Linear cross-entropy benchmarking with Clifford circuits

Jianxin Chen, Dawei Ding, Cupjin Huang, and Linghang Kong

Phys. Rev. A 108, 052613 – Published 20 November 2023

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.

Received 16 May 2023
Revised 25 August 2023
Accepted 1 November 2023

DOI:https://doi.org/10.1103/PhysRevA.108.052613

Physics Subject Headings (PhySH)

Information scrambling Quantum benchmarking Quantum circuits Quantum protocols

Quantum Information, Science & Technology

Authors & Affiliations

Jianxin Chen¹, Dawei Ding ², Cupjin Huang ¹, and Linghang Kong³

¹Alibaba Quantum Laboratory, Alibaba Group USA, Bellevue, Washington 98004, USA
²Alibaba Quantum Laboratory, Alibaba Group USA, Sunnyvale, California 94085, USA
³Alibaba Quantum Laboratory, Alibaba Group, Hangzhou, Zhejiang 311121, People's Republic of China

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Vol. 108, Iss. 5 — November 2023

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Images

Figure 1
Illustration of one iteration of a Clifford XEB experiment. We start with an initial state, apply $m$ cycles of noisy implementations of Clifford elements, and finally perform a measurement. The measured bit string's noiseless probability is then classically computed. This value is averaged over multiple measurements and random circuits.
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Figure 2
Illustration of different quantum device topologies we consider in our paper: (a) 1D chain, (b) 2D grid, and (c) star topology. In (a) and (b), qubit connections with different colors indicate different layers of parallel two-qubit gates to be applied. The Clifford approximate twirl can be naturally applied on a star topology depicted in (c)
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Figure 3
Linear XEB as a function of the number of cycles on (a) 1D chain, (b) 2D grid, and (c) Clifford approximate twirl (star topology) (the $x$ axis is twice the cycle number since we need to apply the circuit construction twice to get a $γ$ -approximate twirl, see Sec. 3 for details), with different qubit numbers and noise levels. Each row corresponds to the labeled topology. The number of qubits simulated on each topology is, from left to right, 25, 100, and 225. The horizontal line is linear XEB $= 1$ for reference. The curves show some deviations from an exponential decay. This can be attributed to rapidly decaying transients and statistical fluctuations. The former can be addressed by a sufficient number of cycles, while the latter can be addressed by increasing the number of samples. See main text for further details.
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Figure 4
Illustration of a typical linear XEB experiment on a $4 \times 4$ grid. Decay of the noisy Clifford XEB curves can be roughly divided into a scrambling phase and a decaying phase with the threshold being about $m = 22$ . We perform a linear fitting for the noisy Clifford XEB curves for $m \in [40, 100]$ and plot the fit as dashed lines. The red crosses illustrate the convergence of the noiseless linear XEB towards 1. In the decaying phase, it can be seen that the noiseless linear XEB value converges to 1 at a certain rate we call the noiseless scrambling rate (NSR) which can computed from the red dashed-dotted line on the bottom. More discussions of the NSR can been found in Sec. 2b.
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Figure 5
Comparison between the extracted error per layer from the linear XEB experiments, the error per layer predicted by MEA, and the NSR, under different topologies and number of qubits. The error per layer is defined as $1 - u$ , where $u$ is the extracted decay rate for linear XEB experiments, and the predicted fidelity assuming MEA. For each noise level, the crosses ( $\times$ ) are predictions given by MEA, and the dots ( $•$ ) are extracted from the numerical experiments. A few data points are missing due to failure of fitting. As we will discuss below, the MEA infidelity matches the fitted error rate when they are below the NSR.
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Figure 6
Clifford XEB on a 2D grid of $35 \times 35 = 1225$ qubits. Data points for curves with noise are evaluated over 3000 random circuits with $100 000$ samples each. Data points for the noiseless curve are evaluated over $300 000$ random circuits. Predictions assuming MEA are plotted in dashed lines for comparison. The prediction assuming MEA with noise level $(10^{- 4}, 10^{- 3})$ is beyond the range of the plot. The red dashed-dotted line is used to illustrate the noiseless scrambling rate (NSR). It again suffers from fluctuations with large depths.
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