Hardness and Complexity Transition of Noisy Random Circuit Sampling

Abstract

Random circuit sampling (RCS) is a leading candidate for demonstrating quantum advantage, supported by strong complexity-theoretic evidence of hardness in the ideal setting and by rapid experimental progress to date. In practice, however, noise is unavoidable, and a central problem is to identify the noise-strength boundary between classically simulable and classically hard regimes. In this work, we establish an architecture-general hardness bound for this boundary for the standard local depolarizing noise of strength $\gamma$. Assuming the standard average-case #P-hardness conjecture for ideal RCS, we show that, for any circuit architecture satisfying this conjecture, noisy RCS on the same architecture remains hard to simulate classically within any inverse-polynomial total variation distance whenever $\gamma=O(\log n/(nd))$ for $n$-qubit circuits of depth $d$, unless the polynomial hierarchy collapses. Crucially, noisy-RCS hardness follows without any additional conjectural or architecture-specific assumption beyond those already entering the ideal-RCS hardness framework. Our proof combines a low-degree polynomial extrapolation with a monotonicity reduction showing that efficient classical simulation at one depolarizing noise strength implies efficient simulation at every larger strength. Together, these ingredients transfer the standard ideal-RCS hardness conjecture to sampling hardness at a prespecified noise strength. Finally, combining the convergence-to-uniformity result of Dalzell et al. [Commun. Math. Phys. 405, 78 (2024)] with our monotonicity reduction yields efficient classical simulation for $\gamma=\omega(\log n/(nd))$ on layered, regularly connected architectures. Thus, wherever the two architectural settings overlap, this identifies $\gamma=\Theta(\log n/(nd))$ as the asymptotic complexity-transition scale.