On the complexity of quantum partition functions

TL;DR

This paper investigates the computational complexity of approximating quantum partition functions and free energy, introduces a polynomial-time classical algorithm for dense 2-local Hamiltonians, and explores the problem's equivalence to quantum approximate counting tasks.

Contribution

It presents a new polynomial-time classical algorithm for dense 2-local Hamiltonians and establishes complexity equivalences with quantum approximate counting problems.

Findings

A polynomial-time classical algorithm approximates free energy for certain dense Hamiltonians.

Approximate counting of witness states in QMA is polynomial-time equivalent to free energy approximation.

State-of-the-art algorithms can be improved in runtime and memory for quantum free energy estimation.

Abstract

The partition function and free energy of a quantum many-body system determine its physical properties in thermal equilibrium. Here we study the computational complexity of approximating these quantities for $n$-qubit local Hamiltonians. First, we report a classical algorithm with $\mathrm{poly}(n)$ runtime which approximates the free energy of a given $2$-local Hamiltonian provided that it satisfies a certain denseness condition. Our algorithm combines the variational characterization of the free energy and convex relaxation methods. It contributes to a body of work on efficient approximation algorithms for dense instances of optimization problems which are hard in the general case, and can be viewed as simultaneously extending existing algorithms for (a) the ground energy of dense $2$-local Hamiltonians, and (b) the free energy of dense classical Ising models. Secondly, we establish polynomial-time equivalence between the problem of approximating the free energy of local Hamiltonians and three other natural quantum approximate counting problems, including the problem of approximating the number of witness states accepted by a QMA verifier. These results suggest that simulation of quantum many-body systems in thermal equilibrium may precisely capture the complexity of a broad family of computational problems that has yet to be defined or characterized in terms of known complexity classes. Finally, we summarize state-of-the-art classical and quantum algorithms for approximating the free energy and show how to improve their runtime and memory footprint.