The complexity of antiferromagnetic interactions and 2D lattices

TL;DR

This paper investigates the computational complexity of quantum Hamiltonian problems with antiferromagnetic interactions on 2D lattices, showing many cases are QMA-complete, highlighting the difficulty of simulating such quantum systems.

Contribution

It classifies the complexity of 2-local Hamiltonian problems with antiferromagnetic interactions on various lattice geometries, identifying many as QMA-complete.

Findings

Antiferromagnetic Heisenberg and XY interactions are QMA-complete.

StoqMA-completeness shown for antiferromagnetic transverse field Ising model.

Most QMA-complete interactions remain complex on 2D lattices.

Abstract

Estimation of the minimum eigenvalue of a quantum Hamiltonian can be formalised as the Local Hamiltonian problem. We study the natural special case of the Local Hamiltonian problem where the same 2-local interaction, with differing weights, is applied across each pair of qubits. First we consider antiferromagnetic/ferromagnetic interactions, where the weights of the terms in the Hamiltonian are restricted to all be of the same sign. We show that for symmetric 2-local interactions with no 1-local part, the problem is either QMA-complete or in StoqMA. In particular the antiferromagnetic Heisenberg and antiferromagnetic XY interactions are shown to be QMA-complete. We also prove StoqMA-completeness of the antiferromagnetic transverse field Ising model. Second, we study the Local Hamiltonian problem under the restriction that the interaction terms can only be chosen to lie on a particular graph. We prove that nearly all of the QMA-complete 2-local interactions remain QMA-complete when restricted to a 2D square lattice. Finally we consider both restrictions at the same time and discover that, with the exception of the antiferromagnetic Heisenberg interaction, all of the interactions which are QMA-complete with positive coefficients remain QMA-complete when restricted to a 2D triangular lattice.