Termwise versus globally stoquastic local Hamiltonians: questions of complexity and sign-curing

TL;DR

This paper investigates the complexity of determining global versus termwise stoquasticity in local Hamiltonians, establishing hardness results and exploring sign-curing transformations, with implications for quantum complexity theory.

Contribution

It proves the complexity of deciding global stoquasticity and extends sign-curing methods using Clifford transformations for certain Hamiltonians.

Findings

Stoquastic local Hamiltonian problem is StoqMA-complete.

Deciding global stoquasticity is coNP-hard in a fixed basis.

Deciding global stoquasticity under single-qubit transformations is Sigma_2^p-hard.

Abstract

We elucidate the distinction between global and termwise stoquasticity for local Hamiltonians and prove several complexity results. We show that the stoquastic local Hamiltonian problem is $\textbf{StoqMA}$-complete even for globally stoquastic Hamiltonians. We study the complexity of deciding whether a local Hamiltonian is globally stoquastic or not. In particular, we prove $\textbf{coNP}$-hardness of deciding global stoquasticity in a fixed basis and $\Sigma_2^p$-hardness of deciding global stoquasticity under single-qubit transformations. As a last result, we expand the class of sign-curing transformations by showing how Clifford transformations can sign-cure a class of disordered 1D $XYZ$ Hamiltonians.