Universality in the Anticoncentration of Noisy Quantum Circuits at Finite Depths

Abstract: We study the anticoncentration properties of quantum circuits in the presence of different sources of noise. In the \textit{weak-noise regime}, we show that different types of noise act in a similar fashion, leading to a universal distribution for the probability of a given bit-string, largely independent of the specific noise channel or circuit architecture. In addition, we can identify three distinct depth-dependent regimes, each signaled by a different scaling of cross-entropy benchmarking (XEB) fidelity over time. In the shallow-depth regime, noise effects are perturbatively small; in the intermediate regime, circuit-induced fluctuations and noise compete on equal footing; and in the deep-depth regime, the output distribution becomes effectively classical, up to corrections that are exponentially small in the noise strength. We provide quantitative predictions for the anticoncentration of generic circuits at finite depth, which we benchmark with numerical simulations giving perfect agreement even for shallow circuits. Our findings are directly applicable to current quantum processors and demonstrate universal behavior beyond random-matrix-theory regimes which are only applicable at large depths.