Generalized cross-entropy benchmarking for random circuits with ergodicity

Cross-entropy benchmarking is a central technique adopted to certify a quantum chip in recent investigations. To better understand its mathematical foundation and develop new benchmarking schemes, the concept of ergodicity was introduced to random circuit sampling and it was found that the Haar random quantum circuit could satisfy an ergodicity condition—the average of certain types of postprocessing function over the output bit strings is close to the average over the unitary ensemble. For noiseless random circuits, it was proven that the ergodicity holds for polynomials of degree t with positive coefficients when the random circuits form a unitary 2t-design. For strong enough noise, the ergodicity condition is violated, which suggests that ergodicity is a property that can be exploited to certify a quantum chip. The deviation of ergodicity was formulated as a measure for quantum chip benchmarking, and it was demonstrated that it can be used to estimate the circuit fidelity for global depolarizing noise and weakly correlated noise. For a quadratic postprocessing function, our framework recovered Google's result on estimating the circuit fidelity via linear cross-entropy benchmarking (XEB), and we gave a sufficient condition on the noise model characterizing when such estimation is valid. The results establish an interesting connection between ergodicity and noise in random circuits and provide new insights into designing quantum benchmarking schemes.