Quantum algorithms for classical lattice models

TL;DR

This paper introduces efficient quantum algorithms for estimating partition functions of various classical lattice models and proves these problems are BQP-complete, linking them to the complexity of quantum computation.

Contribution

The paper develops quantum algorithms for classical lattice models and establishes their BQP-completeness, highlighting their computational hardness.

Findings

Quantum algorithms efficiently estimate partition functions.

Problems are proven to be BQP-complete.

Results apply to complex parameter regimes.

Abstract

We give efficient quantum algorithms to estimate the partition function of (i) the six vertex model on a two-dimensional (2D) square lattice, (ii) the Ising model with magnetic fields on a planar graph, (iii) the Potts model on a quasi 2D square lattice, and (iv) the Z_2 lattice gauge theory on a three-dimensional square lattice. Moreover, we prove that these problems are BQP-complete, that is, that estimating these partition functions is as hard as simulating arbitrary quantum computation. The results are proven for a complex parameter regime of the models. The proofs are based on a mapping relating partition functions to quantum circuits introduced in [Van den Nest et al., Phys. Rev. A 80, 052334 (2009)] and extended here.