case of the J(7,2)-configuration x=y=3,4. These lines in general would be defined as:

    x=y=3:    {13,23,34,…,3(N-1),3N},
    
        x=y=4:    {14,24,34,…,4(N-1),4N}.

The analogous matrix to M_3,4 would have 3(N-2) zeros. The zeros that will be removed are the ones placed in (a,b) = (N-1,1),(N-1,N-1), and (3,N-1). Therefore, a linear combination of saturating points would have at least 3(N-3) zeros that correspond to the zeros in (x,y) = (3,4). Thus the total number of zeros, including the transposition (x,y) = (4,3), will be at least 6(N-3). Consequently, the dimension of the hyperplane will be lower than dim(NS)-6(N-3), with N≥ 7. Hence Eq. (<ref>) does not define a facet.





§ ADVANTAGES AND WEAK POINTS
 



Here, we summarize the advantages of the colored odd-star PQSs with respect to other bipartite PQSs.





 §.§ Bipartite PQSs based on qudit-qudit maximally entangled states and KS sets




Existing bipartite PQSs based on qudit-qudit maximally entangled states and KS sets <cit.> are constructed by one of two methods: 
either both Alice and Bob's measurement settings are all the bases of the corresponding KS set or the settings of one of the parties are all the bases while the settings of the other party are all the vectors.  This implies that the resulting bipartite PQSs largely fail to have a “small” input cardinality,
since, e.g., the simplest known KS sets in d=3 are the the Conway-Kochen 31-vector KS set <cit.>, which has 17 orthogonal bases, and the Peres 33-vector set <cit.>, which has 16 orthogonal bases. In contrast, each of the KS sets that we are using have just d+1 orthogonal bases and the number of local settings of our PQSs is d-1. 





 §.§ Bipartite PQSs based on magic sets




A simple way to achieve bipartite PQS with small input cardinality and high local dimension d is to consider parallel repetitions <cit.> of bipartite PQS of small input cardinality <cit.> based on magic sets.
For example, N parallel repetitions of the magic square produce bipartite PQS with d=4^N, using only 3^N settings. Compared to this, the advantages of the approach presented in this paper are the following: (i) It generates bipartite PQSs with small input cardinality in every dimension d = 2^kp^m with m ≥ k ≥ 1 for p prime, d =8p for p>19 prime, and also in other sporadic cases, rather than only when d=(2^n)^N, with n ≥ 2. (ii) It produces “genuinely” d-dimensional PQSs in the sense that they cannot be achieved as parallel repetitions of the same smaller dimensional PQSs.





 §.§ Bipartite PQSs based on state-independent contextuality sets of vectors




Existing bipartite PQSs based 
on state-independent contextuality sets of vectors whose corresponding orthogonality graph has fractional packing number equal to the Lovász number and larger than the independence number, specifically those corresponding to
Pauli and Newman states <cit.>, have two disadvantages: (i') They only exist in specific dimensions. (i”) They have a very large input cardinality (see <cit.> for details).





 §.§ Noise
 



The arguably main problem of all existing bipartite PQSs is their high sensitive to noise. 
Any noise rapidly pushes us out of perfection. Not only that: even small noise rapidly kills the quantum advantage. 

Here, we first compute the visibility (as defined below) required to beat the classical winning probability of the PQSs for the colored odd-star games. As we will see, it is very high. Then, we present two ways to mitigate this problem.





  §.§.§ Visibility for the colored odd-star game




Suppose that the initial state is not the required maximally entangled state in local dimension d but, instead, the Werner state

    ρ(V_d) = V_d |ψ⟩⟨ψ| + 1-V_d/d^21_d^2,

where 1_d^2 denotes the d^2 × d^2 identity matrix, 0 ≤ V_d ≤ 1 is the visibility, and

    |ψ⟩ = 1/√(d)∑_k=0^d-1|k⟩⊗|k⟩.


Then, if Alice measures A_v = |v⟩⟨v|⊗1_d, and Bob measures B_w = 1_d ⊗|w⟩⟨w|, the probabilities of the four possible outcomes are


    
    P(A_v=1, B_w=1 | ρ)    = [(|v⟩⟨v|⊗|w⟩⟨w|)ρ], 
    
        P(A_v=1, B_w=0 | ρ)    = [(|v⟩⟨v|⊗ (1_d - |w⟩⟨w|))ρ],   
    
        P(A_v=0, B_w=1 | ρ)    = [((1_d - |v⟩⟨v|) ⊗|w⟩⟨w|)ρ], 
    
        P(A_v=0, B_w=0 | ρ)     = [((1_d - |v⟩⟨v|) ⊗ (1_d - |w⟩⟨w|))ρ].



In order to simplify the computation, we note that this computation is only necessary for our purposes when v = w or when v and w are orthogonal. Thus, by symmetry, we only need to do computations using the two vectors |s⟩=|0⟩^⊗ d and |t⟩=|0⟩^⊗ d-1⊗|1⟩, so that the above probabilities need only be computed with v=s, w=s and with v=s, w=t. We now derive one of the eight possibilities by hand, noting that the others follow in a similar manner:

    P(A_s=1, B_s=1| ρ) 
    
        =   [(|s⟩⟨s|⊗|s⟩⟨s|)ρ] 
    
       =   V_d[(|s⟩⟨s|⊗|s⟩⟨s|)|ψ⟩⟨ψ|] 
       +1-V_d/d^2[(|s⟩⟨s|⊗|s⟩⟨s|)] 
    
        =   (d-1)V_d + 1/d^2.


Using this (and appropriate expressions for the other probabilities), we compute the quantum winning probability. As in the computation of the Bell inequality (<ref>), we consider the two cases:

    
  * Alice and Bob receive the same line as input,
    
  * Alice and Bob receive different lines as input.


In Case (1), Alice and Bob win exactly when they make the same choice of point from the d choices. This is equal to d · P(A_s=1, B_s=1 | ρ), which is equal to

    P_1 := (d-1)V_d+1/d.


Now let us consider Case (2). Here, we compute the probability of failure. This happens in two different ways: either (2a) Alice and Bob disagree on their shared point (one of Alice or Bob outputs the intersection point of their two lines, and the other does not) or (2b) Alice and Bob choose points that are in a common (third) line.

The probability of failing in Case (2a) is 

    P(A_s=1, B_s=0|ρ) + P(A_s=0, B_s=1|ρ) 
    
     =    2(d-1)(1-V_d)/d^2.

The probability of failing in Case (2b) is the sum of probabilities that Alice and Bob measure orthogonal vectors over each of the d-1 pairs of points that occur in a common third line (for example, if Alice is given Line 1 of the J(7,2)-configuration and Bob is given Line 2 of the J(7,2)-configuration, then these pairs of points would be (13,23), (14,24), (15,25),(16,26),(17,27)). Thus, the probability of failing in Case (2b) is

    (d-1)P(A_s=1, B_t=1|ρ) 
    
        =    (d-1)(1-V_d)/d^2

remarking again that |s⟩ and |t⟩ are orthogonal vectors.
Therefore, the probability of success in Case 2 is 

    P_2 := 1 - 3(d-1)(1-V_d)/d^2.

Finally, Alice and Bob have d-1 lines each, d-3 of which are in common, and so there are:

    
  * d-3 input choices where Alice and Bob share a common line,
    
  * (d-1)^2 - (d-3) input choices where Alice and Bob are given different lines.

Assuming that any pair of lines is equally likely to be chosen, we find that quantum winning probability is

    W := (d-3)P_1 + ((d-1)^2 - (d-3))P_2/(d-1)^2.


On the other hand, there is a classical strategy that wins the game with probability 1 - 1/(d-1)^2, losing a single time out of the possible (d-1)^2 outcomes. 

In order to simplify our explanation, we assume that Alice may be given all of the lines except for N-3 and N-2 as input and that Bob may be given any of the lines except for N-1 and N. For each odd 1 ≤ i ≤ N-1, Alice and Bob choose the point {i,i+1} whenever either of them are given Line i or i+1 as input. Finally, if Alice is given Line N, then she chooses the point {N-1, N}.

One may check that with the above strategy Alice and Bob only fail if Alice is given Line N and Bob is given Line N-2 since Alice then chooses {N-1, N} and Bob chooses {N-2, N-1}. These both lie on the common line N-1, and so Alice and Bob lose in this case. By the Bell inequality (<ref>), it follows that this classical strategy is optimal. 

Therefore, in order for the quantum strategy to improve on the classical strategy, we must have 

    W ≥ 1 - 1/(d-1)^2,

which is accomplished exactly when

    V_d ≥(d-2)(4d^2-9d+6)/4(d-1)(d^2 - 3d + 3).

In the cases of d = 6,8,10, we get 32/35 ≈ 0.914, 285/301 ≈ 0.947, and 632/657 ≈ 0.962, respectively for the RHS, and for d large, the RHS approaches 1.





  §.§.§ Visibility for the non-colored odd-star line-line game




In this subsection and the following we compare the colored odd-pointed star game against two natural games which could be played on the star. In the current subsection, we consider the same game that we have described, except that both Alice and Bob may each receive any of the N lines as input. We call this the non-colored odd-star line-line game.

Here, Alice and Bob can win with probability 1-4/(d+1)^2 by using the following rule when receiving Line i as input:

    
  * if i is odd and i ≠ N, then they both choose point {i,i+1},
    
  * if i is even or i=N, then they both choose point {i, i-1}.

With this strategy Alice and Bob lose exactly in the four cases when one of them is given Line N-2 or N-1 and the other is given Line N. They win in all other cases. 

In order to show that this strategy is optimal, we consider two cases:

    
  * when Alice and Bob use the same strategy (i.e., choose the same point for each line),
    
  * when Alice and Bob do not use the same strategy (i.e., there is at least one line for which Alice and Bob choose different points).

Let us consider the first case. Since N is odd, the set {1,2,…,N} cannot be partitioned into sets of size 2. Therefore, we see that there must be lines i,j for which Alice and Bob choose the point {i,j} in Line i, but do not choose the point {i,j} in Line j. Let us say that Alice and Bob instead choose point {j,k} in Line j for some k ≠ i. We immediately see that Alice and Bob lose if Alice is asked Line i and Bob is asked Line j and vice-versa. Now consider Line k. No matter what point we choose for Line k, we see that Alice and Bob must fail in at least one of the following two scenarios:

    
  * Alice is given Line i and Bob is given Line k (and vice-versa),
    
  * Alice is given Line j and Bob is given Line k (and vice-versa).

Therefore, we see that Alice and Bob must indeed fail at least 4 times in any shared strategy.



Now let us consider the second case. Here Alice and Bob disagree on some Line i, and so must fail if both asked for this line. Without loss of generality we may assume that the line is Line 1, and that Alice chooses 12 and Bob chooses 13 when both are given this line. The following fact allows us to simplify matters significantly:

    Assume that Alice chooses 12 and Bob chooses 13 when given Line 1 as input. 
    Let 2 ≤ i ≤ N. For Line i if Alice or Bob make any choice from the points {1,i}, {2,i}, {3,i}, then Alice and Bob will fail for
    some choice of lines which includes Line i and Line j for some j ∈{1,i}.



    If Alice and Bob choose different points when both given Line i, then they will fail in this case, therefore we may assume that they choose the same point {i,k} with k ∈{1,2,3}. However, one of the intersections {i,k}∩{1,2} or {i,k}∩{1,3} must have size 1. Therefore, we see that there is indeed a failure involving Line i, and that the other line is either Line 1 or Line i.


Therefore, we see that Alice and Bob must fail at least three times: once when both given Line 1, once for some choice including Line 2 and Line 1 or 2, and once for some choice including Line 3 and Line 1 or 3.

Clearly, then, in order to have fewer than 4 failures, Alice and Bob must use a shared strategy for the lines {4,…, N}, and for each of these lines they must not choose a point that intersects {1,2,3}. By the same token, for each Line i ∈{1,2,3}, any choice of point {i,j} with j ∉{1,2,3} for either Alice or Bob would lead to a failure when the other is given Line j. Therefore, we see that each choice of both Alice and Bob from lines 1,2,3 must be a subset of {1,2,3}. Then we immediately have three failures: when Alice and Bob are both given Line 1, when Alice is given Line 1 and Bob is given Line 3, and when Alice is given Line 2 and Bob is given Line 1. Therefore, they must use a shared strategy for Lines 2 and 3 in order to avoid a fourth failure. One may quickly check by brute force that any such strategy does lead to a fourth failure. Finally, we conclude that an optimal classical strategy for Alice and Bob is the aforementioned shared strategy leading to exactly 4 failures.

On the other hand, we find that the quantum winning probability is 

    (d+1)P_1 + ((d+1)^2-(d+1))P_2/(d+1)^2,

which is higher than the classical winning probability when

    V_d ≥d^2 - d - 1/(d+1)(d-1).

For d=6,8,10, we thus need V_d to be at least 29/35 ≈ 0.829, 55/63 ≈ 0.873, and 89/99 ≈ 0.899, respectively. 





  §.§.§ Visibility for the non-colored odd-star point-line game




Here we consider the classical KS game, where Alice may receive any of the N lines as input, and Bob may receive any of the N-1 points on Alice's line as input. In this case Alice chooses any of the N-1 points on the line and indicates this by labeling the point with a 1, and labeling the remaining points on the line with 0 (therefore Alice has N-1 choices of output). Bob then labels his point with either 0 or 1 (therefore Bob has 2 choices of output). They win if and only if they agree on Bob's point. We call this instance of the classical KS game the non-colored odd-star point-line game.

In the classical KS game, Alice and Bob can win with probability 1 - 1/d(d+1) as follows. For each odd 1 ≤ i < N, Alice labels with 1 the point {i,i+1} with 1 for Line i and Line i+1. Then, for the final line, Alice can choose any point to label with 1. With the exception of the final point, Bob also labels with a 1 each of Alice's points that are labeled with 1. Bob labels the final point with a 0. Then the only losing combination is if Alice is given Line N and Bob is given Alice's final point. This strategy is optimal since there is no perfect classical strategy. Here, the quantum value is easy to compute: it is given by

    P(A=1, B=1 | ρ) + P(A=0, B=0 | ρ) 
    
        =    d^2 - 2(d-1)(1-V_d)/d^2

when Alice and Bob both use v to measure, and so in this case, we find that the quantum winning probability is higher than the classical winning probability when

    V_d ≥2d^2-d-2/2(d+1)(d-1),

which, for d=6,8,10, yields 32/35 ≈ 0.914, 59/63 ≈ 0.937, and 94/99 ≈ 0.949, respectively.





§ SUMMARY AND OPEN PROBLEMS





In this article, we have introduced two related results. On the one hand, we have extended a family of KS sets in dimension d, with d+1 orthogonal bases. Now the family covers any:

    
  * d = 2^kp^m for p prime, m ≥ k ≥ 0 (with the exception of k=m=0),
    
  * d = 8p for p ≥ 19,
    
  * d = pk whenever there exists a classical Hadamard matrix of order k and we have p > ((k-2)2^k-2)^2,
    
  * any d large enough, satisfying the congruence d+1 ≡ d'+1  (d'+1)(d')) for some d' already covered by the family. For example, any d > 294427, d ≡ 7  42,
    
  * any dimension d that may be constructed using <cit.>,
    
  * any d satisfying d+1 = (d'+1)^m where d' is already covered, d'+1 is a prime power, and m is any positive integer,
    
  * d=8,
    
  * other sporadic examples.

 On the other hand, we have shown how to distribute each of these KS sets between two parties and produce a PQS in such a way that the product of the number of settings of Alice times the number of settings of Bob is minimum. As a result, we have presented a family of PQSs in any (2,d-1,d) Bell scenario. The KS sets are the ones having the smallest number of bases known. For dimensions different than 4^N and 8^N (which correspond to the parallel version of the games based on magic sets for two and three qubits), the corresponding PQSs are the ones with the smallest number of settings known.

Here, we list here some problems for future research:

 
  * Find ORs of dimension d for the J(d+1,2) for the even dimensions not covered by the results of this paper. For d = 4, it is easy to see that such ORs do not exist <cit.>. The first case that remains open is d=32. It has been conjectured that for each even dimension d and p prime dividing d, there is a generalized Hadamard matrix GH(p,λ) with pλ = d <cit.>. This would resolve each case when the dimension that is not a power of 2 (since Lisoněk's construction does not work for p=2). The smallest open case is dimension 30 with p=5.

  * Identify (2,d-1,d) PQSs for odd d ≥ 5. For, d=3, it can be proven that (2,2,3) PQSs do not exist <cit.>. Moreover, the simplest qutrit-qutrit PQS known requires 9 settings in one party and 7 settings in the other <cit.>.

  * An open problem is, given d, what is the bipartite PQS with the smallest input cardinality. For d=3 and 4, it has been conjectured that the solution is (2,9-7,3) and (2,3,4), respectively <cit.>. For d=4^N, the PQSs with the smallest known input cardinality, 3^N for both parties, are achieved by taking N parallel repetitions of the magic square correlations. For d=8^N, the parallelization of the Mermin star <cit.> game due to Mancinska <cit.> has input cardinality 5^N for both parties. In fact, this may be reduced to 5^N for Alice and 4^N for Bob by noting that in the original formulation of the game, one can remove one of the inputs from either player while still not allowing a perfect classical strategy. This seems to be the smallest known input cardinality in dimension 8^N.
Of course, there are “trivial” PQSs in any (2,n ≥ 3,d ≥ 4) Bell scenario, so the question is rather what is the bipartite PQS with the smallest input cardinality for the corresponding d in the sense that it cannot be traced back to a PQS of smallest d as in, e.g., <cit.>, or in the previous examples. 

 
  * We noted in Sec. <ref> that the colored odd-pointed star game does not depend on the KS set given, only on the orthogonalities of the J(N,2) graph. 
 In principle, by taking account the additional orthogonalities of the KS set, the classical winning probability could potentially be lower. More formally, for some KS set 𝒦 = (𝒱, ℬ), one can play the following variation of the colored odd-pointed star game. Give Alice an orthogonal basis b ∈ S_A and Bob an orthogonal basis b' ∈ S_B for bipartite KS (S_A,S_B). Alice and Bob choose vectors v ∈ b, v' ∈ b'. They win if and only if v and v' are not orthogonal. Investigating these variations may be of interest.
    
  * For the KS sets of Sec. <ref>, the classical value of this game (with S_A, S_B chosen as in the colored odd-pointed star game) is at most the classical value of the colored odd-pointed star game. However, even with the additional orthogonalities, the classical value may be the same. It is then natural to ask if there is
good way to modify the KS sets of Sec. <ref> (by adding additional vectors) in order to ensure that the classical value is lowered. Such constructions would increase the resistance to noise. This connects to the most general open problem of constructing bipartite PQSs with larger resistance to noise.






§ ACKNOWLEDGMENTS




This work was supported by EU-funded project 10.3030/101070558FoQaCiA. A.C. acknowledges support from the 10.13039/501100011033MCINN/AEI (Project No. PID2020-113738GB-I00).



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§ FN, FNS, AVN
 



Here, we present the definitions of FN correlations, FNS correlations, and AVN proofs of Bell theorem.

Consider the (|X|, |A|; |Y|, |B|) Bell scenario, where Alice has |X| settings with |A| outcomes, and Bob has |Y| settings with |B| outcomes. A correlation p(a,b|x,y) may be viewed as a point in ℝ^|A||B||X||Y|. For any choice of A,B,X,Y, we obtain different sets of correlations by considering different theories.

A correlation p(a,b|x,y) is  deterministic if each output is a function of the corresponding input, i.e., a = f(x), b = g(y) for some deterministic functions f,g. A local correlation is any correlation that can be written as convex combination of deterministic correlations. Therefore, the set of local correlations forms a polytope, called the local polytope, whose vertices are  the deterministic correlations.

A correlation p(a,b|x,y) is quantum if it can be realized as

    p(a,b|x,y) = [(M_a|x⊗ M_b|y)ρ]

for some quantum state ρ and (POVM) measurements M_a|x and M_b|y. 

A correlation p(a,b|x,y) is non-signaling (NS) if, for every a,x,y,y',

    ∑_b p(a,b | x,y) = ∑_b p(a,b|x,y')

and, for every b,x,x',y,

    ∑_a p(a,b | x,y) = ∑_a p(a,b|x',y).

The set of all NS correlations is a polytope, the NS polytope. 

The local polytope is a subset of the set of quantum correlations, which in turn, is a convex subset of the NS polytope <cit.>.



Full nonlocality: Any NS correlation p(a,b|x,y) can be decomposed as

    p(a,b|x,y) = q_L p_L(a,b|x,y) + (1 - q_L)p_NL(a,b|x,y),

where p_L(a,b|x,y) is a local correlation and p_NL(a,b|x,y) is a nonlocal NS correlation, with respective weights q_L (called the local weight) and 1 - q_L (called the nonlocal weight), for some 0≤ q_L≤ 1. The local content or local fraction q_L^max of p(a,b|x,y) is the largest possible local weight over all such decompositions <cit.>. Formally, q_L^max := max_{p_L,p_NL} q_L. Conversely, the nonlocal content is the smallest possible nonlocal weight – thus, q_NL^min := 1-q_L^max.
The correlation p(a,b|x,y) is local if and only if the nonlocal content is zero (i.e., q_NL^min=0). 
A correlation p(a,b|x,y) is fully nonlocal (FN) <cit.> or strongly nonlocal <cit.> whenever q_NL^min=1.



Face nonsignaling correlations: A nonlocal correlation p(a,b|x,y) is a point in the NS polytope that lies outside of the local polytope <cit.>. A nonlocal correlation p(a,b|x,y) is face nonsignaling (FNS) if it is on a face of the NS polytope containing no local points.



All-versus-nothing proofs: 
The support s of a correlation p(a,b|x,y) is the set of indices for which p(a,b|x,y) is non-zero. An all-versus-nothing (AVN) proof of Bell theorem <cit.> is a quantum correlation for which there is no local correlation whose support is a subset of s.





§ CONNECTION BETWEEN PERFECT QUANTUM STRATEGIES AND KOCHEN-SPECKER SETS
 



Here, we summarize the one-to-one connection between PQSs and KS sets proven in <cit.>. The definition of KS set is in Sec. <ref> (see Definition <ref>). The definition of B-KS set is in Sec. <ref> (see Definition <ref>).

Every B-KS set defines a PQS for the following bipartite game: Alice (Bob) receives an orthogonal basis x (y) chosen at random from S_A (S_B) in Definition <ref>. Alice (Bob) outputs a vector of x (y). Alice and Bob win if and only if they output non-orthogonal vectors. 
The PQS is realized as follows: Alice and Bob share a qudit-qudit maximally entangled state in dimension d. For each x (y), Alice (Bob) measures the orthogonal projectors onto the vectors {s_a|x}_a ({s_b|y}_b) and outputs the vector associated to the result that she (he) has obtained.

Reciprocally, every bipartite PQS defines a B-KS set. 
Every PQS consists of a quantum correlation. As shown in, e.g., <cit.>, every quantum correlation can be realized as

    p(a,b|x,y) = ⟨ψ | Π_a|x⊗Π_b|y | ψ⟩,

where |ψ⟩∈ℋ = ℋ_A ⊗ℋ_B is a pure state, Π_a|x are projective measurements for Alice (i.e., Π_a|xΠ_a'|x= δ_a a'Π_a|x and ∑_aΠ_a|x = I_A, where I_A is the identity matrix in ℋ_A), and Π_b|y are projective measurements for Bob (i.e., Π_b|yΠ_b'|y= δ_b b'Π_b|y and ∑_bΠ_b|y = I_B, where I_B is the identity matrix in ℋ_B). 
For each a|x, we define 

    |ψ_a|x⟩ :=(Π_a|x⊗ I_B) |ψ⟩/√(⟨ψ | (Π_a|x⊗ I_B) | ψ⟩).

Similarly, for each b|y, we define 

    |ψ_b|y⟩ :=(I_A ⊗Π_b|y) |ψ⟩/√(⟨ψ | (I_A ⊗Π_b|y) | ψ⟩).

Then, the following result is proven in <cit.>.

 
    A correlation p(a,b|x,y) allows for a PQS in the (|X|, |A|; |Y|, |B|) Bell scenario if and only if there is a B-KS set S=S_A ∪ S_B, where S_A := {|ψ_a|x⟩ : a ∈ A, x ∈ X} and S_B := {|ψ_b|y⟩ : b ∈ B, y ∈ Y}.







>

 Paper Title: Universal Euler-Cartan Circuits for Quantum Field Theories.



 

ananda.roy@physics.rutgers.edu
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019 USA

Division of Condensed Matter Physics and Material Science, Brookhaven National Laboratory, Upton, NY 11973-5000, USA

Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019 USA

§ ABSTRACT 
Quantum computers can efficiently solve problems which are widely believed to lie beyond the reach of classical computers. In the near-term, hybrid quantum-classical algorithms, which efficiently embed quantum hardware in classical frameworks, are crucial in bridging the vast divide in the performance of the purely-quantum algorithms and their classical counterparts. Here, a hybrid quantum-classical algorithm is presented for the computation of non-perturbative characteristics of quantum field theories. The presented algorithm relies on a universal parametrized quantum circuit ansatz based on Euler and Cartan's decompositions of single and two-qubit operators. It is benchmarked by computing the energy spectra of lattice realizations of quantum field theories with both short and long range interactions. Low depth circuits are provided for false vacua as well as highly excited states corresponding to mesonic and baryonic excitations occurring in the analyzed models. The described algorithm opens a hitherto-unexplored avenue for the investigation of mass-ratios, scattering amplitudes and false-vacuum decays in quantum field theories.




Universal Euler-Cartan Circuits for Quantum Field Theories
    David Rogerson
    August 5, 2024
==========================================================






  
*Introduction.
Quantum field theory (QFT), which reconciles quantum mechanics and the special theory of relativity, is fundamental to the description of a vast array of physical phenomena. The latter range from confinement and asymptotic freedom in theories of elementary particles to high-temperature superconductivity and topological order in condensed matter systems. The investigation of non-perturbative characteristics of generic QFTs remains an outstanding challenge with ab-initio computations likely to stay beyond the reach of the most powerful classical computers. Quantum computation <cit.> provides an alternate paradigm for the analysis of QFTs with quantum algorithms boasting exponential speedup over classical ones for the computation of scattering amplitudes in massive scalar <cit.> and fermionic <cit.> theories. 

Despite their potential, it is a daunting challenge to implement quantum algorithms on state-of-the-art quantum simulators and solve QFT problems of widespread interest. This is because the current quantum simulators contain only a few hundred qubits with modest error rates. In the current noisy quantum era, hybrid quantum-classical algorithms, which combine the power of a quantum processor with that of a classical one, are crucial in bridging the enormous divide between purely-quantum algorithms and their classical counterparts. Notable examples include the variational quantum eigensolver <cit.> and its modifications <cit.> as well as the quantum approximate optimization algorithm <cit.>, which have been successful in analyzing problems in quantum chemistry <cit.> and combinatorial optimization <cit.>. 

There are two main elements of a hybrid algorithm that govern its efficacy: i) a parametrized quantum circuit ansatz built with a predetermined pool of unitary operators, ii) an efficient optimization procedure for the parameters specifying the unitary operators. While the second element is typically based on variations of gradient descent methods <cit.>,  there is no natural prescription for the first. The circuit ansatz either relies on the Hamiltonian whose eigenstate is being sought alongside a heuristically-chosen mixer Hamiltonian <cit.> or variations of the coupled cluster theory relevant mostly for chemistry problems <cit.>. Neither of the aforementioned are suitable for generic QFT problems whose Hamiltonians, when mapped to registers of qubits in a quantum computer, could involve long-range couplings <cit.> or might not even be Hermitian <cit.>. Furthermore, many interesting properties of QFTs such as mass-ratios and scattering amplitudes remain encoded in highly-excited states <cit.>, which have largely eluded existing hybrid algorithms. Last but not least, restricting the circuit ansatz to a given pool of unitary operators that does not encompass the entire manifold of possible unitary operators necessarily implies that the result so obtained may not be the globally optimum solution which could potentially be found by considering a larger ansatz. 



In light of the above, it is natural to ask: is there a quantum-classical algorithm based on a universal parameterized quantum circuit ansatz that is tractable with established optimization strategies and applicable to a broad set of QFT problems? The goal of this work is to answer this question. The presented solution computes the optimal circuit over all possible two-qubit unitary rotations required to realize an arbitrary L-qubit unitary rotation <cit.>. The unitary operators are chosen to couple neighboring qubits, which is imposed for potential implementation on majority of quantum computing architectures and could be relaxed in a more general setting. The model under investigation affects the parameters of the circuit, but not the ansatz itself. Importantly, the proposed ansatz explores the largest possible manifold of allowed unitary operators and is closer to the globally optimal solution than what could be obtained by considering a more restrictive ansatz. The potential of the algorithm is demonstrated by computing not only ground states of strongly-interacting QFTs, but also highly excited states that correspond to bound-states of topological excitations occurring in these theories. 




  
*Algorithm. The algorithm is implemented on a quantum processor coupled with a classical one (Fig. <ref>). The former is involved in  applying the parametrized circuit and conducting measurements required for the optimization process. The classical processor receives the measurement outcomes, performs the optimization and feeds the outcome back into its quantum counterpart. 

The algorithm begins with the application of N (see Figs. <ref>, <ref> for representative choices of N) layers of unitary operators on L qubits of the quantum processor initialized in the state |ψ_ i⟩. Each of the N layers is built, in principle, out of two sublayers of two-qubit operators [orange rectangles in Fig. <ref>(b)]. Each two-qubit operator is an element of the group SU(4). It can be written, using Cartan's KAK decomposition <cit.>, as a product of four single-qubit SU(2) rotations and an SU(4)-entangler [Fig. <ref>(c)]. The latter when involving qubits i, i+1 equals e^-i(α X_iX_i+1 + β Y_iY_i+1 + γ Z_iZ_i+1) where X_i, Y_i and Z_i are the Pauli operators of the i^ th qubit. The SU(4) entangler can be realized by three controlled-NOT gates and five single-qubit rotations <cit.> [Fig. <ref>(d)]. Finally, each of the four single-qubit rotations depicted in panel Fig. <ref>(c) can be further decomposed into three single qubit rotations by the well-known Euler angles <cit.>. Using the above, it can be shown that the general circuit ansatz is constructed with two-qubit operators [orange ellipses in Fig. <ref>(b, e)] with ten undetermined angles. 


After the application of the N layers, measurements are performed to determine the relevant gradients and the cost-function L(θ⃗), where θ⃗ contains all parameters of the N layer quantum circuit. To obtain the ground state of a given Hamiltonian H, the cost-function is chosen to be L(θ⃗) = ⟨ψ_ f(θ⃗)|H|ψ_ f(θ⃗)⟩. To obtain the n^ th excited state, the cost-function is augmented to L(θ⃗)→ L(θ⃗) + ∑_m < nλ_m|⟨ψ_m|ψ_ f⟩|^2, with λ_m>0, m = 0,…, n-1. This choice amounts to simultaneously minimizing the energy and the overlap to the n-1 lower-energy states that have been already obtained.  

The measurement results are communicated to the classical processor which checks for the convergence of the cost-function for the given choice of N (iteration convergence in Fig. <ref>). If the condition is not satisfied, the parameters of the circuit are updated using an optimization routine. In this work, the quantum natural gradient optimizer <cit.> is used due to its superior performance in obtaining eigenstates of many-body Hamiltonians <cit.>. In contrast to ordinary gradient descent, this  ensures that optimization follows the steepest descent taking into account the geometry of the space of wavefunctions <cit.> and can viewed as the quantum analog of the classical information geometric approach <cit.>. The parameters at the (t+1)^ th iteration are related to the t^ th one by

    θ⃗_t + 1 = θ⃗_t - η g^-1∂ L/∂θ⃗_t,

where η is the learning rate (typically a real number < 1) and g is the Fubini-Study metric <cit.>. The latter is given by the real part of the quantum geometric tensor G:

    G = ⟨∂ψ_ f/∂θ_i|∂ψ_ f/∂θ_j⟩ - ⟨∂ψ_ f/∂θ_i|ψ_ f⟩⟨ψ_ f|∂ψ_ f/∂θ_j⟩.

Note that both the gradients need to be evaluated using measurements for both the cost-function and the Fubini-Study metric <cit.>.
Subsequently, the updated set of parameters are fed back to the quantum processor and the process is repeated until the iteration convergence criterion is satisfied. Once the latter is satisfied, a second layer convergence criterion is verified. This ensures that the number of layers chosen is sufficient to attain the desired degree of convergence. If this condition is not satisfied, the number of layers is grown by 1 and the earlier steps are repeated. Else, the algorithm terminates. When the number of layers is increased from N to N+ 1, the optimized parameters obtained for the N-layer circuit, augmented by an initial guess for the added layer, were used as the choice for the optimization process for N+ 1 layers. This improved the stability of the algorithm.

For N layers and a ring of L qubits, the total number of parameters determined by the above algorithm is 10LN, where the factor 10 originates from the number of free parameters in the two-qubit unitary operator of Fig. <ref>(e). Despite being tractable [see Fig. <ref>(c) for demonstration], conservation of symmetries present in the target Hamiltonian can drastically reduce the number of parameters to be determined. Symmetries can be conserved by modifying the general circuit ansatz of Fig. <ref> and allowing only operators that respect this symmetry. Alternatively, the cost-function can be augmented to ensure that the obtained state lies in the correct symmetry sector. For example, states which are invariant under translation by r lattice site(s) can be obtained by starting with a state and circuit ansatz respecting this symmetry. For r = 1(2), the number of undetermined parameters for N layers reduces to 10N (20N). Alternatively, in order to obtain a ±1 eigenstate of the translation operator T, the cost-function can be augmented as: L(θ⃗)→ L(θ⃗) ∓μ⟨ψ_ f(θ⃗)|T|ψ_ f(θ⃗)⟩, where μ is a positive real number. Similar considerations apply to symmetry operators related to parity/particle number conservation. 





  
*Results. The proposed algorithm is benchmarked using three paradigmatic one-dimensional QFTs. The results were obtained by modifying standard time-evolution routines for matrix product states. In all three cases, the quantum processor considered comprises a periodic ring with a fixed number of qubits with nearest-neighbor connectivity. This is independent of the model under consideration. The results obtained using the optimized quantum circuit are compared with those from density matrix renormalization group (DMRG) and exact computations. 

First, the critical Ising model in a longitudinal magnetic field is analyzed. In the continuum limit, the QFT contains eight mesonic excitations which scatter with a diagonal scattering matrix <cit.>. These mesons can be viewed as an extremal (g→ 1) limit of bound states formed by the Ising domain walls in the ferromagnetic phase g < 1. The magnetic field leads to a linearly growing energy of separation of two domain walls <cit.>, akin to what happens in quarks in quantum chromodynamics <cit.>, see Fig. <ref>(c). The QFT describing this model is integrable and the number of mesons and their mass ratios are given by the rank and the exponents of the Lie algebra E_8 respectively <cit.>. The relevant lattice Hamiltonian, imposing periodic boundary condition, is:

    H_ I = -∑_j = 1^L X_jX_j+1 - g∑_j=1^LZ_j - h∑_j=1^LX_j,

where g and h were chosen to be 1 and ≃0.156 respectively. Figs. <ref>(a) and <ref>(a) together present the results for the lowest eight translation-invariant states obtained using the proposed algorithm. The circuit ansatz was chosen to be invariant under translation by 2 sites. Furthermore, the cost-function was augmented to ensure that only +1 eigenstates of the operator for single-site translation were obtained. Fig. <ref>(a) shows the results for the ground state energy for system-sizes between L = 8 and 20 in steps of 4. The number of layers required to obtain a reasonable approximation of the ground state do not scale with the system-size, which is compatible with the observations of Ref. <cit.>. In fact, N=2 is sufficient to obtain  the ground state with error in the energy < 1% [Fig. <ref>(d)]. Fig. <ref>(a) shows the results for the first seven excited states for L = 8. For the chosen convergence criterion of 10^-3, the maximum error in the obtained energies was 0.58%. In contrast to the ground state, the number of layers required to achieve this precision ranged between L and 3L. The different translation-invariant excited states can be classified as lattice counterparts of the single and multiple meson states existing in the QFT. The mass ratio for the lowest two mesonic excitations [square and diamond markers in Fig. <ref>(a)] obtained for L = 8 obtained using the proposed algorithm is 1.39 which is close to the expected value of 1.41 obtained using exact diagonalization of  the corresponding spin-chain. This should be compared with the QFT prediction <cit.> given by 2cos(π/5)≃ 1.618. 




Next, the three-state Potts model in the presence of a magnetic field <cit.>  is analyzed. In contrast to its two-state counterpart described above, this model exhibits a richer state of excitations. The three states of each Potts spin are encoded in the triplet of states of two qubits: 

    |0⟩ = |↑↑⟩, |1⟩ = 1/√(2)(|↑↓⟩ + |↓↑⟩), |2⟩ = |↓↓⟩.

The relevant Hamiltonian is

    H_ P = -∑_j  = 1^L'(σ_jσ_j+1^† + gτ + h σ +  h.c.),

where σ and τ are the shift and clock matrices respectively <cit.>. Periodic boundary condition was imposed and L' = L/2 is the number of Potts spins encoded in L qubits. For g< 1, h = 0, the model exhibits a ferromagnetic phase with three degenerate vacua and six domain walls [the three shown in Fig. <ref>(d) and those obtained by reflection about the domain wall]. The presence of the magnetic field breaks the degeneracy of the ground space, giving rise to one true vacuum and two false vacua. The domain-walls experience a confining potential as in the Ising case. However, now there are not only mesonic excitations formed by two domain walls, but also baryonic excitations formed by three domain walls [Fig. <ref>(d)]. Notice that H_ P commutes with the ℤ_2 charge operator C which acts trivially on the state |0⟩ and interchanges |1⟩ and |2⟩. Figs. <ref>,<ref> together show the results for g = h = 0.1 for the lowest eight translation-invariant states with eigenvalue +1 for the operator C. The 2-site translation-invariant circuit ansatz was chosen with the C-eigenvalue imposed using the cost-function. As in the Ising case, the ground state is reached with depth independent of the system-size, with errors in energy less than 0.1% for two layers [Fig. <ref>(b, d)]. The excited states were obtained using between L and 3L layers with maximum error in the obtained energies being 2.26% [Fig. <ref>(b)]. The first excited state is one of the false vacua, with the rest comprising mesonic and baryonic excitations. This is verified by counting the average number of domain-walls in the corresponding state. 

Finally, results are presented for the canonical model for quantum electrodynamics in two space-time dimensions: the one-flavor massive Schwinger model <cit.>. After mapping to an array of qubits, the resulting Hamiltonian with open boundary conditions is given by <cit.>:

    H_ MS   = 1/2∑_j = 1^L  - 1(X_jX_j+1 + Y_jY_j+1) + m∑_j = 1^LP_j 
         + g^2/2∑_j = 1^L - 1[∑_k =1^j (-1)^kP_k]^2,

where P_j = [1 + (-1)^jZ_j]/2. In contrast to the Ising and Potts models, this model involves an all-to-all coupling Hamiltonian. To obtain the ground state of this model, the general circuit ansatz of Fig. <ref> is used. Note the existence of the conserved U(1) charge Q, [H, Q] = 0, where Q = ∑_j=1^LZ_j. To test the algorithm under the most challenging of conditions, the ground states were obtained for system sizes between L = 8 and 16 without conserving this charge. For these simulations, m = 0.5, g = 0.3 were chosen. Despite the long-range nature of the model Hamiltonian, reasonable approximations of the ground-state were obtained with circuit-depths that are independent of the system-size with energies within 1% accuracy obtained within application of 2 layers. Excited states were also obtained similar to the Ising and Potts models and are not shown here for brevity.



  
*Conclusion. To summarize, a hybrid quantum-classical algorithm is presented for the investigation of strongly-interacting QFTs. In contrast to existing approaches that rely on heuristic arguments for choosing the parametrized quantum circuit ansatz, a universal ansatz is described relying on Euler's and Cartan's decomposition of single and two-qubit operators. The algorithm is benchmarked by computing eigenstates of lattice models for three paradigmatic one-dimensional QFTs: the Ising, Potts and one-flavor massive Schwinger models. Quantum circuits were obtained for the realization of ground states, false vacua, mesonic and baryonic excitations occurring in these models. 

The presented results demonstrate the power of geometric control theory <cit.> for the analysis of QFTs. While geometric quantum control is known to be the central concept in the search for an optimal circuit for quantum computation <cit.>, previous proposal are centered around minimization of the distance in the Finsler space of unitary operators. In contrast, the current work presents an alternative framework based on the Fubini-Study metric and a universal circuit ansatz. The proposed framework is not only numerically-tractable, but also readily-implementable on current quantum machines, opening the door to hybrid quantum-classical analysis of a large class of QFTs. 




  
*Acknowledgements.
This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Contract No. DE-SC0012704. 





 Paper Title: Full classification of Pauli Lie algebras.







Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

 
 Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
 
	

 Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

 
 Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
	




§ ABSTRACT 
    Lie groups, and therefore Lie algebras, are fundamental structures in quantum physics that determine the space of possible trajectories of evolving systems. However, classification and characterization methods for these structures are often impractical for larger systems. In this work, we provide a comprehensive classification of Lie algebras generated by an arbitrary set of Pauli operators, from which an efficient method to characterize them follows. By mapping the problem to a graph setting, we identify a reduced set of equivalence classes: the free-fermionic Lie algebra, the set of all anti-symmetric Paulis on n qubits, the Lie algebra of symplectic Paulis on n qubits, and the space of all Pauli operators on n qubits, as well as controlled versions thereof. Moreover, out of these, we distinguish 6 Clifford inequivalent cases and find a simple set of canonical operators for each, which allow us to give a physical interpretation of the dynamics of each class.
    Our findings reveal a no-go result for the existence of small Lie algebras beyond the free-fermionic case in the Pauli setting and offer efficiently computable criteria for universality and extendibility of gate sets. These results bear significant impact in ideas in a number of fields like quantum control, quantum machine learning, or classical simulation of quantum circuits.

Full classification of Pauli Lie algebras
    [black]https://orcid.org/0000-0003-1626-2761Lennart Bittel
    August 5, 2024
==============================================================





§ INTRODUCTION


Lie groups emerge as a fundamental mathematical structure in quantum physics due to their intrinsic connection to the Schrödinger equation. The set 
of possible trajectories of an evolving quantum system is confined to the Lie group associated with the generators of the dynamics. Consequently, the study and classification of continuous symmetries and dynamics in quantum physics can be framed in terms of the analysis of such objects. Often, this analysis can be simplified by instead examining the tangent spaces of these Lie groups, namely Lie algebras, where the commutation relations of their elements encode the essential information about the evolution. 

This connection has been studied extensively in several fields within quantum information theory and quantum computation. In quantum control theory, universality and controllability questions are entirely ones of reachability, and hence Lie-algebraic in nature, making the study of these structures crucial for the design of universal quantum gate-sets and simulators <cit.>. Lie algebras have also been shown to play a crucial role in understanding and designing variational quantum algorithms <cit.> and quantum machine learning methods <cit.>, where in certain scenarios have been directly linked to phenomena like barren plateaus and overparametrization <cit.>. Additionally, approaches such as geometric quantum machine learning <cit.> also rely on Lie algebraic notions in order to simplify training by embedding problem specific symmetries into the circuits. Finally, ideas in quantum circuit complexity, like measures based on operators spread, are also strictly related to the Lie algebra spanned by the circuit generators <cit.> , and hence also the classical simulability of some families of quantum circuits <cit.>. 

Classification and characterization of Lie algebras are thus crucial for the further development of these fields. In a general setting however, this task often becomes intractable. Consequently, progress in this regard has been restricted to specific instances, where it is possible to rely on certain algebraic structural properties, like simplicity <cit.>, or to construct an explicit basis of the Lie algebra, which is only feasible under strict assumptions on the generators <cit.>, like locality or periodicity.

This work adds to this literature by giving a comprehensive classification of all  Pauli Lie algebras, these being all Lie algebras generated by any set of Pauli operators, and also providing an efficient method to determine the Lie algebra of an arbitrary set of Pauli generators. Pauli operators are particularly interesting because they form a basis for all Hermitian operators. As such, they appear as natural generators in many popular gate sets in the context of digital quantum computing and become relevant in a plethora of tasks, like whenever one considers Hamiltonian simulation techniques based on product formulas <cit.>.  Moreover, their particular commutation structure resembles that of Majorana operators in fermionic systems, thus making them directly useful for their study.

Due to these commutation and anti-commutation rules of Pauli operators, which we will leverage to simplify the problem, our work bears big resemblance to studies on (quasi-)Clifford Lie algebras in the mathematics literature <cit.>. 
While these studies focus on the mathematical structure of the Lie algebra, we aim to characterize these sets in a way that allows us to make further remarks about the embedding Hilbert space and the physics of their evolution.
The key component in our analysis will be to map the classification problem to a graph reduction problem. 
This approach will be similar in spirit to previous works like Bouchet's on graph equivalence under local complementations <cit.> – that has important implications
on assessing whether two graph states
<cit.>
are equivalent under sequences of local Clifford
conjugations <cit.> – or others considering the vertex minor problem, where vertex deletion is also considered <cit.>. In our case, we will find a set of universal equivalence classes for graphs under conditioned local complementations. This will allow us to identify the existence of 6 total Clifford inequivalent Lie algebras. Having that any given collection of initial operators can be mapped through Clifford operations to one of these six simpler sets, we will then derive their Lie algebra type and provide a physical interpretation of their structure along with their relationships.

This reveals that the biggest polynomially-sized Pauli Lie algebra is that of free-fermions together with their parity operator. All other non-free-fermionic cases correspond to the full set of Paulis on n-qubits as well as an embedding thereof in an (n+1)-qubit Hilbert space, the space of all anti-symmetric Pauli operators, and the space of symplectic Pauli operators, each exhibiting a dimension growing exponentially with system size. Beyond these cases, only controlled versions of these dynamics exist.

The remainder of this work is structured as follows. In <Ref> we define the structures we are interested 
in, as well as the operations required for the reduction. In <Ref> we introduce the main theorems of this work regarding graph reductions and their mapping to Lie algebra types. We add the distinction between Clifford inequivalent Lie algebras in <Ref>. With all Lie algebra types layed down, we discuss their physical interpretation in <Ref>. In <Ref>, 
we make a final remark regarding disconnected graphs for completeness, and we finalize in <Ref> with a brief discussion of the implications of our results. We prove most of the statements in this work in the Supplemental Material.



§ DEFINITIONS AND SETTING


The statement of our results will require some preparation. Let us first define our main object of interest.

    
        
        We define a Pauli Lie algebra 𝔤 as the Lie algebra whose elements are Pauli strings of some n-qubit system, which is closed under the Lie bracket, ([·,·]_L:𝔤×𝔤→𝔤), given by the imaginary commutator of its elements, P_iP_j_L iP_iP_j =  i(P_iP_j-P_jP_i).
        Then, a set of Paulis 𝒢={P_1,…,P_n_G} is a generating set of 𝔤 if they span all elements in 𝔤 through Lie algebraic operations. Then we write ⟨𝒢⟩_,[·,·]=𝔤.   A generating set is minimal, if removing any P_i would change the size of the Lie-algebra. 
    

    Pauli Lie algebras have a very appealing structure since Pauli strings have simple commutation relations, namely given any two Paulis they either commute or anti-commute and the resulting operator is again a Pauli up to a constant. For two non-commuting Paulis P,Q, it holds that  
    
    [P,Q]= 2  P Q

    where P Q is itself another Pauli. Moreover, Pauli operators are particularly interesting since they form an orthonormal basis for the space of operators.
    A natural question to ask then, is whether one can characterize all Pauli Lie algebras generated by some arbitrary set 𝒢={P_1,…, P_n_G}.
     Like we already mentioned in the introduction, one common approach to address this, has been to explicitly construct a basis for the Lie algebra by repeatedly taking nested commutators of the generating elements <cit.>. However, since the size of the Lie algebra could be exponential, this method is in many cases not scalable. 

    Here, we will propose a new strategy that overcomes this by mapping the problem to a graph problem. Exploiting some of the already mentioned properties of Pauli operators, we will be able to characterize Pauli Lie algebras for an arbitrary set of generators. This connection is expected to be
    fruitful, in a way that is reminiscent of 
    the
    way graph states <cit.> 
    constitute a theoretical
    laboratory to study questions in entanglement theory.

    Since the basis of the Lie algebra is generated through the commutator of its elements, and the commutator of two Paulis is either 0 or their product, all the information of the basis elements is already encoded exclusively in the commutation relations of the elements in 𝒢, so long as the generators in 𝒢 are Lie algebraically 
    independent, that is, no generator can be computed through commutators of others. This is satisfied if the generators form a minimal set. For now we assume that we are given a minimal set of generators, but then we lift this assumption in <Ref>.
    This allows us to study the Lie algebra by the anti-commutation relations of the generators 𝒢.
    
    
        Given a set of Pauli operators 𝒢={P_i}_i=1^n_G, we call its anti-commutation graph the graph Γ=(𝒱,ℰ), where every vertex corresponds to some P_i, and two vertices are connected by an edge if their corresponding Paulis anti-commute, i.e.,
        
    𝒱 = {P_i}_i=1^n_G,  ℰ = {(P_i,P_j)|[P_i,P_j]≠ 0}.

    
    In general, this can be any possible graph for sufficently many qubits.
    In 
    <Ref>(a, left),
    we show the anti-commutation graph for the Lie algebraically independent set 𝒢={Z_0,X_0,Y_1X_2,X_1X_3,Z_3,Z_4X_3Z_1,X_4}.

    
   
    
    Our goal will be to find a set of universal equivalence classes for graphs, that allows us to determine the Lie algebra associated to any given graph. To that end, we will define graph operations that leave the underlying Lie algebra invariant, and restate our problem as a graph reducibility problem under such operations. The operation in question will be a conditional local complementation. Namely, given two vertices (p,q), we will complement edges (p,v) conditioned on q sharing an edge with v or not. On the level of the Paulis, this will correspond to replacing, in the generating set, {P,Q}↦{± i PQ,Q}, i.e., replacing a generator by its commutator with another generator. 
    
    
        Given two non-commuting generators P,Q∈𝒢, a contraction of Q onto P maps P↦ P'=±1/2i PQ and 𝒢↦𝒢'∋ P'. This leaves the Lie algebra invariant 𝔤=𝔤'.
    
    It is easy to see that indeed this operation does not change the Lie algebra because of the fact that P=±1/2 iP'Q. As such, the set generated from nested commutators of 𝒢' contains all those generated by 𝒢.
     Like we said however, for the anti-commutation graph Γ, this transformation will result in a new graph Γ' with modified connectivity. In more concrete terms, for the associated vertices p,q, corresponding to P,Q∈𝒢, a contraction of q onto p amounts to a complementation of the edges E_p={(p,v) | (p,v)∈ E  (q,v)∈ E}. 
     In terms of the adjacency matrix of the graph A_Γ, this is performed by adding the the columns and rows corresponding to vertex q onto the columns and rows corresponding to vertex p in A_Γ.
    
    An example of such a contraction is shown in 
    <Ref>(a, right).
    In the following section we will describe universal equivalence classes under these contractions and how can that help us determining the Lie algebra in the most general case.
    
    

§ CANONICAL GRAPH TYPES AND LIE ALGEBRAS

    
    
    We establish that using graph contractions, every anti-commutation graph is reducible to one of a few canonical types.
     
        
    For any given connected graph Γ with at least two vertices, there exists a sequence of contractions which result in one of four graphs:
    
        
  * A line graph with with n_L vertices as well as n_c∈ℕ_0 single vertices connected to the second to last vertex.
        
  * A star graph with legs of length of at most 4 and where the number of legs of length 1 and 2 are n_c+1, with n_c∈ℕ_0 and n_2∈ℕ_+, respectively. Additionally, they can present the following:
        
            
  * no legs of length 3 or 4,
            
  * one leg of length 4 and no leg of length 3,
            
  * one leg of length 3 and no leg of length 4.
        
    
    The canonical 
    types are shown in <Ref>(b).
    
    (Sketch)
        The full proof is in  <Ref>.
        The statement is proven by induction. The procedure works by selecting some initial vertex at random and iteratively adding new vertices. 
        We show that if the starting graph is one of the considered here, if one adds a new vertex with some arbitrary connectivity to it, there is always a sequence of contractions that transforms the new graph to one of these types.
    
    One important remark is that for any given initial graph Γ, the set of contractions that map it to one of these canonical representatives can be found efficiently. Since the proof works inductively in the number of vertices, applying a known set of transformations, one can find an algorithm that is efficient in the number of vertices of Γ that finds the corresponding class. We leave the explicit algorithm for future work.

    With this, one can then classify the Lie algebras generated by a minimal set of Paulis by only looking at their graph and the canonical representative it maps to.
    This brings us to the main result providing  the  classification of the Lie algebra types.

    
        
        Given a set of Lie algebraically independent Paulis 𝒢, with connected anti-commutation graph Γ, then its Pauli Lie algebra corresponds to
        one of the following cases.
        
            
  * ⊕_i=1^2^n_c𝔰𝔬(n_L+1)   when Γ maps to some Γ in class A.
            
  * ⊕_i=1^2^n_c𝔰𝔭(2^n_2)   when Γ maps to some Γ in B1.
            
  * ⊕_i=1^2^n_c𝔰𝔬(2^n_2+3)   when Γ maps to some Γ in B2.
            
  * ⊕_i=1^2^n_c𝔰𝔲(2^n_2+2)   when Γ maps to some Γ in B3.
        
    


    
        (Sketch) The full proof can be found in the appendices. In <Ref> we prove the ⊕_i=1^2^n_c terms 
        .The main idea will be to show that several legs of length 1 act as a control register, thus splitting 𝔤 into two equal, smaller Lie algebras. The rest of the statement is proven by construction in <Ref>. The key will be to come up with a good set of canonical Pauli operators, for which it is easy to show, and since the Lie algebra type is uniquely determined by the graph, argue for the general case. For graphs in class A, we will explicitly construct a set of free-fermionic majorana operators and show that this is indeed equivalent to 𝔰𝔬(n_L+1). For graphs in B1 and B2, we will show that some set of canonical generators obey the properties defining 𝔰𝔭,𝔰𝔬 respectively, and thus they span a sub-algebra of those. On the other hand, we will come up with a set of rules to compute non-zero commutators on these graphs, and show that every element in 𝔰𝔭,𝔰𝔬 can actually be written as such a commutator, thus showing equality with the Pauli Lie algebra spanned by our set. Finally, for graphs in B3 we will just show that every Pauli can be computed through nested commutators of our generators.
    
    Crucially, the classification of the Lie algebras does not depend directly on the number of qubits of the system. For our purposes, however, besides the structure of the algebra, it will also be relevant to see how a Lie algebra acts within the physical Hilbert space, which we will address in the following sections.
    
    

§ CLIFFORD EQUIVALENCE OF PAULI LIE ALGEBRAS

    
    So far we have discussed the anti-commutation relations for a minimal set of Pauli generators. However, Paulis can also have algebraic dependencies, meaning that even if no generator can be written as a non-zero commutator of others, we could have some generators being a product of others. This is relevant for our next theorem
    (Clifford equivalence lemma)
        Two Pauli Lie algebras are Clifford equivalent if they can be mapped to the same anti-commutation graph Γ, where the generators share the same algebraic dependencies.
    
   
    
        This follows from the general properties of Cliffords <cit.>.
    
    In general it also holds that if two Lie algebras are not Clifford equivalent, then they are also not unitarily equivalent since no unitary can change commutation relations or algebraic dependencies.
    As such, we would like to determine all possible algebraic dependencies within the Pauli Lie algebra of some minimal generating set. We can establish the following theorem.
    
    
        Every minimal generator Pauli Lie algebra has at most one algebraic dependence. Algebraic dependencies can only occur for graphs of type A and B3. These can be mapped by contraction to only one possible case each which are shown in <Ref> (c).
        As such, there is a total of 6 Clifford inequivalent families of Pauli 
        Lie algebras.
    
    
    
        (Sketch) The full proof can be found in <Ref>, but we 
        state the main idea here.
        For this it is enough to check algebraic dependencies within the generating set of the graphs in <Ref>, as contractions preserve algebraic and Lie algebraic dependencies. Given these specific anti-commutation relations, the sets of vertices that can be algebraically dependent are limited to subsets {v_i}_i=1^m⊆Γ involving legs of odd length, where the v_i's appear in an alternating fashion in Γ. Moreover, we will show that since algebraic dependence between legs of length 1 also implies Lie algebraic dependence, the only cases left are the ones stated in the theorem. 
     
    

 §.§ Finding the minimal set of generators

    
    The scenario described above assumes that we have a minimal set of generators. In general, this might not be the case however. If the set is not minimal, then a generator can be removed. The strategy is to still perform all the graph transformations to get to a canonical form which leaves the Lie algebra invariant even it is not the minimal set. By Gaussian elimination one can identify algebraic dependencies between generators, which is a necessary condition for Lie algebraic dependence. For the canonical types, algebraic dependencies can only occur between legs of length 1 and on a long leg of odd length in an alternating structure (as in <ref>). Since the latter is an allowed dependence, a redundant generator will necessarily result in an algebraic relation involving only legs of length 1. In this case one of the involved vertices can be safely removed without changing the Lie algebra. This process can be (efficiently) repeated until all Lie-algebraic dependencies are removed.
    

§ GENERAL DESCRIPTION OF ALL PAULI LIE-ALGEBRAS

    
    In this section, we will give an intuition behind our classification result. First of all, we show in <Ref> that the direct sums in <Ref>, translate to an actual direct sum within the physical Hilbert space. Namely, given several legs of length 1, using Cliffords, we can always map the Lie algebra to be of the form
    
    𝔤=span({Z_1^i_1⊗⋯⊗ Z_n_c^i_n_c⊗ P_𝔤'}|i⃗∈{0,1}^n_c)

    where i⃗∈{0,1}^n_c labels each block in the direct sum, and P_𝔤 corresponds to operators in the Lie algebra where all additional legs of length 1 are removed (n_c=0). Then, from <Ref> it is clear
    that these additional legs of length 1 always introduce a 
    new qubit register acting as a control on the dynamics given by P_𝔤. 

    With this, we can now move on to considering each of the 6 types from <Ref> without additional legs of length 1 (and hence qubit registers).
    From now on, we will refer to n as being the minimal number of qubits required to span their dynamics (hence the total number of minimal qubits will actually correspond to n=n+n_C). We will show in <Ref> that this number n, like for the case of controls, can easily be derived from the number of vertices of the graph.
 
    For type A_I, the number of qubits corresponds to n=⌊ n_L/2 ⌋. Then the Lie algebra corresponds to 𝔰𝔬(2n) or 𝔰𝔬(2n-1) depending on the parity of |𝒢|. In both cases, this corresponds to the free-fermionic Lie algebra, where 𝔰𝔬(2n) can be seen as generated by all Majorana operators on n modes, i.e., both first and second order operators
    
    𝔤 = span( {γ_iγ_j}_i< j ∈ [2n]∪{γ_i}_i∈ [2n])

    while 𝔰𝔬(2n-1) can be thought as being generated just by second order operators
    
    𝔤=span({γ_iγ_j}_i< j ∈ [2n])
            .

    For the algebraic dependent case A_D, the generating set in terms of Majorana operators becomes 𝒢={γ_1,…,γ_2n, γ_1⋯γ_2n} which corresponds to the set of free-fermionic operations on n modes, together with their parity operator have a Lie algebra of type 𝔰𝔬(2n+1)
    
    𝔤    = span( {γ_i}_i∈ [2n]∪{γ_iγ_j}_i< j ∈ [2n]
         ∪{∏_j≠ iγ_j }_i∈ [2n]∪{∏_j∈ [2n]γ_j })

    which is then the largest connected Pauli Lie algebra on n qubits of polynomial size. In this case, these new terms arising from having the parity operator as a generator lead to dynamics allowing more independent dynamics between both fermionic parity operators, compared to the A_I case. A Lie algebra of this class is discussed in Ref. <cit.>, as arising from some many-body systems like the Kitaev honeycomb model in two dimensions <cit.>.
    
    Moving on to the exponentially big Lie algebras,  the number of qubits n can be given as a function of the number of legs of length 2. Then, for B1 the Lie algebra corresponds to 𝔰𝔭(2^n), which describes the space of all Paulis on n qubits (here n=n_2+1) satisfying the symplectic condition, i.e., given some Q:Q^T=-Q
    
    𝔤=span({ P∈𝒫| P^T Q+QP=0 }) .

    This Lie algebra has recently been motivated, and its properties have been further studied in Ref. <cit.>. The Lie algebra given by B2 instead, corresponds to 𝔰𝔬(2^n), i.e., the span of all purely imaginary Paulis (P^T=-P) in the same Pauli basis on n qubits (here n=n_2+2). Then, for Q^T=Q, we have
    
    𝔤=span({ P∈𝒫| P^TQ+QP=0 }).


    This subalgebra has been previously studied in relation to Haar random circuits and benchmarking tasks, when one needs to restrict to some subspace of the full unitary group, for instance, when working with certain fault-tolerant error correcting codes <cit.>. Moreover, both 𝔰𝔭(2^n) and 𝔰𝔬(2^n) can arise when considering operators in commonly studied 1-dimensional spin models, such as the Kitaev chain or the XY-model under certain interaction fields <cit.>.
    For the last two cases, for B3_D we have 𝔰𝔲(2^n), the full Lie algebra of trace-less Hermitian matrices on n qubits (n=n_2+2) 
     
    𝔤=span(𝒫\{1})  .

    On the other hand, B3_I is equivalent to all Hermitian matrices on n-1 qubits 𝔰𝔲(2^n-1) (n=n_2+3), which can be understood as the embedding of a complex vector space into a real one (ℂ^d≃^2d) <cit.>. The Paulis take the form P↦ iIm(P)⊗1+Re(P)⊗ Y. Then
    
    𝔤=span({ P∈𝒫\Q |  [Q,P]=0, P^TK+KP=0}),
    where  Q^TK+KQ=0 .

    Note that all the definitions given above for the exponentially big Lie algebras are given in a Clifford invariant form, i.e., under Clifford transformations we just have Q↦ Q' preserving the (anti-) symmetric property.
   
    

§ DISCONNECTED PAULI LIE-ALGEBRAS

     
    Hitherto the discussion centered around connected anti-commutation graphs. However, one could also have a generating set with a disconnected anti-commutation graph. In that case, in general, different connected components are Lie algebraically independent and, therefore, their Lie-algebras are direct sums (in terms of their representations) of the Lie algebras of the subgraphs
    
    𝔤∼⊕_i=1^n_d𝔤_𝔦

    where n_d is the number of connected subgraphs and g_i is the Lie algebra of a connected subgraph.
    If there is no algebraic dependence between their generators, 
    then they are Clifford equivalent to Lie algebras acting on the tensor product of their Hilbert spaces.
    In general, generators of different sub Lie-algebras can still have an algebraic dependence between them, creating unitarily inequivalent Lie-algebras.  
    For this to occur, we need to consider the non-trivial center of a Lie algebra
    
    Z(𝔤)={X∈ A(𝔤)| ∀ Y∈𝔤: [X,Y]_L=0}

    where A(𝔤)⊃𝔤 is the algebra generated by the elements of 𝔤. To have an algebraic dependence, two sub Lie algebras need to share their non-trivial center 
    (Z(𝔤_𝔦)∩ Z(𝔤_𝔧)≠span(1)). The non-trivial center of the whole 
    Lie algebra we call Z(𝔤):=
    span({Z(𝔤_𝔦)}_i∈ [d]). The non-trivial center arises from two sources:
    
        
  * The product of two Paulis corresponding to vertices of legs of length 1.
        
  * If the graph has a non control symmetry. This is the case if 𝔤 does have an algebraic dependence but does not, i.e., the graph is of a structure admissible to A_D or B3_D, but no algebraic dependence.
    
     As such, the number of independent generators of the algebra of the center is given by
    
    n_Z(𝔤_i)=    n_c(𝔤_i)+n_s(𝔤_i)

    where n_s∈{0,1} describes the presence of the additional symmetry.
    With this the total number of generators in the non-trivial center is
    
    n_Z(𝔤)≤∑_i=1^n_dn_Z(𝔤_𝔦)n̅_Z(𝔤) ,

    where the difference between both quantities n_Z(𝔤) and n̅_Z(𝔤) comes from algebraic dependencies between disconnected components.
    In that case, it suffices to describe the algebraic relation between a chosen set of generators of the respective centers of Z(𝔤_𝔦) and the generators of Z(𝔤) (ℬ={P_1,… ,P_n_Z}).
     The full set of dependencies can therefore be described by a matrix M ∈{0,1}^n_Z,n̅_Z where Q_j, corresponding to a generator of a non-trivial center of a sub Lie algebra, is given by Q_j:=∏_i∈[n_Z]P_i^M_i,j. 
      For connected anti-commutation graphs, there are only polynomially many (in the number of vertices) Clifford inequivalent Lie algebras, corresponding to the 6 canonical types. 
      If we allow for disconnected graphs, due to the freedom to reuse controls and potential symmetries in the case of A and B3, the number of Clifford inequivalent Pauli Lie algebras becomes exponentially big in general.
    
    

§ DISCUSSION AND OUTLOOK

    
    
    In this work, we have presented a full classification of Pauli Lie algebras, as well as a sketch on how to algorithmically compute them. 
    Having found a universal set of equivalence classes for graphs under conditioned complementation, our results ultimately serve as a rigorous proof showing that every Pauli Lie algebra is, up to direct sums, either free-fermionic or exponentially large, including all types 𝔰𝔭(2^n), 𝔰𝔬(2^n) and 𝔰𝔲(2^n). Additionally, we find an interpretation for direct sums in terms of the physical Hilbert space, as either tensor products, or as actual direct sums coming from gates acting as controlled operations. These considerations regarding the Hilbert space also reveal the existence of 6 Clifford inequivalent Lie algebras. In particular, we show that for 𝔰𝔬(2^n) and 𝔰𝔭(2^n), 
    one can always find a Clifford mapping any generating set to another. In contrast, for the free-fermionic and 𝔰𝔲(2^n), one can have two Clifford inequivalent versions. In the former case they are related through inclusion of the fermionic parity operator, and in the latter, by the embedding of the Lie algebra into a larger Hilbert space.
  
    We believe our results can be taken and interpreted in basically two ways. On the one hand, one can see them as a no-go result for the existence of sub-exponential Pauli Lie algebras other than the mentioned free-fermionic ones. 
    Free fermions been used as toy models and studied in a wide range of fields, including quantum machine learning and variational methods <cit.>, as well as simulation techniques of quantum systems <cit.>, and hence their behavior is well understood. This is relevant as our results limit any further avenues for methods within these fields relying on polynomially sized Pauli Lie algebras, like is the case for 𝔤-sim, geometric QML  and others.
   
    While our result only holds for Pauli generators, it would be interesting to study whether these restrictions hold for more general operators, e.g.,  when considering linear combinations of Paulis. In recent work, for instance, similar graph tools have been used to identify free-fermionic Hamiltonians given by arbitrary sums of Paulis <cit.>. It would be relevant to find out if this could be extended to identify all possible Lie algebras in a similar way as we do here.
   
    On a more positive note, our work can be viewed as a new framework for efficiently computing Lie algebraic criteria for universality and controllability of Pauli gate sets. Our techniques not only provide an answer to this, but also elucidate a clear way to extend some given operator pool to transition from one Lie algebra to another, offering valuable insights into the structure and capabilities of these gate sets. We believe that this can 
    be used to study properties of quantum circuits when doped with some additional set of gates outside their Lie algebra.
    
    To summarize, efficient and generic tools for classifying and characterizing Lie algebras are crucial in quantum information theory. This work then provides a comprehensive classification of Pauli Lie algebras without any additional assumption on the generators.  In doing so, we highlight the limitations that some quantum machine learning and simulation methods might face,  while also offering a new tool to address questions regarding reachability and equivalence of gate sets generated by Pauli operators. We believe this work can help in assessing questions that might have remained elusive to date, as well as motivate new algebraic methods in order to deal with other families of operators.
    
    

§ ACKNOWLEDGMENTS

    We thank Lorenzo Leone, Greg White and Hakop Pashayan
 for useful discussions. 
This work has been supported by the BMBF (FermiQP, MuniQCAtoms, DAQC), the 
Munich Quantum Valley (K-8),
the Quantum Flagship (PasQuans2, Millenion),
the DFG (CRC 183), the Einstein Research Unit, 
Berlin Quantum, and the ERC (DebuQC).
 






§ PRELIMINARIES


 In this section, we will  introduce some basic notions about the spaces and objects we work with. 

 
 
       We define a Lie algebra 𝔤 as a vector space over some field K endowed with a  Lie product ([·,·]_L:𝔤×𝔤→𝔤), under which it is closed, i.e., ∀ a,b∈𝔤: [a,b]_L∈𝔤, such that the product satisfies
        
            
  * Bilinearity.
            
  * Anti-symmetry: ab_L=-ba_L.
            
  * Jacobi identity: abc_L_L+bca_L_L+cab_L_L=0.
        
        


In our case we will focus on the space of N × N complex matrices  Mat_N(ℂ), where we will take the Lie product to be given by the complex commutator of two matrices, i.e., AB_L=iAB=i(A· B-B· A). This Lie algebra is denoted by 𝔤𝔩(N,ℂ). If one would remove the last two requirements from the previous definition, and simply impose bilinearity on the product, that would define an algebra instead. In many cases, we will be interested in looking at products of elements in 𝔤𝔩(N,ℂ), which might not be in the Lie algebra, but in their algebra.

In this work, we will have a particular focus on the 
Pauli group, which is a subgroup of Mat_2^n(ℂ), defined as

    𝒫_n = {. e^iϕπ/2σ_i_1⊗…⊗σ_i_n| ϕ∈{0,1,2,3},i_k∈{0,1,2,3}}

for 

    σ_0=1=[ 1 0; 0 1 ],
        σ_1=X=[ 0 1; 1 0 ],
        σ_2=Y=[  0 -i;  i  0 ],
        σ_3=Z=[  1  0;  0 -1 ].

Another particularly interesting group that will be discussed over this work is, in fact, the Clifford group, which is the normalizer of 𝒫_n
given by

    𝒞_n = { C∈ U(2^n)|CPC^†∈𝒫_n },

i.e., the set of all unitary operations on n qubits that 
leaves the Pauli group invariant. It will be interesting because given two elements of 𝒫_n, their Lie product will be preserved under Clifford operations, and thus we will be able to transform between sets without changing the underlying Lie algebra.

There are many properties of 𝒫_n which make Paulis interesting, one being that they form a basis for the space of all operators on n qubits. In particular, if one restricts to coefficients over ℝ, they span the set of all Hermitian operators. When one excludes the σ_0^⊗ n term, this gives a basis for all (Hermitian) traceless operators. 
Pauli commutation and anti-commutation rules are particularly interesting, given that for two Paulis P_1,P_2∈𝒫_n, P_1P_2=± P_2P_1 and hence, they either commute or anti-commute. These properties will be instrumental for this work. 

It is easy to check that the hermicity and traceless properties are preserved under the Lie product and hence, these two conditions define an actual sub-algebra of 𝔤𝔩(N,ℂ), namely

    𝔰𝔲(N) ≡{P ∈Mat_N(ℂ) | P^† = P,  Tr(P) = 0}

i.e., the Lie algebra spanned by all non-identity Paulis over real coefficients. If one would add 1 to this set then the Lie algebra is called 𝔲(N). In our case, we will be interested in 𝔰𝔲(N) and sub-algebras thereof. One such sub-algebra can be constructed by restricting to anti-symmetric matrices, which for Hermitian matrices (of which Pauli matrices is a subset of) is equivalent to purely imaginary ones

    𝔰𝔬(N) ≡{P∈𝔰𝔲(N)|P^T=-P }⊂𝔰𝔲(N).

On the other hand, for even N, one could restrict to the matrices in 𝔰𝔲(N) satisfying the equation

    P^TΩ=-Ω P

for Ω some non-degenerate, anti-symmetric bilinear form. Typically, however, Ω is defined as

    Ω≡[    0  1_N; -1_N    0 ] = iY_0⊗1_N

which then gives a canonical definition of the symplectic Lie algebra

    𝔰𝔭( N/2) ≡{ P∈𝔰𝔲(N)|P^TY_0=-Y_0 P }⊂𝔰𝔲(N).

One particular construction of a Lie algebra which we will be interested in is when this is defined with respect to some generating set.

    We say that some set of Paulis 𝒢={P_1,…,P_n_G} is a generating set of some Lie algebra 𝔤 if they span all elements in 𝔤 through Lie algebraic operations. Then we write ⟨𝒢⟩_,[·,·]=𝔤. A generating set is minimal, if removing any P_i would change the size of the Lie algebra. 


When some Lie algebra 𝔤 is defined as being generated by some set 𝒢, this is sometimes referred as the Dynamical Lie algebra of 𝒢.

One of the main questions of interest regarding Lie algebras is their classification. When a Lie algebra 𝔤 is defined as being generated by a set 𝒢 one would also be interested in methods that efficiently characterize these spaces. When dealing with Paulis, and hence sub-algebras of 𝔰𝔲(2^n), certain results exist. In particular, when one considers sub-algebras of 𝔲(2^n) (i.e., including σ_0^⊗ n), a classification in terms of direct sums of 𝔰𝔲,𝔰𝔭,𝔰𝔬, plus some Abelian component has been developed. These sub-algebras are usually called reductive, and in particular semisimple when the Abelian component is null. While this is a useful result that can help characterize some of these Lie algebras, it is in general practically unfeasible to find such a decomposition, given a set of generators, and thus the task remains difficult in spite of these sparse results.

In this work, we will sometimes refer to the center of a Lie algebra, Z(𝔤) as the set of operators commuting with every element in 𝔤. If there is some Abelian component in the decomposition of 𝔤 it will be a subset of it. If we think of 𝔤 as a sub-algebra of some bigger space, we can also have some elements z∈ Z(𝔤) such that z∉𝔤, but potentially in the algebra.
We refer to Ref. <cit.> for a more in depth discussion about these objects and their properties. 

Finally, a particular class of interesting systems that will repeatedly appear in this work are free-fermionic systems. These are many-body systems consisting of a set of non-interacting particles. In our context, they are particularly interesting for two reasons. Firstly, they can be described by a set of Hermitian operators with similar commutation structure to Pauli matrices, namely Majorana fermion modes {γ_i}_i∈[2n], which satisfy


    {γ_i,γ_j}=2δ_i,j1.

As such, there are several ways one can map these operators to Paulis, one of the most common ways being through the Jordan-Wigner transformation, which maps

    γ_2j-1=Π_k=1^j-1Z_k⊗ X_j, γ_2j=Π_k=1^j-1Z_k⊗ Y_j.

Secondly, they are known to be solvable by classical methods, and, in fact, are one of the only known systems to span a polynomially large Lie algebra. To see this, it can be shown that such free-fermionic Hamiltonians can be written as quadratic in the Majorana modes

    H=2i∑_i,j∈[2n]h_i,jγ_iγ_j iγ· h·γ^T

with γ a vector of the Majorana operators, and h a real anti-symmetric coefficient matrix. Due to the canonical anti-commutation relations,

    γ· h·γ^Tγ_i = -4(h·γ^T)_i

for some individual mode γ_i. Hence, these evolve under H as

    e^iHtγ_ie^-iHt=∑_j (e^4ht)_i,jγ_j.

Since h is anti-symmetric and real, e^4ht∈ SO(2n,ℝ), and  there ∃ W∈ SO(2n,ℝ) s.t.

    W^T· h· W  = ⊗_j=1^n[    0  λ_j; -λ_j    0 ]

with real {λ_j}. The special orthogonal group
SO(2n,ℝ)
preserves the parity of fermion number.
W itself can be represented as the exponential of a quadratic Majorana fermion operator as well, so that W e^4ω. Finally, the Hamiltonian H can be solved by exact diagonalization as 

    e^-γ· w·γ^THe^γ· w·γ^T=2∑_j=1^nλ_jγ_2j-1γ_2j=2∑_j=1^nλ_j Z_j.






§ GRAPH EQUIVALENCES 


In this section,  we prove <Ref> and introduce some of the tools required for it. Since our goal is to classify Pauli Lie algebras in all generality, we cannot rely on constructive arguments like in Ref. 
<cit.>, where locality constraints make explicit computation of 𝔤 feasible. Instead, we map the problem to a graph reduction problem and find all equivalence classes under contraction operations as defined in the main text. Recall that contractions correspond to performing a shuffle of the set of generators on the Lie algebra level, which on the graph will act as a conditioned local complementation. In order to facilitate discussion of sequences of contractions on the graph we will introduce the following tool verbatim to Ref. <cit.>.


 Given a graph Γ=(𝒱,ℰ), and a vertex V∈𝒱 onto which we wish to perform some contractions, we define the lightning χ=Γ(V) with respect to V as the induced sub-graph Γ∖{V} with labeled vertices: lit (unlit) if on Γ they are connected to V (if on Γ they are not connected to V). We then say we toggle a lit vertex ω whenever we perform a contraction of ω onto V. Since this operation on Γ complements edges (n_ω,V), for every n_ω in the neighbourhood of ω, in terms of the lightning, it changes the lit / unlit state of all n_ω. 


Graphically we will represent lit vertices as white vertices and unlit ones as black vertices.
We show in an example of a lightning in <Ref>.


With this, our approach will be tightly related to the lit-only σ-game, where given two lightnings on a graph, the goal is to determine equivalence through some adequate sequence of toggles <cit.>. Big part of our result will then be based on finding equivalent lightnings for arbitrary initial ones on our set of canonical graphs.

From now on, we will say a vertex V is central whenever deg(V)≥ 3. Typically, in our graphs we will only have one such vertex which we will call O. Then, we will refer to every connected component of Γ∖{O} as a leg of O. Given a leg L, we will use L_i for the vertex in L at distance i from O.

With this, we can now move on to proving our statement. To that end, we will introduce first two lemmas that will be instrumental.


    
    Given a lightning on a star graph Γ (i.e., only one central vertex) and at least one leg of length 1, where the central vertex O is lit. Take two legs L and M of length l_L∈{2,3,4} and l_M=2 respectively. Then, as long as L_2 and M_2 are initially unlit, there is a sequence of toggles that inverts the state of L_1,M_1 and L_3 if exists, while leaving the rest of the lightning invariant.



    Assume first that both L_1 and M_1 are in the same intial state, say for now that both are unlit. Then we just need to toggle 
    
    O,M_1,L_1,O,M_2→ 1,L_2→ 1.

    If they are both initially lit then toggling according to the same sequence in reverse order unlights them both. Besides this, all other vertices remain unchanged, except for L_3 if it exists since we toggled L_2 only once, and hence L_3 is flipped.

    On the other hand, wlog we can assume L_1 is lit and M_1 is unlit. Then, for ω a leg of length 1, which for now we assume is lit, we toggle according to

    
    L_1,L_2,ω,O,L_1,M_1,O,M_2,M_1,ω.

    After this, L_1 is unlit, M_1 is lit and since we toggled O an even number of times, the rest of the lightning remains the same, except again for L_3 potentially. For the previous sequence, we assume that ω is initially lit. If it was not, we just need to toggle O once before, apply the same sequence, and toggle O once more at the end of it.



With this we can prove the following lemma about equivalence between graphs with one single center and our canonical graphs.


    
    

    Given a star graph Γ (i.e., only one central vertex) with at least one leg of length 1, and any number of legs of arbitrary length, they can always be transformed by contractions into one of our canonical graphs.





    Let us call O the central vertex, and ω one leg of length 1 (we ask for Γ to have at least one such leg). First of all, if the central vertex O has N_L legs, of which at least N_L-1 are of length 1, then Γ is of type A.

    Otherwise, there are at least two legs of length l≥ 2. In that scenario, 
    if there is a leg L of length l>4 we can transform it into a leg of length 4 and one of length l-4. To see this, take a lightning χ=Γ(L_5). Call M the other leg of length bigger than 1, and P any other neighbour of O (which must exists since O has degree at least 3). Then toggle 
    
    L_4,L_3,L_2,L_1,O,M_1,P,O,L_1,M_2,M_1,O,L_2,L_1,L_3,L_2,L_4,L_3,P,O,L_1,L_2,M_1,M_2,O,L_1,M_1,O.

    This leaves only the O vertex lit, i.e., we removed the edge between L_4 and L_5, and created an edge between O and L_5.
    Repeating this we reduce the maximum length of a leg in the graph to 4.

    Moreover, if Γ has several legs of length 3, it can always be transformed into a graph with only one leg of length 3. Let L,M be two such legs. Then take a lightning χ=Γ(L_3) and toggle according to
    
    L_2,L_1,O,M_1,M_2,M_3,ω

    applying <Ref> to L and M lights L_1,M_1 and unlights L_3. Toggling O once more unlights all vertices but itself. Hence, we managed to transform L into a leg of length 2 and a leg of length 1. This procedure can be repeated until only one leg of length 3 is left.

    Finally, we will prove that legs of length 4 can be transformed into legs of length 2 under certain conditions. On the one hand, if our graph has no leg of length 3, then let L and M be two of the legs of length 4. Then take a lightning χ=Γ(L_3) and toggle according to 
    
    L_2,L_1,O,M_1,M_2,M_3,M_4,ω,O,L_1,L_2,M_1,O,ω,L_1,M_4,M_3,O,M_1,M_2.

    This transformation results in M_2 having deg(M_2)=3 and two legs of length 2, which we call P and Q. Now, through an adequate sequence of contractions we will remove these legs from M_2 and append them to O. In particular, we contract
    
    M_2    ↦M_2M_1, 
    
                O    ↦OM_2.

    In particular, this creates an edge between O and P_1, and O and Q_1. We can now take a lightning χ=Γ(M_2), where O,P_1 and Q_1 are all lit. Applying <Ref>, onto P and Q, unlights them without changing the rest of the lightning. This just disconnected P and Q from M_2, and connects them to O. 
    At the end of these transformations, L and M have both been broken down into two legs of length 2 each. This transformation can be repeated for every pair of legs of length 4. After this transformation we will always have reduced our Γ to a graph either in B1 or B2, depending on the parity of legs of length 4.
    On the other hand, if Γ has one leg of length 3, then let M be the leg of length 3 and L a leg of length 4. We can take a lightning χ=Γ(L_3) and toggle according to 
    
    L_2,L_1,O,M_1,M_2,M_3,ω,O,L_1,L_2,M_1,O,ω,L_1,M_3,ω,M_3,M_2,M_1,O.

    This breaks down L into two legs of length 2. Repeating this for every leg of length 4, results in a graph B3.
 

By making use of these two lemmas, we will be able to first prove that any lightning on one of our canonical graphs, is equivalent to a lightning where only one vertex is lit. From this it will follow that there will always be a sequence of contractions mapping back to one of the canonical graphs.



    
    Given a lightning χ on some graph Γ of a canonical type, it is always equivalent to some lightning where only one vertex is lit.



    We will first make a comment regarding multiple legs of length 1.
    

  
*Legs of length 1 in different initial lit states.
    First, assume that not all legs of length 1 have the same lit configuration in χ. Let P be one of the lit legs, and Q an unlit one. Then, given any generator g∈𝒢 other than P or Q we can find a sequence of contractions that maps g↦ g'=PQ· g and leaves the graph invariant (this is clearly the case since PQ commutes with every element in the Lie algebra). If g=O then we can contract P and Q onto O since both anti-commute with it. For g=L_j any other vertex on a leg L, we contract onto L_j the following sequence of vertices
    
    L_j-1,…,L_1,O,P,Q,O,L_1,…,L_j-1.

    It is easy to check that all of these contractions are allowed, and transform g so that the final generator is g· PQ. Now, since P was lit, the transformation g↦ g· PQ changes the lit state of g. Then, we can perform this operation freely to every lit generator g≠ P, so that after that, all vertices are unlit but P. 
    
    

  
*Legs of length 1 in the same initial lit state. If all legs of length 1 are in the same initial state, since this can only change by toggling O, which flips all legs, they will always be in the same state. Then, for graphs in class A let us establish that legs of length 1 are on the left end of the graph, and the central vertex O corresponds to the second leftmost vertex. Then we will proceed by always toggling the second leftmost vertex in Γ. Repeating this procedure we always end up moving the leftmost lit vertex one position to the right. Thus, ultimately all lit vertices concentrate in a smaller region of the path and hence eventually cancel all out but one. In case only the legs of length 1 are lit, we toggle one of them, in order to light O, and then proceed in the same way. For all other cases, i.e., graphs of type B1,B2,B3, we will show that we can easily take care of at least all but one legs of length 2, and map the discussion to a particular case of the above, namely a small path graph. In this scenario, we can assume wlog that the center of the graph O and all legs of length 1 are lit. If not, we simply need to take some other leg L with at least one lit vertex, and toggle the innermost lit vertex until O is lit. Toggling O then lights the legs of length 1. 
    
    Moreover, for any leg L of length 2, with at least one lit vertex, we can always assume only L_1 is lit. If both were lit, we can toggle L_1. This unlights L_2 and O, which we can light back by toggling ω. If only L_2 was lit, we can toggle it and we are in the previous case. For the leg of length 3 or 4 in B3 and B2 respectively, the same logic follows. From this we can assume that the second vertex of every leg is always unlit, which in turn allows us to apply <Ref> freely between any pair of legs. Hence, let us call L either the leg of length 4 in B2, the leg of length 3 in B3 or some randomly chosen leg of length 2 in B1. For any unlit leg M of length 2, applying <Ref> lights up M_1, while inverting the state of L_1 and possibly L_3. Repeating this for every unlit leg of length 2 and toggling O, unlights all neighbours of O except L_1 potentially. If after this step only O is lit we are done. Otherwise, we just need to solve a small instance of the previous case for graphs in A, where again, we will toggle the second innermost lit vertex in L repetitively. By doing so, we always end up with just one lit vertex in L.


With this, we can now prove <Ref>.

thm:types[Canonical graphs]
       
    For any given connected graph Γ, there exists a sequence of contractions which result in one of three graphs:
    
        
  * A line graph with with n_L vertices as well as n_c∈ℕ_0 single vertices connected to the 
        second to last vertex.
        
  * A star graph with legs of length of at most 4 and where n_c,n_2∈ℕ_+ appear at least once but can be arbitrarily often and additionally there can be 
        
            
  * no legs of length 3 or 4,
            
  * one leg of length 4 and no leg of length 3,
            
  * one leg of length 3 and no leg of length 4.
        
    
    

    
        Given any initial graph Γ we will prove such a sequence exists by induction on its vertices. It is clear that if we pick any vertex of Γ at random, and then add some vertex in its neighbourhood, this trivially is a graph in A. Assume now that after having added N vertices one has a graph Γ_N of the form of one of our canonical types. We show that Γ_N+1=Γ_N∪{V}, for some V∈𝒩(Γ_N), can always be mapped to a canonical graph as well. Take a lightning χ=Γ_N+1(V). By <Ref> this is always equivalent to a lightning where only one vertex is lit, such that V shares just one edge with Γ_N. For the case where several legs of length 1 where initially in different lit configurations in χ, the new vertex creates a new leg of length 2. By <Ref> this can then be transformed into one of the canonical graphs. If all legs are in the same state, for graphs in A, if O or the outermost vertex of the long leg L are lit, then the graph is still in class A. Otherwise, if V is connected to some vertex L_j∈ L other than the one at its end, then deg(L_j)=3. In this case, we can take a lightning χ'=Γ_N+1(L_j+1) and toggle according to 
        
    L_j,L_j-1,…,L_1,O,ω,V,L_j,…,L_1,O

        with ω one of the legs of length 1 of O. This removes the edge (L_j,L_j+1) and creates the edge (O,L_j+1). The resulting graph is a star graph with just one central vertex, which by <Ref> can be transformed into one of our canonical graphs. For all other cases, namely for Γ_N of type B1,B2,B3, whenever V is connected to O, Γ_N+1 remains in B1,B2,B3 respectively. Otherwise, V is connected to some vertex in a leg of length 2, the leg of length 4 or the leg of length 3, respectively. If this is actually the end vertex of these legs, either Γ_N+1 is already of our canonical types, or by <Ref> it can be converted to it. Instead, if V connects to some other vertex L_j, such that deg(L_j)=3 in Γ_N+1, we can reproduce the same argument as before. After taking a lightning χ'=Γ_N+1(L_j+1) and repeating the sequence in <Ref> we transform Γ_N+1 back into a star graph, which again by <Ref> can be transformed into one of our canonical graphs.
    




§ ELEMENTS OF GRAPH LIE-ALGEBRA

In order to describe Paulis within the Lie algebra spanned by some generators we introduce the concept of colouring. 
In general, we are interested in the Paulis in the algebra. As such, we can associate to a particular Pauli

    P_C=∏_P_i∈𝒢 P_i^C_i

a vertex colouring C∈{0,1}^n_G of the graph, such that a vertex v_i is coloured if C_i=1. We say that P_C is in the Lie algebra if C is a valid colouring. The map C↦ P_C does not need to be injective, since the P_i can have algebraic relations between them.

When we consider elements of the Lie-algebra, we require that they can be written as a linear combination of nested commutators of the 
generators

    P=[⋯ [[P_i_1,P_i_2],P_i_3]⋯ P_i_n_L] .

We do not need to consider commutators of commutators since by the Jacobi identity

    [[A,B],[C,D]]=[[[A,B],C],D]]-[[[A,B],D],C]].

In particular, for Paulis, in order for the term in the lhs to be non-zero, one of the two terms in rhs has to vanish, due to the rule that for 3 anti-commuting Paulis, the commutator of 2 of them commutes with the third. This then means that there is always a way to rewrite some non-zero nested commutator into the form of <Ref>. As such, we can simply consider how do graph colourings change when taking commutators with the generators. In order for the commutator to be non-trivial, we need that the Pauli anti-commutes with the generator. As such, the rule is that a colour of a vertex can be flipped, if an odd number of its neighbouring vertices are coloured.

Having found the previous fundamental graphs, now we can try to exploit their cycle-less structure to find a set of rules to determine which colourings are valid, i.e., which generators give non-zero commutators.


    
    For any graph in its canonical form, we have that
    
        
  * for a path, any colouring that has one connected component and potentially an additional even numbers of legs of length 1 that are coloured corresponds to valid colouring,
        
  * for a star, any colouring with an odd number of connected components is a valid colouring, except for the case with a single leg of length 3 (B3), when the first and third vertex of the long leg and an odd number of legs of length 1 are coloured.
    
    Here, connected component refers to the connected component of the sub-graph induced by only coloured vertices.




We first show that every colouring that can be generated by the allowed transformations needs to obey these conditions.
We use that the graphs do not have loops. Therefore, connectivity is locally preserved, meaning that two coloured vertices in the neighbourhood of an uncoloured vertex always belong to two different connected components.
If a vertex has only one coloured neighbour, then flipping it only adds/removes the vertex from the respective connected component, but never changes the number of components. In general flipping a vertex with 2k+1 coloured neighbours then merges/disconnects 2k+1 components. As such, the parity of the number of connected components can never change (since flipping a vertex with an even number of coloured neighbours is not allowed). Since by <Ref> we always start our colouring from a single vertex, the number of connected components will always be odd. Additionally, for the canonical types, this means that the number of connected components only changes by flipping the central vertex. 
In particular, this implies that for graphs in class A we cannot have different connected components along the long leg. Given <Ref> one can check, that in order to create several connected components on the long leg, on should start from a colouring with one single connected component spanning over some vertices of the long leg, the center, and potentially some even number of legs of length 1. Uncolouring O, would then create an odd number of connected components, one of which on the long leg. Given that colour flips on the long leg, will not create new connected components by themselves, the only way to create more connected components, would be to uncolour the innermost vertex of this leg, and colour back the center. This however, cannot be done, since the center would then just have an even number of coloured neighbours, preventing us from colouring it back.

Finally, for the exception in B3, where an odd number of legs of length 1 and alternating vertices of the leg of length 3 are coloured, we note, that this corresponds to a configuration where no additional vertices are allowed to be flipped. As such, this colouring cannot be constructed from some sequence of vertex colourings.

To show that every such colouring can also be generated, we will give an explicit algorithm. 

    
  * Start with a coloured central vertex.
    
  * Colour the first vertex of all legs which in the final configuration have at least one coloured vertex. 
    
  * Potentially colour a leg of length 1 ω, then uncolour the center.
    
  * Move each connected component into the correct position, by colouring vertices adjacent to it. 
    
  * If there is a second connected component in the long leg L (3 or 4), first move the outer into position. Then, take some leg T of length 2 with some coloured vertex. Depending on the parity of the number of neighbours of O that are coloured, flip the colours in T. Colour the central vertex O and L_1. Uncolour the leg of length 1 ω and uncolour O again.
    
    
  * If desired, colour the center back on and flip the leg of length 1 again.
   

This algorithm will prepare any valid graph colouring as long as there is a connected component on a leg of length 2 or no two connected components on the same leg. The two remaining cases are when the long legs L have two connected components and only legs of length 1 are coloured. For the case  B3 (one leg of length 3), these are not able to be prepared as argued above. 
For B2 (one leg of length 4), however, we can prepare them. In order to show this, we just need to argue that a colouring where the central vertex and the second and fourth vertex of the long leg are coloured, is a valid colouring. 
Starting from the coloured center we achieve this by flipping the colours according to

    L_1,L_2,L_3,L_4 ,T_1,ω,O, T_2,T_1, L_1,L_2,L_3, O,ω ,L_1,O,L_2,L_1 T_1,T_2,O,T_1

where L refers to the long leg, T to a leg of length 2 and ω to a leg of length 1. After this, we can colour the rest of legs of length 1, which is an odd number, and uncolour O. Finally, on L, we can trivially go from this colouring to any other with two connected components.





§ ALGEBRAIC DEPENDENT GENERATORS



Like we discussed in the main text, we are not only interested in classifying Pauli Lie algebras, but also in determining which can be mapped to each-other through Clifford operations. To do so, we introduced <Ref> to characterize algebraic dependencies between generators, which ultimately will determine whether or not two sets of Paulis are Clifford equivalent. We prove this theorem here.


thm:clif_algdeb[Limits to algebraic dependencies]
        Every minimal generator Pauli Lie algebra has at most one algebraic dependence. Algebraic dependencies can only occur for graphs of type A and  B3. The two possibilities are shown in <Ref> (c)
        As such, there is a total of 6 Clifford inequivalent families of Pauli Lie algebras.



    It is enough to prove this just for graphs in the types of <Ref>, as contractions preserve algebraic and Lie algebraic dependencies. Then, we will show that algebraic dependencies on these graphs need to involve a colouring where the end vertices of some legs of odd length are colored, have alternating structure, and can only involve one leg of length 1. Overall, this will just leave the two cases mentioned in the statement. 
    
    Assume we have an algebraically dependent subset of 𝒢, that is {P_i}_i=1^m such that P_1… P_m=1. Since 1 commutes with everything, the colouring of {P_i}_i=1^m on the graph, must share an even number of edges with every vertex in it.
    Then, for some leg L⊆Γ (which for class A might be the path itself), if some element in L is coloured, i.e.,  {P_i}_i=1^m∩ L≠{∅}, then the end vertex of L must be coloured. Otherwise, we could find some vertex in L sharing just one edge with the colouring. From this, it is immediate to see that the colouring must have an alternating structure (coloured, then uncoloured). If the end vertex of a leg is coloured and its neighbour as well, then the end vertex itself would share just one edge with the coloured set. The alternating structure then follows from carrying on with this argument. 
    Finally, the center can never be coloured, as otherwise the vertex of a leg of length 1 would only have one coloured neighbour.
    It follows that an algebraic relation cannot have a colouring involving legs of even length, as there is no alternating colouring that colours the outer vertex, but not the center.
    
    If we have a colouring involving more than one leg of length 1, we can show that this either violates the condition that the generators are Lie-algebraically independent, or that the graph can be mapped to a form where the algebraic relation involves at most one leg of length one. 
    Let {L_i}_i=1… 2k+1 be coloured legs of length 1. We can contract L_2,…, L_2k+1 onto O, yielding O'=OL_2… L_2k+1. Contracting now O' onto L_1, results in L_1'=L_1L_2… L_2k+1O. While this modifies the connectivity of the graph, if we now contract again all L_2,…, L_2k+1 onto O', and then again the new O”=O onto L_1', this undoes the change, and leaves all vertices the same, except for L_1 which is mapped to L_1”=L_1,… , L_2k+1. After this, the only coloured leg of length 1 is L_1”.
    Since the total number of coloured legs need to be even, this transformation allow us to get to two scenarios. The first is when only two legs of length one are coloured, in which case P_1P_2=1 P_1=P_2, which contradicts the Lie algebraic independence. As such, only the second case is relevant, when one leg of length 1 and a longer leg of odd length are coloured. This is only possible in case A, when the length of the long leg is odd and B3.
    One last remark, is that there cannot be several such lightnings involving different legs of length 1, as this would imply an algebraic dependence between the legs of length 1, which we already showed is not allowed.
   





§ CONTROLLED LIE-ALGEBRAS AND REMOVING LEGS OF LENGTH 1



In <Ref> in the main text, we show that in the presence of several legs of length 1, 2^n_c direct sum terms appear in the description of the Lie algebra. In this appendix, we show why this happens and what these several legs of length 1 on the graph represent in physical terms.


    Given a graph Γ with two legs of length 1 (P_1,P_2), its Lie algebra splits into a direct sum of two smaller Lie algebras, in particular those with anti-commutation graph Γ∖{P_2}. This also corresponds to a physical direct sum over the qubits upon which Z_i P_1P_2 acts.



    It first should be noted that the product Z=P_1P_2 is a symmetry in the Lie algebra, since it shares an even number of edges with every generator. Now, we can use the fact that in order to find the minimal number of qubits for a set of algebraically independent generators, we count the number of disconnected points and pairs in the algebra graph.
    Namely, it is easy to see that if we allow ourselves to perform contractions between disconnected points (which one can do if interested just in the algebra, not the Lie algebra) any given graph Γ can be transformed into the disjoint union of some pairs of connected vertices and isolated vertices. Then if all generators are independent, we necessarily need to add a qubit for each such connected component, as connected pairs can be mapped to (X_i,Z_i) on some qubit i, and single points to (Z_j) on some different register j. Hence, since there cannot be an algebraic dependence between legs of length 1 (shown in <Ref>), and P_1P_2 is a disconnected point, this means that this must necessarily involve adding a new qubit register. This in turn means that we can always find a Clifford operation that maps Z=P_1P_2↦ Z_N, where N is the new register. 

    Now, given some arbitrary element P∈𝔤(Γ), we can show that Z· P∈𝔤(Γ). Using the terms that we introduced in previous sections, we need to show that given a colouring on the graph such that the resulting P is in the Lie algebra, then adding P_1,P_2 to this colouring is also in 𝔤. However, from <Ref> this follows directly, since flipping the colouring of P_1,P_2 does not change the parity of the number of connected components, and thus also is a valid colouring. This then splits the Lie algebra in
    
    {P⊗1_N}⊕{P⊗ Z_N}=𝔤.

    Now, since wlog the only term containing Z_N is P_2, for every element P⊗1_N∈𝔤, its colouring must be in Γ∖{P_2}. On the other hand, for any element P⊗ Z_N∈𝔤, its colouring must be in Γ∖{P_1}, with 𝔤(Γ∖{P_1})≅𝔤(Γ∖{P_2}) because the graphs are the same. 


Thus, the two takeaways from this are that, one the one hand, we can characterize graphs with just one leg of length 1, as all other Lie algebras will be direct sums of these and, on the other hand, that these direct sums, are effectively also a direct sum in Hilbert space, i.e., they entail a whole new register which is acting as a control on the dynamics. By repeated application of <ref>, it follows that the Lie algebra decomposes into 2^n_c many direct sums in Hilbert space, which are Clifford equivalent to 
    Z_1^i_1⊗⋯⊗ Z_n_c^i_n_c⊗ P_𝔤

for i⃗∈{0,1}^n_c.
Finally, since a Lie algebra is a vector space an equally valid basis is

    |i⃗⟩⟨⊗|P_𝔤

making explicit the effect of registers 1 to n_c.




§ LIE ALGEBRA CLASSIFICATION


In this section, we will constructively prove the rest of <Ref>. As the key ingredient to do so, will be introducing a canonical set of Pauli labels for each graph family. Then, since the Lie algebra type just depends on the graph, we will just need to characterize these canonical sets in order to have a full classification. We show in <Ref> all 6 Clifford inequivalent graphs, together with 
their corresponding canonical labels.



Let us start with graphs of type A, both with algebraic dependence and without. In this case, we simply need to choose a set of closest neighbour interacting Paulis, 

    S_A={Z_0,X_0,Z_0X_1,Z_1,X_1Z_2,X_2,… P_m}∪{Z_0Z_C^i}_i= 1,…,n_C,

together with a set of labels for additional control legs,
where P_m depends on the parity of n_L and whether we are in the (A_D) or (A_I) instance. For n_L=2k all terms must be algebraically independent, so that P_m∈{Z_n_L/2-1X_n_L/2,X_n_L/2-1Z_n_L/2} depending on the parity of n_L/2. For n_L=2k+1>3 we can either have algebraic dependence or not. For (A_I), P_m∈{Z_(n_L-1)/2-1X_(n_L-1)/2,X_(n_L-1)/2-1Z_(n_L-1)/2}, whereas in (A_D), P_m∈{Z_(n_L-1)/2,X_(n_L-1)/2}. With this, we can now prove that indeed this corresponds to the free-fermionic Lie algebra. We can limit to characterizing the Paulis lying on the path graph without legs of length 1, as controls can be removed at this point. From these, it is easy to define a set of n_L majorana operators. In particular, wlog we use Paulis in (A_I) to do so, as 
this will not change the Lie algebra. Thus

    γ_0    = Z_0,
    γ_1    = Y_0,
    γ_2    = X_0X_1,
    …
    γ_n_L-1   = Π_i=0^m 
            P_i,

    which satisfy
    
    γ_iγ_j = ± 2iγ_iγ_j,∀ i,j

    and thus
    
    γ_kγ_iγ_j = 0,∀ k≠ i,j.

    Hence, a basis for the Lie algebra is given by {γ_i}_0≤ i≤ n_L-1∪{γ_iγ_j}_0≤ i<j≤ n_L-1, where quadratic terms satisfy 
    
    γ_iγ_jγ_rγ_s = ± 2i(δ_j,rγ_iγ_s-δ_s,iγ_rγ_j).

    The size of this basis then gives us that 
    (𝔤)=n_L(n_L+1)/2,

    which coincides with the dimension of 𝔰𝔬(n_L+1). Finally, taking a basis of anti-symmetric matrices of size (n_L+1)×(n_L+1), it is easy to check that the structure constants of 𝔤 and 𝔰𝔬(n_L+1) are also the same, thus proving equality.
    The difference between (A_I) and (A_D) then comes from the fact that for the latter, the last Majorana 
    operator γ_n_L-1 is equal to the global parity operator.

    For the rest of star-graph families, before giving the labeling, we will first introduce the following lemma regarding properties of certain colourings on the graph. With this lemma, proving the Lie algebra type will follow almost directly.


    
    If all generators P∈𝒢 of a star-type Pauli Lie algebra satisfy 
    
    P^T Q+QP=0

    for some Pauli Q, then every Pauli K=∏_i∈ s⊂𝒱, |s|≥ 1 P_i
    it holds that
    
    the colouring corresponding to  K  has an odd number of connected components⇔ K^TQ+QK=0.






    For the set of S={∏_i∈ s P_i| s⊂𝒱, s≠∅} of all Paulis generated by some colouring,
    we define the two subsets 
    
    S_i ={P∈ S | P^TQ+(-1)^i QP=0},   for  i ∈{0,1}.

    In general, we have for L=V· W, where V∈ S_i_V and W∈ S_i_W
    
    L^TQ± QL   =(VW)^TQ± QVW=W^TV^T Q± QVW
            =-(-1)^i_V W^TQV± QVW
       =(-1)^2(-1)^i_V+i_WQWV ± QVW=Q((-1)^i_V+i_WWV ± VW).

    As a consequence, we have that if V,W∈ S_0 and {V,W}=0, then L=VW∈ S_0.
    This means that if we have a connected component C_i⊂𝒱, it follows that P_C_i∏_j∈ C_i P_j∈ S_0. This is because we can generate this by a nested sequence of generators. Since the graph has no loop, we can start at any element, and then take a commutator with a Pauli from C_i that anti-commutes, until all Paulis were used exactly once. 
    Now we consider the disconnected components. We have from <Ref> that 
    P_C_1P_C_2∈ S_1 since [P_C_i,P_C_j]=0. By iterative application we have that 
    
    K=P_C_1… P_C_n_cc∈ S_n_cc+1 2.

    This means that the Pauli K satisfies the condition if the number of connected components n_cc is odd, and satisfies K^TQ- QK=0 if n_cc is even. This gives the if and only if relationship that is desired by the lemma.



Having stated this, we can start discussing graphs B1. In this case, there is just one Clifford inequivalent class and we choose the labeling to be given by

    S_B1={Y_0,Z_0}∪{Z_0Z_i,Y_i}_i=1,… , n_2∪{Z_0Z_C^i}_i=1,…,n_C

where these sets label the central vertex and the first leg of length 1, all legs of length 2, and controls, respectively. Note that the central Pauli Y_0, is chosen so that Q-Ω as in the canonical definition in <Ref>  in the definition of 𝔰𝔭. Moreover, every vertex adjacent to the center is labeled by a symmetric Pauli Z_0,Z_0Z_i, whereas non-adjacent vertices are labeled with an anti-symmetric Pauli Y_i. This precisely matches the defining condition for 𝔰𝔭, i.e.,

    P^TY_0=-Y_0P.

Hence, since all our generators satisfy this condition, which is preserved under Lie product, all elements in our Lie algebra will, i.e.

    span(⟨ S_B1⟩)⊆⊕_i=1^2^n_c𝔰𝔭(2^n_2).

For us to have equality as stated by <Ref>, one would have to check that any element in ⊕_i=1^2^n_c𝔰𝔭(2^n_2) can be generated by our Paulis. This, however, follows from <Ref>. Given some K∈𝔰𝔭(2^n_2), this can be written as a product of our generators, since these form a basis for the algebra of symplectic matrices of size (n_2+1)×(n_2+1). This follows from the same arguments as for finding the minimal number of qubits. Since 2n independent Paulis are required to span the full algebra on n qubits, in order to span 𝔰𝔭 one also needs at most 2n symplectic generators, which is what we have. Thus, there is some colouring on the graph corresponding to K. Since K satisfies

    K^TQ+QK=0

with Q=Y_0, then the colouring of K must have an odd number of connected components. Now by <Ref> we know that such colourings can indeed be computed through nested commutators, thus concluding the proof that the Lie algebra spanned by this set of Paulis is indeed ⊕_i=1^2^n_c𝔰𝔭(2^n_2).

Given this labeling for the symplectic Lie algebra, one can easily find one for the family of graphs in B3, both for B3_I and B3_D. For the case of algebraic dependent generators, we can simply get our canonical labeling by adding Z_n, for n∈{1,…,n_2}.  It is easy to see that if one takes the product of Paulis in the leg of length 1 and the innermost and outermost vertices of the newly created leg of length 3

    Z_0· Z_0Z_i· Z_i=1.

Adding this additional Pauli allows us to generate any Pauli on n_2+1 qubits, i.e., promotes the Lie algebra to 𝔰𝔲(2^n_2+1). That is because given a Pauli on n_2+1 qubits, we can always find a colouring which is computable, since in the case where P has a colouring with an even number of connected components, then Z_0· Z_0Z_i· Z_i· P is a colouring with an odd number of connected components that yields the same Pauli.

If the generators are not algebraically dependent, by the rules that determine the minimal number of qubits, we must add a new qubit as well. Hence, instead of adding Z_n, we will introduce Z_nY_n+1. Moreover, since every other Pauli commutes with Y_n+1, we can freely modify them so that the overall labeling for this case is

    S_B3_I={Y_0,Z_0Y_n+1}∪{Z_0Z_iY_n+1,Y_i}_i=1,… , n_2-1∪{Z_0Z_C^iY_n+1}_i=1,…,n_C∪{Z_0Z_nY_n+1,Y_n,Z_nY_n+1},

 it is clear that now 

    Z_0Y_n+1· Z_0Z_iY_n+1· Z_iY_n+1=Y_n+1

so we cannot use the same argument as before. However, it suffices to prove it for one case, since the graph is the same for both. Furthermore, the last change was useful to see that as we stated in the main text, the map between B3_D and B3_I is given by P↦ iIm(P)⊗1_n+1+Re(P)⊗ Y_n+1, so that indeed the Lie algebra is the same in both cases, albeit in B3_I it is embedded in a bigger Hilbert space.

The other reason why appending Y_n+1 to symmetric generators was because we can now easily find our canonical labeling for the last class of graphs B2. Since we want to argue that this corresponds to the Lie algebra of anti-symmetric matrices 𝔰𝔬, and all generators in <Ref> are already anti-symmetric, we just need to add another anti-symmetric Pauli that breaks the symmetry with respect to Y_n+1. To that end, our new set of canonical generators will be given by 

    S_B2={Y_0,Z_0Y_n+1}∪{Z_0Z_iY_n+1,Y_i}_i=1,… , n_2-1∪{Z_0Z_C_iY_n+1}_C_i∈ I∪{Z_0Z_nY_n+1,Y_n,Z_nY_n+1,Y_0X_n+1}.

Then, since all generators satisfy the anti-symmetric condition, which is preserved under Lie product, we have

    span(⟨ S_B2⟩)⊆⊕_i=1^2^n_c𝔰𝔬(2^n_2+3).

Like for the symplectic case, checking that any element in ⊕_i=1^2^n_c𝔰𝔬(2^n_2+3) can be computed through nested commutators of our canonical generators is easy to see from <Ref> with Q=1. Again, for any Pauli K satisfying 

    K^T1_n_2+3+1_n_2+3K=0

its colouring on the graph must have an odd number of connected components, and thus must be computable through nested commutators. 
This concludes our construction of all canonical Pauli sets for every Clifford inequivalent class, and serves as a proof of <Ref>.



 §.§ Clifford invariant form


In the previous section we mainly worked with the canonical definition for 𝔰𝔭 and 𝔰𝔬. However, since Clifford operations do not change the Lie algebra, one could come up with a form that makes this explicit.
Before, we introduced the canonical forms as

    𝔤=span({ P∈𝒫| P^T Q+QP=0 }) ,

where Q=1 for 𝔰𝔬 and Q=𝐘_0
for 𝔰𝔭.
If we perform a Clifford transformation, this turns into 

    P↦ P'=C^† P C.

As such, the statement becomes 

          0   =(C P' C^†)^T Q+Q (C P' C^†)

             =C^* P^'T C^TQ+Q C P' C^†

    ⇒       0   =P^'T C^T QC+C^TQ C P',

using C^† C=C^TC^*=1.
With this the action maps

    Q↦ Q'= C^T QC.

This map still maps Paulis to Paulis upon conjugation up to phase. To see this, we can look at a particular generating set of the Clifford group, e.g.,  (S,H,CNOT). On the one hand, since H^T=H^† and CNOT^T=CNOT^†, these act as ordinary Cliffords, mapping Paulis to Paulis. On the other hand, for the map Q↦  S^TQS, we have

    1↦ Z,   X↦ i X,   Y↦ iY,   Z↦1.

However, for the relation that concerns us, this global phase becomes irrelevant. 
Additionally, we have that

    Q'^T=C^T Q^TC = Q'×
            1    Q^T=Q,
    
            -1     Q^T=-Q,

i.e., the symmetric/anti-symmetric property is preserved under the 
action of this map. Additionally, we can map any (anti-)symmetric Pauli to any other (anti-)symmetric Pauli. For this, we can change 1↦ Z by applying S, Z↦ X by applying H and  X↦ I, by applying HS. In order to change the number of Y Paulis we can use CNOTs to map Y⊗ Y↦ X⊗ Z. 
This then motivates the definition that is given in the main text.

Another interesting case in this regard, is that of the class B3_I where the canonical form is defined by

    𝔤=span({ P∈𝒫\{1}|[Y_n+1,P]=0 , P^T+ P=0 }) .

Again, applying a Clifford to the argument this becomes 

    0   =(C P' C^†)Y-Y(C P' C^†),    0=(C P' C^†)^T +(C P' C^†),
    
        0   = P' C^† Y C-C^† YC P',     0=P^'T C^T C+C^T C P',

meaning 

    K C^TC  ,   Q C^† YC.

Thus, when the condition is satisfied

    Q^TK+KQ   =(C^† YC)^T (C^TC)+(C^TC)(C^† YC)
       =C^T Y^TC^*C^TC+C^TCC^† YC
       =-C^T YC+C^T YC=0.

Naturally, Q can become every non identity Pauli. Using only H,CNOT, it is possible to create every symmetric Pauli: C=CNOT_i,n to set it to X_n, H_n to set X_n to Z_n, and CNOT on X⊗ Z to create Y⊗ Y. Moreover, since all gates are real C^T=C^†, which means that 1↦ C_1^T C_1 =1 as well.

Next, we choose a C_2, such that K=C_2^TC_2. Naturally, C=C_1C_2 maps K↦ K, Q↦ C_2^† C_1^† YC_1C_2. Now, choosing C_1 such that C_1^† YC_1=C_2 Q C_2^† we get to the result we wanted. However, by design (C_1^† YC_1)^T=-C_1^† YC_1, meaning 

    (C_2 Q C_2^†)^T   =-C_2 Q C_2^†,
    
        C_2^* Q^T C_2^T   =-C_2 Q C_2^†,
    
         Q^T C_2^TC_2   =-C_2^TC_2 Q,
    
         Q^T K   =-K Q .

The current reduction requires K=K^T. However, we can rewrite the commutation relation in an equivalent form

    0   =[Q,P]=QP-PQ=QP+KP^TKQ, 
    
        0   =KQP+P^TKQ,

as well as

    0   =Q^TK+KQ=(KQ)^T+KQ

using that K=K^T.
So if we define L∝ KQ, we get that L^T=-L and the conditions that ∀ P∈𝒫\ L:

    P^TL+LP   =0  ,   P^TK+KP=0 ,

which shows that B3_I corresponds to a system with both a symplectic and an orthogonal relation.
If we now use the symplectic relation we get

    P^T   =-LPL, 
    
        0   =-LPLK+KP, 
    
        0   =[P,LK].

As such, defining Q'∝ LK, K'=L
we have

    K^T   =K, 
    
        (K'Q')^T   =K'Q', 
    
        0   =(Q')^T K'+ K' Q' ,

using that (K')^T=-K'.
This means that the requirement for K to be symmetric can be removed, by the outlined transformations. This concludes the condition of the Clifford invariant form given in the main text. 





