Simultaneous Stoquasticity

TL;DR

This paper investigates whether multiple Hamiltonians can be transformed into a simultaneously stoquastic form, revealing that such a transformation is generally NP-hard to determine and has implications for quantum simulation complexity.

Contribution

It establishes the NP-hardness of deciding simultaneous stoquasticity and provides a geometric perspective using generalized Bloch vectors.

Findings

Almost no Hamiltonians can be made simultaneously stoquastic via a unitary.

Deciding simultaneous stoquasticity is NP-hard.

Geometric interpretation in terms of generalized Bloch vectors.

Abstract

Stoquastic Hamiltonians play a role in the computational complexity of the local Hamiltonian problem as well as the study of classical simulability. In particular, stoquastic Hamiltonians can be straightforwardly simulated using Monte Carlo techniques. We address the question of whether two or more Hamiltonians may be made simultaneously stoquastic via a unitary transformation. This question has important implications for the complexity of simulating quantum annealing where quantum advantage is related to the stoquasticity of the Hamiltonians involved in the anneal. We find that for almost all problems no such unitary exists and show that the problem of determining the existence of such a unitary is equivalent to identifying if there is a solution to a system of polynomial (in)equalities in the matrix elements of the initial and transformed Hamiltonians. Solving such a system of equations is NP-hard. We highlight a geometric understanding of this problem in terms of a collection of generalized Bloch vectors.