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arXiv:2206.08293 · Physical Review A

Linear cross-entropy benchmarking with Clifford circuits

Jianxin Chen, Dawei Ding, Cupjin Huang, Linghang Kong

Published Nov 20, 2023• 7 citations
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Abstract

With the advent of quantum processors exceeding 100 qubits and the high engineering complexities involved, there is a need for holistically benchmarking the processor to have quality assurance. Linear cross-entropy benchmarking (XEB) has been used extensively for systems with 50 or more qubits but is fundamentally limited in scale due to the exponentially large computational resources required for classical simulation. In this work we propose conducting linear XEB with random Clifford circuits of constant to logarithmic depth, a scheme we call Clifford XEB. Since Clifford circuits can be simulated in polynomial time, Clifford XEB can be scaled to much larger systems. To validate this claim, we run numerical simulations for the classes of Clifford circuits we propose with noise and observe exponential decays. When noise levels are low, the decay rates are well correlated with the noise of each cycle assuming a multiplicative error accumulation, i.e., where the fidelities of individual gates multiply. We perform simulations of systems up to 1225 qubits, where the classical processing task can be easily dealt with by a workstation. Furthermore, using the theoretical guarantees in Chen et al. [PRX Quantum 3, 030320 (2022)], we prove that Clifford XEB with our proposed Clifford circuits must yield exponential decays under a general error model for sufficiently low errors. Our theoretical results explain some of the phenomena observed in the simulations and shed light on the behavior of general linear XEB experiments.

DOI
10.1103/physreva.108.052613

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With the advent of quantum processors exceeding 100 qubits and the high engineering complexities involved, there is a need for holistically benchmarking the processor to have quality assurance. Linear cross-entropy benchmarking (XEB) has been used extensively for systems with 50 or more qubits but is fundamentally limited in scale due to the exponentially large computational resources required for classical simulation. In this work we propose conducting linear XEB with random Clifford circuits of constant to logarithmic depth, a scheme we call Clifford XEB. Since Clifford circuits can be simulated in polynomial time, Clifford XEB can be scaled to much larger systems. To validate this claim, we run numerical simulations for the classes of Clifford circuits we propose with noise and observe exponential decays. When noise levels are low, the decay rates are well correlated with the noise of each cycle assuming a multiplicative error accumulation, i.e., where the fidelities of individual gates multiply. We perform simulations of systems up to 1225 qubits, where the classical processing task can be easily dealt with by a workstation. Furthermore, using the theoretical guarantees in Chen et al. [PRX Quantum 3, 030320 (2022)], we prove that Clifford XEB with our proposed Clifford circuits must yield exponential decays under a general error model for sufficiently low errors. Our theoretical results explain some of the phenomena observed in the simulations and shed light on the behavior of general linear XEB experiments.
The linear cross-entropy benchmark (Linear XEB) has been used as a test for procedures simulating quantum circuits. Given a quantum circuit C with n inputs and outputs and purported simulator whose output is distributed according to a distribution p over {0,1}ⁿ, the linear XEB fidelity of the simulator is ℱ_C(p) = 2ⁿ 𝔼_{x ∼ p} q_C(x) -1, where q_C(x) is the probability that x is output from the distribution C |0ⁿ⟩. A trivial simulator (e.g., the uniform distribution) satisfies ℱ_C(p) = 0, while Google’s noisy quantum simulation of a 53-qubit circuit C achieved a fidelity value of (2.24 ±0.21)×10^{-3} (Arute et. al., Nature'19). In this work we give a classical randomized algorithm that for a given circuit C of depth d with Haar random 2-qubit gates achieves in expectation a fidelity value of Ω(n/L⋅15^{-d}) in running time poly(n,2^L). Here L is the size of the light cone of C: the maximum number of input bits that each output bit depends on. In particular, we obtain a polynomial-time algorithm that achieves large fidelity of ω(1) for depth O(√{log n}) two-dimensional circuits. This is the first such result for two dimensional circuits of super-constant depth. Our results can be considered as an evidence that fooling the linear XEB test might be easier than achieving a full simulation of the quantum circuit.
Cross-entropy benchmarking (XEB) with single-qubit reference sequences is widely used to characterize multi-qubit gates in large-scale quantum processors, despite the lack of a rigorous theoretical justification. Here we show that the commonly employed additive single-qubit errors approximation underlying this approach breaks down and leads to a systematic overestimation of gate fidelities. We derive an analytical expression for the joint decay of simultaneous single-qubit reference sequences, identify when interleaved circuits generate sufficient randomization for the standard depolarizing approximation, and obtain a refined expression for the interleaved gate fidelity estimation. Experiments on a superconducting quantum processor validate the theory and demonstrate that fidelities obtained using XEB with single-qubit references agree with those extracted from standard interleaved randomized benchmarking (IRB), while achieving higher precision due to reduced reference-sequence errors. Our results establish theoretical foundation for interleaved benchmarking protocols employing single-qubit reference sequences and show that, with appropriate post-processing, they provide a simple and accurate alternative to multi-qubit Clifford reference circuits.
The linear cross-entropy benchmark (Linear XEB) has been used as a test for procedures simulating quantum circuits. Given a quantum circuit C with n inputs and outputs and purported simulator whose output is distributed according to a distribution p over {0,1}ⁿ, the linear XEB fidelity of the simulator is ℱ_C(p) = 2ⁿ 𝔼_{x ∼ p} q_C(x) -1, where q_C(x) is the probability that x is output from the distribution C |0ⁿ⟩. A trivial simulator (e.g., the uniform distribution) satisfies ℱ_C(p) = 0, while Google’s noisy quantum simulation of a 53-qubit circuit C achieved a fidelity value of (2.24 ±0.21)×10^{-3} (Arute et. al., Nature'19). In this work we give a classical randomized algorithm that for a given circuit C of depth d with Haar random 2-qubit gates achieves in expectation a fidelity value of Ω(n/L⋅15^{-d}) in running time poly(n,2^L). Here L is the size of the light cone of C: the maximum number of input bits that each output bit depends on. In particular, we obtain a polynomial-time algorithm that achieves large fidelity of ω(1) for depth O(√{log n}) two-dimensional circuits. This is the first such result for two dimensional circuits of super-constant depth. Our results can be considered as an evidence that fooling the linear XEB test might be easier than achieving a full simulation of the quantum circuit.
Recently, Google announced the first demonstration of quantum computational supremacy with a programmable superconducting processor. Their demonstration is based on collecting samples from the output distribution of a noisy random quantum circuit, then applying a statistical test to those samples called Linear Cross-Entropy Benchmarking (Linear XEB). This raises a theoretical question: how hard is it for a classical computer to spoof the results of the Linear XEB test? In this short note, we adapt an analysis of Aaronson and Chen [2017] to prove a conditional hardness result for Linear XEB spoofing. Specifically, we show that the problem is classically hard, assuming that there is no efficient classical algorithm that, given a random n-qubit quantum circuit C, estimates the probability of C outputting a specific output string, say 0^n, with variance even slightly better than that of the trivial estimator that always estimates 1/2^n. Our result automatically encompasses the case of noisy circuits.
Demonstrations of quantum computational advantage and benchmarks of quantum processors via quantum random circuit sampling are based on evaluating the linear cross-entropy benchmark (XEB). A key question in the theory of XEB is whether it approximates the fidelity of the quantum state preparation. Previous works have shown that the XEB generically approximates the fidelity in a regime where the noise rate per qudit ε\varepsilonε satisfies εN≪1\varepsilon N \ll 1εN≪1 for a system of NNN qudits and that this approximation breaks down at large noise rates. Here, we show that the breakdown of XEB as a fidelity proxy occurs as a sharp phase transition at a critical value of εN\varepsilon NεN that depends on the circuit architecture and properties of the two-qubit gates, including in particular their entangling power. We study the phase transition using a mapping of average two-copy quantities to statistical mechanics models in random quantum circuit architectures with full or one-dimensional connectivity. We explain the phase transition behavior in terms of spectral properties of the transfer matrix of the statistical mechanics model and identify two-qubit gate sets that exhibit the largest noise robustness.
The computational advantage of noisy quantum computers has been demonstrated by sampling the bitstrings of quantum random circuits. An important issue is how the performance of quantum devices could be quantified in the so-called "supremacy regime". The standard approach is through the linear cross entropy benchmark (XEB), where the theoretical value of the probability is required for each bitstring. However, the computational cost of XEB grows exponentially. So far, random circuits of the 53-qubit Sycamore chip were verified up to 10 cycles of gates only; the XEB fidelities of deeper circuits were approximated with simplified circuits instead. Here we present a multi-tensor contraction algorithm for speeding up the calculations of XEB for quantum circuits, where the computational cost can be significantly reduced through some form of memoization. As a demonstration, we analyzed the experimental data of the 53-qubit Sycamore chip and obtained the exact values of the corresponding XEB fidelities up to 16 cycles using only moderate computing resources (few GPUs). If the algorithm was implemented on the Summit supercomputer, we estimate that for the supremacy (20 cycles) circuits, it would only cost 7.5 days, which is several orders of magnitude lower than previously estimated in the literature.