A Sublinear-Time Quantum Algorithm for Approximating Partition Functions

TL;DR

This paper introduces a quantum algorithm that estimates Gibbs partition functions in sublinear time, offering significant speed-ups over classical methods and improving various computational tasks related to statistical physics and combinatorics.

Contribution

The authors develop the first sublinear-time quantum algorithm for partition functions, utilizing novel quantum estimation techniques that reduce variance and preserve quantum states.

Findings

Achieves sublinear time complexity for partition function approximation.

Provides polynomial improvements for Ising model and graph counting problems.

Introduces new quantum mean estimation methods with low variance.

Abstract

We present a novel quantum algorithm for estimating Gibbs partition functions in sublinear time with respect to the logarithm of the size of the state space. This is the first speed-up of this type to be obtained over the seminal nearly-linear time algorithm of \v{S}tefankovi\v{c}, Vempala and Vigoda [JACM, 2009]. Our result also preserves the quadratic speed-up in precision and spectral gap achieved in previous work by exploiting the properties of quantum Markov chains. As an application, we obtain new polynomial improvements over the best-known algorithms for computing the partition function of the Ising model, counting the number of $k$-colorings, matchings or independent sets of a graph, and estimating the volume of a convex body. Our approach relies on developing new variants of the quantum phase and amplitude estimation algorithms that return nearly unbiased estimates with low variance and without destroying their initial quantum state. We extend these subroutines into a nearly unbiased quantum mean estimator that reduces the variance quadratically faster than the classical empirical mean. No such estimator was known to exist prior to our work. These properties, which are of general interest, lead to better convergence guarantees within the paradigm of simulated annealing for computing partition functions.