Estimating Gibbs partition function with quantumClifford sampling

TL;DR

This paper introduces a hybrid quantum-classical algorithm using Clifford sampling to estimate the partition function of quantum many-body systems efficiently with shallow quantum circuits, suitable for NISQ devices.

Contribution

The paper proposes a novel Clifford sampling technique enabling accurate partition function estimation with shallow quantum circuits, reducing quantum resource requirements compared to prior methods.

Findings

Requires only shallow $ ext{O}(1)$-depth quantum circuits.

Achieves $ ext{O}(1/\epsilon^2)$ repetitions for a given accuracy.

Comparable accuracy to previous methods with significantly lower quantum circuit depth.

Abstract

The partition function is an essential quantity in statistical mechanics, and its accurate computation is a key component of any statistical analysis of quantum system and phenomenon. However, for interacting many-body quantum systems, its calculation generally involves summing over an exponential number of terms and can thus quickly grow to be intractable. Accurately and efficiently estimating the partition function of its corresponding system Hamiltonian then becomes the key in solving quantum many-body problems. In this paper we develop a hybrid quantum-classical algorithm to estimate the partition function, utilising a novel Clifford sampling technique. Note that previous works on quantum estimation of partition functions require $\mathcal{O}(1/\epsilon\sqrt{\Delta})$-depth quantum circuits~\cite{Arunachalam2020Gibbs, Ashley2015Gibbs}, where $\Delta$ is the minimum spectral gap of stochastic matrices and $\epsilon$ is the multiplicative error. Our algorithm requires only a shallow $\mathcal{O}(1)$-depth quantum circuit, repeated $\mathcal{O}(1/\epsilon^2)$ times, to provide a comparable $\epsilon$ approximation. Shallow-depth quantum circuits are considered vitally important for currently available NISQ (Noisy Intermediate-Scale Quantum) devices.