Quantum Speed-up for Approximating Partition Functions

TL;DR

This paper demonstrates a quantum algorithm that significantly accelerates the approximation of partition functions by combining quantum techniques with classical methods, reducing complexity in both spectral gap and accuracy parameters.

Contribution

It introduces a novel quantum approach that integrates Grover's search, quantum walks, and quantum counting to improve the efficiency of estimating partition functions.

Findings

Quadratic speed-up in spectral gap dependence

Quadratic reduction in accuracy-related complexity

Efficient quantum encoding of Markov chain distributions

Abstract

We achieve a quantum speed-up of fully polynomial randomized approximation schemes (FPRAS) for estimating partition functions that combine simulated annealing with the Monte-Carlo Markov Chain method and use non-adaptive cooling schedules. The improvement in time complexity is twofold: a quadratic reduction with respect to the spectral gap of the underlying Markov chains and a quadratic reduction with respect to the parameter characterizing the desired accuracy of the estimate output by the FPRAS. Both reductions are intimately related and cannot be achieved separately. First, we use Grover's fixed point search, quantum walks and phase estimation to efficiently prepare approximate coherent encodings of stationary distributions of the Markov chains. The speed-up we obtain in this way is due to the quadratic relation between the spectral and phase gaps of classical and quantum walks. Second, we generalize the method of quantum counting, showing how to estimate expected values of quantum observables. Using this method instead of classical sampling, we obtain the speed-up with respect to accuracy.