Computing partition functions in the one clean qubit model

TL;DR

This paper introduces a quantum algorithm for approximating partition functions of quantum systems using the one clean qubit model, potentially outperforming classical methods for large systems.

Contribution

It presents a novel quantum approach to estimate partition functions with near-linear expected runtime and establishes the problem's complexity within the DQC1 class.

Findings

Method has expected runtime almost linear in system size and inverse precision.

Partition function estimation is complete for the DQC1 complexity class.

Classical approximation procedures developed for desired precision.

Abstract

We present a method to approximate partition functions of quantum systems using mixed-state quantum computation. For positive semi-definite Hamiltonians, our method has expected running-time that is almost linear in $(M/(\epsilon_{\rm rel}\mathcal{Z} ))^2$, where $M$ is the dimension of the quantum system, $\mathcal{Z}$ is the partition function, and $\epsilon_{\rm rel}$ is the relative precision. It is based on approximations of the exponential operator as linear combinations of certain operators related to block-encoding of Hamiltonians or Hamiltonian evolutions. The trace of each operator is estimated using a standard algorithm in the one clean qubit model. For large values of $\mathcal{Z}$, our method may run faster than exact classical methods, whose complexities are polynomial in $M$. We also prove that a version of the partition function estimation problem within additive error is complete for the so-called DQC1 complexity class, suggesting that our method provides a super-polynomial speedup for certain parameter values. To attain a desired relative precision, we develop a classical procedure based on a sequence of approximations within predetermined additive errors that may be of independent interest.