Polynomial-time classical simulation of quantum ferromagnets

TL;DR

This paper demonstrates that the partition functions of certain quantum ferromagnetic spin models can be efficiently approximated classically, enabling polynomial-time estimation of free energy and ground energy.

Contribution

It introduces a classical randomized algorithm for approximating the partition function of a family of quantum ferromagnets with special graph structures.

Findings

Efficient classical approximation of partition functions for quantum ferromagnets.

Polynomial-time algorithms for free energy and ground energy estimation.

Utilization of perfect matching sums and specialized graph structures.

Abstract

We consider a family of quantum spin systems which includes as special cases the ferromagnetic XY model and ferromagnetic Ising model on any graph, with or without a transverse magnetic field. We prove that the partition function of any model in this family can be efficiently approximated to a given relative error E using a classical randomized algorithm with runtime polynomial in 1/E, system size, and inverse temperature. As a consequence we obtain a polynomial time algorithm which approximates the free energy or ground energy to a given additive error. We first show how to approximate the partition function by the perfect matching sum of a finite graph with positive edge weights. Although the perfect matching sum is not known to be efficiently approximable in general, the graphs obtained by our method have a special structure which facilitates efficient approximation via a randomized algorithm due to Jerrum and Sinclair.