Classical algorithms for quantum mean values

TL;DR

This paper presents classical algorithms for estimating expectation values of tensor product observables in shallow quantum circuits, with complexity depending on the observable type and circuit geometry, advancing the understanding of classical simulability.

Contribution

It introduces new classical approximation algorithms for quantum mean values with different error bounds, based on polynomial interpolation, quantum query complexity, and MPS techniques.

Findings

Classical algorithms with polynomial and subexponential runtimes for different observable types.

Efficient approximation of mean values with small relative or additive errors.

A technical lemma on zero-free regions of associated polynomials.

Abstract

We consider the task of estimating the expectation value of an $n$-qubit tensor product observable $O_1\otimes O_2\otimes \cdots \otimes O_n$ in the output state of a shallow quantum circuit. This task is a cornerstone of variational quantum algorithms for optimization, machine learning, and the simulation of quantum many-body systems. Here we study its computational complexity for constant-depth quantum circuits and three types of single-qubit observables $O_j$ which are (a) close to the identity, (b) positive semidefinite, (c) arbitrary. It is shown that the mean value problem admits a classical approximation algorithm with runtime scaling as $\mathrm{poly}(n)$ and $2^{\tilde{O}(\sqrt{n})}$ in cases (a,b) respectively. In case (c) we give a linear-time algorithm for geometrically local circuits on a two-dimensional grid. The mean value is approximated with a small relative error in case (a), while in cases (b,c) we satisfy a less demanding additive error bound. The algorithms are based on (respectively) Barvinok's polynomial interpolation method, a polynomial approximation for the OR function arising from quantum query complexity, and a Monte Carlo method combined with Matrix Product State techniques. We also prove a technical lemma characterizing a zero-free region for certain polynomials associated with a quantum circuit, which may be of independent interest.