Random Quantum Circuits Transform Local Noise into Global White Noise
Alexander M. Dalzell, Nicholas Hunter-Jones, Fernando G. S. L. Brandão · Communications in Mathematical Physics · 2024
Abstract We study the distribution over measurement outcomes of noisy random quantum circuits in the regime of low fidelity, which corresponds to the setting where the computation experiences at least one gate-level error with probability close to one. We model noise by adding a pair of weak, unital, single-qubit noise channels after each two-qubit gate, and we show that for typical random circuit instances, correlations between the noisy output distribution $$p_{\text {noisy}}$$ p noisy and the corresponding noiseless output distribution $$p_{\text {ideal}}$$ p ideal shrink exponentially with the expected number of gate-level errors. Specifically, the linear cross-entropy benchmark F that measures this correlation behaves as $$F=\text {exp}(-2s\epsilon \pm O(s\epsilon ^2))$$ F = exp ( - 2 s ϵ ± O ( s ϵ 2 ) ) , where $$\epsilon $$ ϵ is the probability of error per circuit location and s is the number of two-qubit gates. Furthermore, if the noise is incoherent—for example, depolarizing or dephasing noise—the total variation distance between the noisy output distribution $$p_{\text {noisy}}$$ p noisy and the uniform distribution $$p_{\text {unif}}$$ p unif decays at precisely the same rate. Consequently, the noisy output distribution can be approximated as $$p_{\text {noisy}}\approx Fp_{\text {ideal}}+ (1-F)p_{\text {unif}}$$ p noisy ≈ F p ideal + ( 1 - F ) p unif . In other words, although at least one local error occurs with probability $$1-F$$ 1 - F , the errors are scrambled by the random quantum circuit and can be treated as global white noise, contributing completely uniform output. Importantly, we upper bound the average total variation error in this approximation by $$O(F\epsilon \sqrt{s})$$ O ( F ϵ s ) . Thus, the “white-noise approximation” is meaningful when $$\epsilon \sqrt{s} \ll 1$$ ϵ s ≪ 1 , a quadratically weaker condition than the $$\epsilon s\ll 1$$ ϵ s ≪ 1