Classical simulation of noisy random circuits from exponential decay of correlation

TL;DR

This work identifies exponential decay of conditional mutual information (CMI) as a central criterion governing the classical simulability of noisy random circuit sampling, challenging anticoncentration as the sole hardness marker under realistic noise. By linking CMI decay to an effective shallow-depth reduction and a region-based patching sampling method, the authors prove that noisy $D$-dimensional random circuits with decaying CMI can be sampled classically in polynomial time for $D=1$ and quasi-polynomial time for $D\ge 2$, provided the average ${\eta(\ell)}$-approximate Markov condition holds. They further establish a practical two-step program: approximate any deep circuit by a shallow one of depth $d^* = O(\log^D(n/\varepsilon))$ and then efficiently sample the shallow circuit via local marginals, with rigorous bounds on the total variation distance. Numerical simulations across multiple noise models corroborate the universality of exponential CMI decay in both one- and two-dimensional noisy random circuits, supporting the proposed tractability framework. Overall, the paper reframes the boundary of quantum advantage in noisy devices, positioning CMI decay as a fundamental determinant of classical simulability.

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.

Figures (9)

• Figure 1: (a) A depth-$d$ noisy random circuit consists of alternating layers of unitary gates (white boxes) and noise channels (red boxes) (b) Assuming exponential decay of CMI, the output distribution can be approximated by a depth-$d^* = O(\log^D (n/\varepsilon))$ circuit.
• Figure 2: Example of coarse-graining a 2D grid. (a) A 2D grid is partitioned into disjoint squares of side length $\ell$. (b) The coarse-grained graph $G = (V, E)$, where each vertex corresponds to a hypercube of side length $\ell$, and edges are drawn between two vertices if the distance between them is less than $\ell$.
• Figure 3: (a) Simulation setup of the 1D Haar random circuits. (b) Simulation setup of the 2D Clifford random circuits. (c) Simulation results of the CMI decay in 1D Haar random circuits (top row) and 2D Clifford random circuits (bottom row). For 1D Haar random circuits, CMI is averaged over $64$ circuit realizations, and CMI is estimated with $1{,}000$ MC samples. For 2D Clifford random circuits, CMI is averaged over $100{,}000$ circuit realizations, and CMI is exactly computed. To reduce fluctuations due to specific gate patterns, we average the CMI over pairs of consecutive depths in 1D (e.g., depths 10--11) and over groups of four consecutive depths in 2D (e.g., depths 10--13). For all plots, solid lines present the results with non-unital noise channels (amplitude damping and heralded reset channel), while dashed lines present the results with unital noise channels (depolarizing channel and heralded depolarizing channel).
• Figure 4: (a) The trace-preserving property of quantum channels. Here, the grounding symbol denotes applying partial trace. (b) The lightcone argument for computing marginal distributions. The white gates denote the ones inside the backward lightcone of $X$, and the gray gates are those outside the backward lightcone of $X$. As we trace out all qubits in $[n]\setminus X$, every gate outside the backward lightcone of $X$ vanishes.
• Figure 5: (a) For 1D Haar random circuits, each unitary layer $\mathcal{U}_i$ consists of two-qubit Haar random gates applied on $i$-th and $(i+1)$-th qubits for $i = 1, 3, \dots, n-1$ for even $t$ (qubits connected by purple lines) and $i = 2, 4, \dots, n-2$ for odd $t$ (qubits connected by blue lines). (b) For 2D Clifford random circuits, we apply alternately apply gates on qubits linked with purple lines (when $t = 1 \pmod 4$), orange lines (when $t = 2 \pmod 4$), blue lines (when $t = 3 \pmod 4$), and black lines (when $t = 0 \pmod 4$).

Theorems & Definitions (30)

• Definition 1
• Proposition 1
• Theorem 1
• proof : Sketch of Proof
• Theorem 2
• Theorem 3: Main Theorem
• Proposition 2: Data processing inequality Wilde_2013
• Proposition 3
• proof
• Proposition 4