Complexity of commuting Hamiltonians on a square lattice of qubits

TL;DR

This paper investigates the complexity of commuting Hamiltonians on a qubit lattice, showing that verifying if a ground state minimizes local energies is in NP, with a classical certificate confirming this property.

Contribution

It introduces a new classical certificate for verifying ground states of commuting Hamiltonians without requiring state preparation instructions.

Findings

Deciding local energy minimization is in NP for these Hamiltonians.

A classical certificate exists that verifies the ground state property.

This approach differs from previous work by not providing a state preparation method.

Abstract

We consider the computational complexity of Hamiltonians which are sums of commuting terms acting on plaquettes in a square lattice of qubits, and we show that deciding whether the ground state minimizes the energy of each local term individually is in the complexity class NP. That is, if the ground states has this property, this can be proven using a classical certificate which can be efficiently verified on a classical computer. Different to previous results on commuting Hamiltonians, our certificate proves the existence of such a state without giving instructions on how to prepare it.