diff --git "a/69AyT4oBgHgl3EQfQfbA/content/tmp_files/load_file.txt" "b/69AyT4oBgHgl3EQfQfbA/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/69AyT4oBgHgl3EQfQfbA/content/tmp_files/load_file.txt" @@ -0,0 +1,674 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf,len=673 +page_content='On the gate-error robustness of variational quantum algorithms Daniil Rabinovich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 Ernesto Campos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 Soumik Adhikary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 Ekaterina Pankovets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 2 Dmitry Vinichenko,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 3 and Jacob Biamonte4 1Skolkovo Institute of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Moscow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Russian Federation 2Moscow Institute of Physics and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Moscow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Russian Federation 3Moscow Engineering Physics Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Moscow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Russian Federation 4Beijing Institute of Mathematical Sciences and Applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' China Variational algorithms are designed to work within the limitations of contemporary devices and suffer from performance limiting errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Here we identify an experimentally relevant model for gate errors, natural to variational quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' We study how a quantum state prepared variationally decoheres under this noise model, which manifests as a perturbation to the energy approximation in the variational paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' A perturbative analysis of an optimized circuit allows us to determine the noise threshold for which the acceptance criteria imposed by the stability lemma remains satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' We benchmark the results against the variational quantum approximate optimization algorithm for 3-SAT instances and unstructured search with up to 10 qubits and 30 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Finally, we observe that errors in certain gates have a significantly smaller impact on the quality of the prepared state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Motivated by this, we show that it is possible to reduce the execution time of the algorithm with minimal to no impact on the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' INTRODUCTION Noisy Intermediate Scale Quantum (NISQ) quantum computing [1] suffers from limited coherence times and opeartion precision [2–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' In practice we are severely lim- ited by the number of qubits and circuit depths that one may implement with reasonable fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' This has piratical implications in that it limits contemporary ex- perimental demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' A host of theoretical results are now emerging, leading to improved understanding of the use of random circuit sampling as the basis of a scalable experimental violation of the extended Church- Turing thesis [6] and on the complexity analysis of NISQ [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' The variational model of quantum computation is designed to work within these practical limitations [8– 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' More generally, the variational model is known to be computationally universal, yet these results are highly idealized and do not account for noise [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Reminiscent of machine learning, a variational algo- rithm makes use of a short parameterized quantum cir- cuit, known as ansatz, in which parameters are itera- tively tuned to minimize a cost function in a quantum-to- classical feedback loop [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' The cost function is typically given in the form of the expectation of a so called prob- lem Hamiltonian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' where the ground state of the problem Hamiltonian encodes the solution of a given problem in- stance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Thus, by the way of cost function (energy) min- imization, a variational algorithm attempts to approx- imate the ground state of a given Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' This strategy, however, does not provide us with a guarantee in regards to the quality of the approximate solution, where the latter is typically quantified as the overlap between the state prepared by the ansatz and the true ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Nevertheless, the overlap can be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' It has been shown using the stability lemma that the bounds can be directly related to the energy, thus allow- ing us to determine the energy threshold (upper bound) required to guarantee a fixed minimum overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' We call this the acceptance threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' a state with energy below this threshold is said to be accepted by the algorithm [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Variational algorithms by their design alleviate the ef- fects of certain systematic limitations of NISQ devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Nevertheless, variational algorithms are not immune to stochastic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' While there exist some evidence that variational algorithms can in fact benefit from certain level of stochastic noise [13], in general, it is detrimental to the performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' stochastic noise leads to decoherence thus typically reducing solution quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' In this paper we study the extent to which errors, in the form of parameter alterations, affects the performance of variational algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' We analytically show that the shift in energy varies quadratically with the strength of noise (for small amounts of noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' We demonstrate this numerically for variational quantum approximate opti- misation in two common problems—3-SAT [14] and un- structured search [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Furthermore we also found the performance to be more resilient to alterations in certain parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' With that in mind we propose avenues to potentially improve performance and reduce the execu- tion time of variational quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' PRELIMINARIES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Variational Quantum Approximate Optimization The quantum approximate optimization algorithm (QAOA) [17], originally designed to approximately solve combinatorial optimization problems [14, 17–28], consists of ansatze circuits expressive enough to (in theory) emu- late any quantum cirucuit [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Consider a pseudo-Boolean function C : {0, 1}×n → R, the objective of the algorithm is to approximate a bit string that minimizes C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' To accomplish this, C is first arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='00048v1 [quant-ph] 30 Dec 2022 2 encoded as a problem Hamiltonian H, diagonal in the computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' The ground state H encodes the solution to the problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' in other words QAOA searches for a solution |g⟩ such that ⟨g|H|g⟩ = min H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' The algorithm begins with an ansatz state |ψp(γ, β)⟩— prepared by a circuit of depth p — parameterized as: |ψp(γ, β)⟩ = p � k=1 e−iβkHxe−iγkH |+⟩⊗n , (1) with real parameters γk ∈ [0, 2π), βk ∈ [0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Here Hx = �n j=1 Xj is the standard one-body mixer Hamil- tonian with Pauli matrix Xj applied to the j-th qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' The cost function is given by the expectation of the prob- lem Hamiltonian with respect to the ansatz state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' The algorith minimizes this cost function to output: E∗ = minγ,β ⟨ψp(γ, β)| H |ψp(γ, β)⟩ (2) γ∗, β∗ ∈ arg minγ,β ⟨ψp(γ, β)| H |ψp(γ, β)⟩ (3) Here, |ψp(γ∗, β∗)⟩ is the approximate ground state of H and hence the approximate solution to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Indeed, the quality of the approximation, quantified as the overlap between the true solution and the approximate solution, is not known a priori from (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Nevertheless one can establish bounds on this quantity using the so called sta- bility lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Stability lemma The stability lemma states that if |g⟩ is the true ground state of H with energy Eg and ∆ is the spectral gap (the difference between the ground state energy and the energy of the first excited state) the following relation holds [11, 29]: 1 − E∗ − Eg ∆ ≤ |⟨ψp(γ∗, β∗)|g⟩|2 ≤ 1 − E∗ − Eg Em − Eg (4) where Em is the maximum eigenvalue of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Thus to guarantee a non-trivial overlap one must ensure that E∗ ≤ Eg + ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' We call the latter the acceptance con- dition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' VARIATIONAL QUANTUM ALGORITHMS IN THE PRESENCE OF REALISTIC GATE ERRORS Implementation of unitary operations depends signif- icantly on the considered hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' However, typically the implementation makes use of electromagnetic pulses, such as in superconducting quantum computers [30, 31], neutral atom based quantum computers [32, 33], and trapped ion based quantum computers [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Such pulses can change the population of the energy levels that constitute a qubit or introduce phases to the quan- tum amplitudes, thus controlling the state of the qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Consequently, the main contribution to gate errors comes from variation in pulse shaping, meaning that amplitude and timing of electromagnetic pulse can stochasticaly vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' In certain experimental setups, such as ground state ion qubits, where entangling operations are per- formed using the radial phonon modes [36], the variabil- ity in pulse shaping is the main source of gate errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Angles of rotation in a typical gate operation depend on time averaged intensity I(t) of the electromagnetic pulse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' θ ∝ � I(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Thus, variations in the pulse shap- ing lead to stochastic deviations of the angles of rota- tions from the desired values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' In other words, if a cir- cuit is composed of the parameterised gates {Uk(θk)}k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' θ ∈ [0, 2π) and one tries to prepare a state |ψ(θ)⟩ = � k Uk(θk) |ψ0⟩, a different state |ψ(θ + δθ)⟩ = � k U(θk + δθk) |ψ0⟩ , (5) is prepared instead due to the presence of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' No- tice here that the perturbation δθ to the parameters is stochastic and is sampled with a certain probability den- sity p(δθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' This implies that the prepared state can be described by an ensemble {|ψ(θ + δθ)⟩ , p(δθ)}, which we can equivalently view as a density matrix ρ(θ) = � |ψ(θ + δθ)⟩⟨ψ(θ + δθ)|p(δθ)d(δθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' (6) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' (6) represents a noise model native to the vari- ational paradigm of quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' For the rest of this paper we systematically study the effect of this noise model on the performance of QAOA for instances of 3-SAT and the unstructured search problem (see ap- pendix A for more details on the considered problems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' In particular we study the energy perturbation around E∗ in different scenarios subsequently recovering the strength of noise under which the acceptance condition continues to be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Perturbative analysis in presence of gate errors Consider a problem Hamiltonian H and a variational ansatz |ψ(θ)⟩ = U1(θ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Uq(θq) |ψ0⟩ used to mini- mize H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Here the gates Uk(θk) have the form: Uk(θk) = eiAkθk, A2 k = 1, (7) A typical example of such an ansatz is the checkerboard ansatz, with Mølmer-Sørensen (MS) gates as the entan- gling two qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Nevertheless, any quantum circuit can admit a decomposition in terms of operations that satisfy (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' this adds generality to this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 3 In the presence of gate errors the prepared quantum state decoheres as |ψ(θ)⟩ → ρ(θ) as per (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' To obtain the analytic form of ρ(θ) we first note that Uk(θk + δθk) = Uk(θk)Uk(δθk) = cos δθkUk(θk) + sin δθkUk � θk + π 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' (8) This follows directly from (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Therefore we get: |ψ(θ + δθ)⟩⟨ψ(θ + δθ)| = 1 � k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=',kq,m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=',mq=0 (cos2 δθ1 tank1+m1 δθ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' (cos2 δθq tankq+mq δθq)|ψk1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='kq⟩⟨ψm1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='mq|, (9) where |ψk1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='kq⟩ = U1(θ1 + k1 π 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Uq(θq + kq π 2 ) |ψ0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' (10) Here we make three realistic assumptions—(a) pertur- bations to all the angles are independent, (b) average perturbation ⟨δθk⟩ = 0 and (c) the distribution p(δθk) vanishes quickly outside the range (−σk, σk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' that is, the error is localized on the scale σk ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Note that if as- sumption (b) does not hold, one can always shift the parameters as θ → θ + ⟨δθ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Substituting (9) in (6) we arrive at the expression: ρ(θ) = |ψ(θ)⟩⟨ψ(θ)| + δρ, (11) where δρ ≈ − q � k=1 ak|ψ(θ)⟩⟨ψ(θ)|+ q � k=1 ak|ψk⟩⟨ψk|+o(σ2 k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' (12) Here |ψk⟩ = |ψ00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='00⟩ with 1 placed in the k-th posi- tion, and ak ≡ ⟨sin2 δθk⟩ = � sin2 δθkp(δθk)d(δθk) ∼ σ2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' (13) Notice that the derivation above does not require θ to be a minimum of the noiseless cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Let us now assume that θ∗ is a vector of parameters such that |ψ(θ∗)⟩ approximates the ground state of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' The noise induced energy perturbation around the optimal energy E∗ is given as: δE = Tr(ρ(θ∗)H) − ⟨ψ(θ∗)| H |ψ(θ∗)⟩ ≤ (Em − E∗) � k ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' (14) For the simplest case where each parameter is sampled from the same distribution (σk = σ) we can roughly es- timate: δE ≤ qσ2(Em − E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' (15) Thus, requesting an energy threshold E ≤ Eg + ∆, we conclude that for σ <∼ � ∆ − (E∗ − Eg) q(Em − E∗) the acceptance condition is still satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' While our perturbative analysis holds for all varia- tional algorithms, we substantiate our findings numer- ically using QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' In particular we solve instances of 3-SAT and unstructured search problems to study the behaviour of energy perturbation around E∗ caused by the presence of gate errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Constant perturbation We begin with a simplified version of the noise model proposed in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' We ran QAOA for 100 uniformly gen- erated 3-SAT instances of 6,8, and 10 variables with 26, 34 and 42 clauses respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' All the instances were selected to have a unique satisfying assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' The in- stances were minimized by QAOA sequences of 15, 25 and 30 layers respectively in order to obtain expected values well below the energy gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' In order to numeri- cally verify the behaviour of the energy perturbation, we vary all optimal parameters by a constant angle δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Fig- ure 1 illustrates the shift in the energy for the minimized instances, which can be seen to have a quadratic depen- dence of the perturbed energy δE with respect to the shift δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' This is natural to expect since the parameters deviate from the local minimum, where linear contribu- tion must have vanished (a rigorous expression showing the quadratic behavior is derived in appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Similar to the case of 3-SAT, for the problem of un- structured search we perturb optimal parameters of the circuit by an angle δ and plot corresponding energy in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Again, as expected, for small values of δ the en- ergy perturbation is quadratic which comes from the fact that the deviation happens around the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0200 δ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='12 δE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 qubits 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='8δ2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 qubits 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='5δ2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 qubits 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='3δ2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Energy shift obtained by perturbing the ansatz state as |ψp(γ∗ + δ, β∗ + δ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' The curves illustrate averages over 100 uniformly generated 3-SAT instances of 6, 8 and 10 qubits with clause to variable ratio of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='2 and unique satisfying as- signment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' The error bars depict standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Polynomial fits of data indicates δ ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02] follow quadratic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 δ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 δE 6 qubits 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='9δ2 8 qubits 1228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='4δ2 10 qubits 4664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='2δ2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Energy shift for the problem of unstructured search obtained by perturbing of the ansatz state as |ψp(γ∗ + δ, β∗ + δ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Polynomial fits for data points of 6, 8 and 10 qubits follow quadratic curves in the ranges δ ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02], [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='01], [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='008] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Stochastic perturbation We now consider the complete noise model in (6) and verify our analytical prediction as shown in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' For each 3-SAT instance, we randomly sample perturbations δ to each of the gates from a uniform distribution on the interval (−σ, σ) and average the obtained energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Then we average energies over instances of the same number of qubits as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' It is seen that for small values of σ the behaviour is quadratic as per (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' It is seen, that the value σ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='075 could never violate the acceptance criteria, as corresponding energy error never exceeds the gap ∆ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' For smaller number of qubits and gates the threshold value of σ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' For unstructured search, we average the energy over δ sampled for each gate from the uniform distribution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='40 σ 0 1 2 3 4 5 δE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 qubits 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0σ2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 qubits 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='5σ2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 qubits 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='8σ2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Average energy shift of 100 uniformly generated 3- SAT instances of 6, 8 and 10 qubits with clause to variable ratio of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='2 and unique satisfying assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' The shifts are obtained by the perturbation of γ∗, β∗ by δ uniformly sam- pled from the range (−σ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Error bars depict standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Polynomial fits of data indicates σ ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1] follow quadratic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' (−σ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' We again recover quadratic behaviour in σ, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' It is seen that the same threshold σ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='075 now increases energy by no more then 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='6, which guaranties 40% overlap with the target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='28 σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 δE 6 qubits 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='4σ2 8 qubits 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='3σ2 10 qubits 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='2σ2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Average energy for the problem of unstructured search obtained by the perturbation of γ∗, β∗ by δ uni- formly sampled from the range (−σ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Error bars de- pict standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Polynomial fits of data points of 6, 8 and 10 qubits follow quadratic curves in the ranges σ ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1], [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='07], [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='05], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Perturbation to individual parameters Here we consider a modified version of (6), where pa- rameters are perturbed one at a time while the rest are kept intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Effect of this model on the energy is illus- trated in Figures 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' The results are numerical and are yet to be explained analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' We observe that per- turbations to certain angles have a significantly smaller 5 tbh γ β n = 6 p = 8 1 2 3 4 5 6 7 8 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0125 ⟨H⟩ δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='05 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 1 2 3 4 5 6 7 8 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='12 ⟨H⟩ δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='05 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 n = 8 p = 15 2 4 6 8 10 12 14 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0210 ⟨H⟩ δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='05 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 2 4 6 8 10 12 14 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='20 ⟨H⟩ δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='05 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 n = 10 p = 25 0 3 6 9 12 15 18 21 24 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0850 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0855 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0860 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0870 ⟨H⟩ δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='05 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 0 3 6 9 12 15 18 21 24 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='30 ⟨H⟩ δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='05 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Energy ⟨H⟩ = ⟨ψ(θ∗ + δθ)| H |ψ(θ∗ + δθ)⟩ from the unstructured search problem, where βk (right column) or γk (left column), from the k-th layer, are perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' effect on the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Thus we can infer that reducing the value of such angles would not have a significant ef- fect on performance but will reduce the execution time of the algorithm, that is texec = �p k=1 βk + γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Conversely, we could limit the execution time as texec ≤ tmax and increase the number of layers, since min ⟨ψp| H |ψp⟩ ≥ min ⟨ψp+1| H |ψp+1⟩ (16) for the same tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Reducing the execution time is important to quantum algorithms, since variational parameters are proportional to the time required to execute a gate experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' NISQ era devices suffers from limited coherence, thus reducing execution times can lead to more efficient hard- ware utilization [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' We test these ideas in the setting of unstructured search, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Here we show the optimized QAOA energies for 6 qubits at mul- tiple depths with execution time limited to tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' The highlighted green and orange rectangles depict the two groups of optimal angles that minimize the energy at each depth, as presented in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Green rectangles also indicate the depth and texec at which an ansatz will not be able to decrease its energy by either increasing depth or tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Following the observations of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 5, by slightly reducing tmax the optimizer will reduce the parameters to which the energy is less sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' This results in a 6 γ β n = 6 p = 15 2 4 6 8 10 12 14 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='14 ⟨H⟩ δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='05 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 2 4 6 8 10 12 14 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='200 ⟨H⟩ δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='05 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 n = 8 p = 25 0 3 6 9 12 15 18 21 24 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='16 ⟨H⟩ δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='05 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 0 3 6 9 12 15 18 21 24 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='25 ⟨H⟩ δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='05 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 n = 10 p = 30 0 4 8 12 16 20 24 28 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='20 ⟨H⟩ δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='05 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 0 4 8 12 16 20 24 28 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='30 ⟨H⟩ δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='02 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='05 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='08 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Average energy ⟨H⟩ = ⟨ψ(θ∗ + δθ)| H |ψ(θ∗ + δθ)⟩ of 100 uniformly generated 3-SAT instances where βk (right column) or γk (left column), from the k-th layer, are perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' The instances are of 6, 8 and 10 qubits with clause to variable ratio of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='2 and unique satisfying assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' slight energy increase as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 7 where to the left of the green rectangles we can observe darkening gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' By contrast, orange rectangles highlight longer execu- tion times corresponding to different sets of angles that also minimize the energy for a given number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Therefore if the optimization routine finds the a solu- tion corresponding to the orange rectangle, setting tmax to be slightly less than the texec of the orange rectangle will lead the optimizer to find angles corresponding to the green rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' This will amount to a considerable reduction in execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Alternatively increasing the number of layers while keeping tmax will reduce the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' In general, for an arbitrary problem Hamiltonian we can not be sure if our optimization has returned the ideal set of angles (green ones in our example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' For this reason, the best strategy would be to reduce tmax or increase depth while fixing tmax until performance stagnates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' DISCUSSION In this study we considered a realistic noise model— one where the variational gate parameters are stochasti- cally perturbed—and demonstrated its effect on the per- 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0 max execution time tmax 8 7 6 5 4 3 2 1 depth p energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Expected value for multiple combinations of depth for maximum execution times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Green and orange rectangles depict the two branches of angles that minimize expectation value for a given depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' formance of variational algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Using a perturba- tive analysis we showed that the change in energy δE (from optimised energy E∗), caused due to the pres- ence of the considered gate errors, behaves quadratically with respect to the angle perturbations for small values of the perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' This allows us to establish upper bounds on the amount of perturbation such that the ac- ceptance condition continues to be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' This guar- antees a fixed overlap between the target sate and the state prepared by the noisy variational circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' We con- firm our analytical findings numerically in QAOA for two common problems—3-SAT and unstructured search, us- ing different modifications of the considered noise model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Moreover we observed form our numerical results that the algorithmic performance is more resilient to pertur- bations of certain variational parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Motivated by this observation we demonstrated that performance of QAOA with a total execution time texec = � k γk + βk is stable if retrained with a maximum execution time tmax = texec ± ϵ for ϵ ≪ texec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' We also show that in some cases (a) reduction in tmax can lead to dramatic reductions in texec, and (b) increasing depth while fixing texec can lead to an energy reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' While our study is primarily focused on energy pertur- bations around the noiseless optimum θ∗, in practice one has to train in the presence of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' This would change optimal angles θ∗ → θ∗ + δθ∗, where shift δθ∗ increases with increase of the strength of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Nevertheless, using perturbation theory around the noiseless optimum one can estimate δθ∗ = O(σ2), and the corresponding change in the energy is Tr(ρ(θ∗+δθ∗)H)−Tr(ρ(θ∗)H) = O(σ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Therefore, working in the regime of weak noise one can safely use noiseless optimum θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' See appendix C for detailed calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' ACKNOWLEDGEMENT D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' acknowledge support from the research project, Leading Research Center on Quantum Computing (agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 014/20).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Gorshkov, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Gong, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Monroe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Observation of a many-body dynami- cal phase transition with a 53-qubit quantum simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Nature, 551(7682):601–604, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' [36] Laird Nicholas Egan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Scaling Quantum Computers with Long Chains of Trapped Ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' PhD thesis, University of Maryland, College Park, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' [37] Zhi-Cheng Yang, Armin Rahmani, Alireza Shabani, Hartmut Neven, and Claudio Chamon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Optimizing vari- ational quantum algorithms using pontryagin’s minimum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Physical Review X, 7(2):021027, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' [38] Mohannad Ibrahim, Hamed Mohammadbagherpoor, Cynthia Rios, Nicholas T Bronn, and Gregory T Byrd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Pulse-level optimization of parameterized quantum cir- cuits for variational quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='00350, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 9 Appendix A: 3-SAT and unstructured search problems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 3-SAT Boolean satifyability, or SAT, is the problem of deter- mining weather a boolean formula written in conjunctive normal form (CNF) is satisfiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' It is possible to map any SAT instance via Karp reduction into 3-SAT, which are restricted to 3 literals per clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' In order to ap- proximate solutions to SAT we embed the instance into a Hamiltonian as HSAT = � j P(j), (A1) where j indexes clauses of an instance, and P(j) is the tensor product of projectors that penalizes bit string as- signments that do not satisfy the j-th clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Unstructured search Consider an unstructured database S indexed by j ∈ {0, 1}×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Let f : {0, 1}×n → {0, 1} be a Boolean function (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' black box) such that: f(j) = � 1 iff j = t 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' (A2) The task is to find t ∈ {0, 1}×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' The corresponding prob- lem Hamiltonian for QAOA is Ht = 1 − |t⟩⟨t|, (A3) thus the expected value is given by ⟨H⟩ = 1 − |⟨t|ψp(γ, β)⟩|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' (A4) QAOA performance for unstructured search is not sen- sitive to the particular target state |t⟩ in the computa- tional basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' For any target state |t⟩ representing a binary string, there is a U = U † composed of X and 1 opera- tors such that U |0⟩⊗n = |t⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' The overlap of an arbitrary state prepared by a QAOA sequence with |t⟩ is then: ⟨t|ψp(γ, β)⟩ = ⟨t| p � k=1 e−iβkHxe−iγk|t⟩⟨t| |+⟩⊗n = ⟨0|⊗n U p � k=1 e−iβkHxe−iγkU(|0⟩⟨0|)⊗nU |+⟩⊗n = ⟨0|⊗n U p � k=1 e−iβkHxUe−iγk(|0⟩⟨0|)⊗nU |+⟩⊗n = ⟨0|⊗n p � k=1 e−iβkHxe−iγk(|0⟩⟨0|)⊗n |+⟩⊗n , which is independent on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Appendix B: Energy variation in presence of constant perturbations to gate parameters Using (9) one can calculate perturbation to the energy caused by a shift of the optimal angles by a constant δθ as δE = ⟨ψ(θ∗ + δθ)| H |ψ(θ∗ + δθ)⟩ − ⟨ψ(θ∗)| H |ψ(θ∗)⟩ = − q � k=1 δθ2 kE∗ + q � m̸=k δθkδθm(⟨ψ(θ∗)| H |ψkm⟩ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=') + q � m,k δθkδθk ⟨ψm| H |ψk⟩ + o(δθkδθm) = 1 2(δθ)T Hδθ + o(δθkδθm), (B1) where |ψmk⟩ = |ψ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content='0⟩ with 1 placed only at m-th and k-th positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' H is the Hessian of the energy at noiseless optimum, Hij = ∂2 ∂θi∂θj ⟨ψ(θ)| H |ψ(θ)⟩ |θ=θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Here we use the fact that at the optimal position linear contribution to the cost function necessarily vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' It is seen now that for the constant perturbation δθk = δ the energy changes as δE ∝ δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Appendix C: Optimal parameters variation in the presence of noise Let us use expressions (11) and (12) to estimate change in the energy if one accounts for shift of optimal param- eters θ∗ → θ∗ + δθ∗: Tr(ρ(θ∗ + δθ∗)H) = (1 − q � k=1 ak) ⟨ψ(θ∗ + δθ∗)| H |ψ(θ∗ + δθ∗)⟩ + q � k=1 ak ⟨ψk(θ∗ + δθ∗)| H |ψk(θ∗ + δθ∗)⟩ + o(σ2 k) (C1) We introduce gradients of the noisy terms Bk = ∂ ∂θ ⟨ψk(θ)| H |ψk(θ)⟩ |θ=θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Notice that gradients of the 10 noiseless function ⟨ψ(θ)| H |ψ(θ)⟩ vanish at optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Then, Tr(ρ(θ∗ + δθ∗)H) ≈ (1 − q � k=1 ak)E∗ + 1 2(δθ∗)T Hδθ∗ + q � k=1 ak[⟨ψk(θ∗)| H |ψk(θ∗)⟩ + (δθ∗)T Bk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' (C2) Minimizing it with respect to δθ∗ one gets δθ∗ = �q k=1 akH−1Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' Thus, if we account for the change of optimal parameters in the presence of noise, the energy shifts by Tr(ρ(θ∗ + δθ∗)H) − Tr(ρ(θ∗)H) ≈ (δθ∗)T Hδθ∗ + q � k=1 ak(δθ∗)T Bk = O(σ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'} +page_content=' (C3)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfQfbA/content/2301.00048v1.pdf'}