Predicting properties of quantum thermal states from a single trajectory

TL;DR

This paper introduces a method to efficiently estimate thermal expectation values in quantum states using a single Gibbs-sampling trajectory, reducing sampling costs by leveraging autocorrelation properties and advanced measurement techniques.

Contribution

It proposes a novel approach combining a single trajectory sampling with Gaussian-filtered quantum phase estimation and a weighted operator Fourier transform to improve efficiency in quantum observable estimation.

Findings

Sampling cost is reduced using a single trajectory after burn-in.

Autocorrelation times can be shorter than mixing times in many settings.

Efficient energy estimation achieved with logarithmic overhead.

Abstract

Estimating thermal expectation values of observables is a fundamental task in quantum physics, quantum chemistry, and materials science. While recent quantum algorithms have enabled efficient quantum preparation of thermal states, observable estimation via sampling remains costly: a straightforward implementation separates successive measurements by a full mixing time in order to ensure samples are approximately independent. In this work, we show that the sampling cost can be substantially reduced by using a single Gibbs-sampling trajectory. After a single burn-in period, we interleave coherent measurements that satisfy detailed balance with respect to the target Gibbs state. The efficiency of this approach rests on the fact that, in many settings, the autocorrelation time can be significantly shorter than the mixing time. For energy estimation (and more generally for observables commuting with the Hamiltonian), we implement the required measurements using Gaussian-filtered quantum phase estimation with only logarithmic overhead. We also introduce a weighted operator Fourier transform technique to mitigate measurement-induced disturbance for general observables.