On the Complexity of Two Dimensional Commuting Local Hamiltonians

TL;DR

This paper proves that a broad class of two-dimensional commuting local Hamiltonians on surfaces have groundstates that can be efficiently generated by quantum circuits, advancing understanding of their computational complexity.

Contribution

It establishes that 2D CLHs on surface complexes are in NP and their groundstates are efficiently preparable, generalizing previous results and addressing a major open problem.

Findings

Class of 2D CLHs on surface complexes is in NP.

Groundstates of these CLHs can be prepared by efficient quantum circuits.

Generalizes previous special case results to broader classes of 2D CLHs.

Abstract

The complexity of the commuting local Hamiltonians (CLH) problem still remains a mystery after two decades of research of quantum Hamiltonian complexity; it is only known to be contained in NP for few low parameters. Of particular interest is the tightly related question of understanding whether groundstates of CLHs can be generated by efficient quantum circuits. The two problems touch upon conceptual, physical and computational questions, including the centrality of non-commutation in quantum mechanics, quantum PCP and the area law. It is natural to try to address first the more physical case of CLHs embedded on a 2D lattice but this problem too remained open, apart from some very specific cases. Here we consider a wide class of two dimensional CLH instances; these are $k$-local CLHs, for any constant $k$; they are defined on qubits set on the edges of any surface complex, where we require that this surface complex is not too far from being "Euclidean". Each vertex and each face can be associated with an arbitrary term (as long as the terms commute). We show that this class is in NP, and moreover that the groundstates have an efficient quantum circuit that prepares them. This result subsumes that of Schuch [2011] which regarded the special case of $4$-local Hamiltonians on a grid with qubits, and by that it removes the mysterious feature of Schuch's proof which showed containment in NP without providing a quantum circuit for the groundstate and considerably generalizes it. We believe this work and the tools we develop make a significant step towards showing that 2D CLHs are in NP.