Complexity of geometrically local stoquastic Hamiltonians

TL;DR

This paper proves that approximating the ground state energy of geometrically local stoquastic Hamiltonians in one and two dimensions remains computationally hard, specifically MA-hard and StoqMA-complete, highlighting their classical complexity.

Contribution

It establishes the MA-hardness and StoqMA-completeness of geometrically local stoquastic Hamiltonians in 1D and 2D, extending understanding of their computational complexity.

Findings

MA-hardness for 1D and 2D geometrically local stoquastic Hamiltonians

StoqMA-completeness of related problems

Hardness persists with high qudit dimension

Abstract

The QMA-completeness of the local Hamiltonian problem is a landmark result of the field of Hamiltonian complexity that studies the computational complexity of problems in quantum many-body physics. Since its proposal, substantial effort has been invested in better understanding the problem for physically motivated important families of Hamiltonians. In particular, the QMA-completeness of approximating the ground state energy of local Hamiltonians has been extended to the case where the Hamiltonians are geometrically local in one and two spatial dimensions. Among those physically motivated Hamiltonians, stoquastic Hamiltonians play a particularly crucial role, as they constitute the manifestly sign-free Hamiltonians in Monte Carlo approaches. Interestingly, for such Hamiltonians, the problem at hand becomes more ''classical'', being hard for the class MA (the randomized version of NP) and its complexity has tight connections with derandomization. In this work, we prove that both the two- and one-dimensional geometrically local analogues remain MA-hard with high enough qudit dimension. Moreover, we show that related problems are StoqMA-complete.