Efficient classical simulation of random shallow 2D quantum circuits

TL;DR

This paper demonstrates that certain shallow 2D random quantum circuits can be approximately simulated efficiently classically, challenging the belief that such circuits are inherently hard to simulate, and introduces algorithms supported by both numerical and analytical evidence.

Contribution

It introduces a class of shallow 2D random quantum circuits that are efficiently simulable approximately, and proposes algorithms with supporting numerical and analytical evidence.

Findings

Efficient approximate simulation for most circuit instances in linear time.

Numerical evidence suggests the algorithms are practical and efficient.

Analytical reduction to 1D dynamics indicates a phase transition in simulability.

Abstract

Random quantum circuits are commonly viewed as hard to simulate classically. In some regimes this has been formally conjectured, and there had been no evidence against the more general possibility that for circuits with uniformly random gates, approximate simulation of typical instances is almost as hard as exact simulation. We prove that this is not the case by exhibiting a shallow circuit family with uniformly random gates that cannot be efficiently classically simulated near-exactly under standard hardness assumptions, but can be simulated approximately for all but a superpolynomially small fraction of circuit instances in time linear in the number of qubits and gates. We furthermore conjecture that sufficiently shallow random circuits are efficiently simulable more generally. To this end, we propose and analyze two simulation algorithms. Implementing one of our algorithms numerically, we give strong evidence that it is efficient both asymptotically and, in some cases, in practice. To argue analytically for efficiency, we reduce the simulation of 2D shallow random circuits to the simulation of a form of 1D dynamics consisting of alternating rounds of random local unitaries and weak measurements -- a type of process that has generally been observed to undergo a phase transition from an efficient-to-simulate regime to an inefficient-to-simulate regime as measurement strength is varied. Using a mapping from quantum circuits to statistical mechanical models, we give evidence that a similar computational phase transition occurs for our algorithms as parameters of the circuit architecture like the local Hilbert space dimension and circuit depth are varied.