Generalized Cross-Entropy Benchmarking for Random Circuits with Ergodicity

TL;DR

This paper introduces ergodicity in random circuit sampling to improve quantum chip benchmarking, connecting mathematical properties with noise effects and recovering known results like Google's XEB.

Contribution

It establishes ergodicity conditions in random circuits, linking them to benchmarking and noise estimation, and generalizes cross-entropy benchmarking methods.

Findings

Ergodicity holds for Haar random circuits and certain polynomials in noiseless cases.

Ergodicity is violated under strong noise, indicating noise impact on benchmarking.

The framework recovers Google's linear XEB and provides noise condition insights.

Abstract

Cross-entropy benchmarking is a central technique used to certify a quantum chip in recent experiments. To better understand its mathematical foundation and develop new benchmarking schemes, we introduce the concept of ergodicity to random circuit sampling and find that the Haar random quantum circuit satisfies an ergodicity condition -- the average of certain types of post-processing function over the output bit strings is close to the average over the unitary ensemble. For noiseless random circuits, we prove that the ergodicity holds for polynomials of degree $t$ with positive coefficients and when the random circuits form a unitary $2t$-design. For strong enough noise, the ergodicity condition is violated. This suggests that ergodicity is a property that can be exploited to certify a quantum chip. We formulate the deviation of ergodicity as a measure for quantum chip benchmarking and show that it can be used to estimate the circuit fidelity for global depolarizing noise and weakly correlated noise. For a quadratic post-processing function, our framework recovers Google's result on estimating the circuit fidelity via linear cross-entropy benchmarking (XEB), and we give a sufficient condition on the noise model characterizing when such estimation is valid. Our results establish an interesting connection between ergodicity and noise in random circuits and provide new insights into designing quantum benchmarking schemes.