Classical simulation of peaked shallow quantum circuits

TL;DR

This paper presents classical algorithms with quasipolynomial and polynomial runtimes for approximately sampling from the output distributions of peaked shallow quantum circuits, enabling efficient probability estimation.

Contribution

The authors develop faster classical algorithms for simulating peaked shallow quantum circuits, improving over previous exponential-scaling methods.

Findings

Quasipolynomial runtime $n^{O( ext{log}n)}$ for general peaked circuits.

Polynomial runtime $n^{O(1)}$ for 2D nearest-neighbor circuits.

Almost-polynomial runtime $n^{O( ext{loglog}n)}$ for higher-dimensional circuits.

Abstract

An $n$-qubit quantum circuit is said to be peaked if it has an output probability that is at least inverse-polynomially large as a function of $n$. We describe a classical algorithm with quasipolynomial runtime $n^{O(\log{n})}$ that approximately samples from the output distribution of a peaked constant-depth circuit. We give even faster algorithms for circuits composed of nearest-neighbor gates on a $D$-dimensional grid of qubits, with polynomial runtime $n^{O(1)}$ if $D=2$ and almost-polynomial runtime $n^{O(\log{\log{n}})}$ for $D>2$. Our sampling algorithms can be used to estimate output probabilities of shallow circuits to within a given inverse-polynomial additive error, improving previously known methods. As a simple application, we obtain a quasipolynomial algorithm to estimate the magnitude of the expected value of any Pauli observable in the output state of a shallow circuit (which may or may not be peaked). This is a dramatic improvement over the prior state-of-the-art algorithm which had an exponential scaling in $\sqrt{n}$.