QMA-complete problems for stoquastic Hamiltonians and Markov matrices

TL;DR

This paper demonstrates that certain eigenvalue problems for stochastic matrices and stoquastic Hamiltonians are QMA-complete, and establishes the universality of adiabatic quantum computation using specific excited states and ground states.

Contribution

It introduces new QMA-complete problems related to Markov chains and stoquastic Hamiltonians, and shows the universality of adiabatic quantum computation in these contexts.

Findings

Finding the lowest eigenvalue of a 3-local symmetric stochastic matrix is QMA-complete.

Finding the highest energy of a stoquastic Hamiltonian is QMA-complete.

Adiabatic quantum computation using certain excited states is universal.

Abstract

We show that finding the lowest eigenvalue of a 3-local symmetric stochastic matrix is QMA-complete. We also show that finding the highest energy of a stoquastic Hamiltonian is QMA-complete and that adiabatic quantum computation using certain excited states of a stoquastic Hamiltonian is universal. We also show that adiabatic evolution in the ground state of a stochastic frustration free Hamiltonian is universal. Our results give a new QMA-complete problem arising in the classical setting of Markov chains, and new adiabatically universal Hamiltonians that arise in many physical systems.