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arXiv:2510.06328

Classical simulation of noisy random circuits from exponential decay of correlation

Su-un Lee, Soumik Ghosh, Changhun Oh, Kyungjoo Noh, Bill Fefferman, Liang Jiang

Published Oct 7, 2025• 0 citations
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Abstract

We study the classical simulability of noisy random quantum circuits under general noise models. While various classical algorithms for simulating noisy random circuits have been proposed, many of them rely on the anticoncentration property, which can fail when the circuit depth is small or under realistic noise models. We propose a new approach based on the exponential decay of conditional mutual information (CMI), a measure of tripartite correlations. We prove that exponential CMI decay enables a classical algorithm to sample from noisy random circuits -- in polynomial time for one dimension and quasi-polynomial time for higher dimensions -- even when anticoncentration breaks down. To this end, we show that exponential CMI decay makes the circuit depth effectively shallow, and it enables efficient classical simulation for sampling. We further provide extensive numerical evidence that exponential CMI decay is a universal feature of noisy random circuits across a wide range of noise models. Our results establish CMI decay, rather than anticoncentration, as the fundamental criterion for classical simulability, and delineate the boundary of quantum advantage in noisy devices.

DOI
10.48550/arxiv.2510.06328

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We study the classical simulability of noisy random quantum circuits under general noise models. While various classical algorithms for simulating noisy random circuits have been proposed, many of them rely on the anticoncentration property, which can fail when the circuit depth is small or under realistic noise models. We propose a new approach based on the exponential decay of conditional mutual information (CMI), a measure of tripartite correlations. We prove that exponential CMI decay enables a classical algorithm to sample from noisy random circuits -- in polynomial time for one dimension and quasi-polynomial time for higher dimensions -- even when anticoncentration breaks down. To this end, we show that exponential CMI decay makes the circuit depth effectively shallow, and it enables efficient classical simulation for sampling. We further provide extensive numerical evidence that exponential CMI decay is a universal feature of noisy random circuits across a wide range of noise models. Our results establish CMI decay, rather than anticoncentration, as the fundamental criterion for classical simulability, and delineate the boundary of quantum advantage in noisy devices.
While quantum computing can accomplish tasks that are classically intractable, the presence of noise may destroy this advantage in the absence of fault tolerance. In this work, we present a classical algorithm that runs in npolylog(n)n^{\rm{polylog}(n)}npolylog(n) time for simulating quantum circuits under local depolarizing noise, thereby ruling out their quantum advantage in these settings. Our algorithm leverages a property called approximate Markovianity to sequentially sample from the measurement outcome distribution of noisy circuits. We establish approximate Markovianity in a broad range of circuits: (1) we prove that it holds for any circuit when the noise rate exceeds a constant threshold, and (2) we provide strong analytical and numerical evidence that it holds for random quantum circuits subject to any constant noise rate. These regimes include previously known classically simulable cases as well as new ones, such as shallow random circuits without anticoncentration, where prior algorithms fail. Taken together, our results significantly extend the boundary of classical simulability and suggest that noise generically enforces approximate Markovianity and classical simulability, thereby highlighting the limitation of noisy quantum circuits in demonstrating quantum advantage.
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We consider quantum circuits consisting of ddd layers of nearest-neighbor two-qubit gates acting on nnn qubits arranged on a line, where every qubit is independently depolarized with a constant probability before each layer. We describe a randomized parallel algorithm which samples from the output distribution of any such circuit to within total variation error δδδ, with parallel runtime 2O(d)log⁡log⁡(n/δ)2^{O(d)}\log\log(n/δ)2O(d)loglog(n/δ) and n2O(d)n 2^{O(d)}n2O(d) elementary real-arithmetic operations. Without noise, the same approach gives an exact sampler with parallel runtime O(log⁡n)O(\log n)O(logn). For constant depth, we further show that the input/output behavior of the noisy quantum circuit is reproduced up to a constant error by a randomized AC0\mathsf{AC}^0AC0-circuit, that is, a Boolean circuit of polynomial size and constant depth with unbounded fan-in AND and OR gates and NOT gates. Consequently, every relation problem solved by a noisy constant-depth quantum circuit in one dimension is also solved, with essentially the same success probability, by a randomized AC0\mathsf{AC}^0AC0-circuit. This rules out, for noisy circuits in one dimension, the unconditional quantum advantage established for noisy shallow circuits in two and three dimensions, where polynomial-size classical circuits over the same gate set require depth Ω(log⁡n/log⁡log⁡n)Ω(\log n/\log\log n)Ω(logn/loglogn). Our algorithm exploits the fact that depolarizing noise cuts a one-dimensional circuit into independent pieces of logarithmic width, each of which can be sampled exactly in parallel.
Quantum circuits with a constant rate of depolarizing noise per qubit per time step are known to converge to the uniform distribution at depth ω(p−1log⁡n)ω(p^{-1}\log n)ω(p−1logn), and hence become trivially classically simulable by uniform sampling. We show that under the physically natural constraint of geometric locality, noisy circuits become classically simulable at shallower depths by substantially more structured classical algorithms. We consider arbitrary geometrically local quantum circuits on nnn qubits, initialized in an arbitrary product state, with nearest-neighbor gates in O(1)O(1)O(1) spatial dimensions and interspersed depolarizing noise of any constant strength ppp. Our first result is that when the depth exceeds a threshold d∗=Θ(p−1log⁡n)d^*=Θ(p^{-1}\log n)d∗=Θ(p−1logn), the output distribution can be approximately sampled in quasipolynomial time. This gives a worst-case simulability result for noisy geometrically local circuits, and matches in its nnn-dependence the best previously known results for noisy random circuits. The proof is based on new information-theoretic bounds showing that in geometrically local noisy circuits, local regions lose correlations and independently converge to maximally mixed strictly before the entire system converges to the uniform distribution. We further prove structural results suggesting a sharper transition at depth Θ~(p−1)\tildeΘ(p^{-1})Θ~(p−1): after coarse-graining the lattice, Pauli weight supported on long connected paths can be truncated with exponentially small error. These results provide evidence for a percolation-like mechanism behind classical simulability at constant depth, and motivate our conjecture that all noisy geometrically local circuits admit quasipolynomial-time approximate sampling once circuit depth exceeds Θ~(p−1)\tildeΘ(p^{-1})Θ~(p−1).
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