The Complexity of Local Stoquastic Hamiltonians on 2D Lattices

TL;DR

This paper proves that the 2D local stoquastic Hamiltonian problem is StoqMA-complete by extending circuit constructions and gadgets that preserve geometry and stoquasticity without increasing particle dimension.

Contribution

It introduces spatially sparse StoqMA circuits and geometrical perturbative gadgets that maintain stoquasticity and spatial sparsity in 2D lattices.

Findings

Proves 2D local stoquastic Hamiltonian problem is StoqMA-complete.

Extends spatially sparse circuit construction to 2D lattices.

Constructs geometrical, stoquastic-preserving perturbative gadgets without increasing particle dimension.

Abstract

We show the 2-Local Stoquastic Hamiltonian problem on a 2D square qubit lattice is StoqMA-complete. We achieve this by extending the spatially sparse circuit construction of Oliveira and Terhal, as well as the perturbative gadgets of Bravyi, DiVincenzo, Oliveira, and Terhal. Our main contributions demonstrate StoqMA circuits can be made spatially sparse and that geometrical, stoquastic-preserving, perturbative gadgets can be constructed, without an increase to particle dimension.