The Complexity of Stoquastic Sparse Hamiltonians

TL;DR

This paper proves that the problem of Stoquastic Sparse Hamiltonians is complete for the complexity class StoqMA, and its separable version is complete for StoqMA(2), advancing understanding of Hamiltonian complexity classes.

Contribution

It establishes the StoqMA-completeness of the Stoquastic Sparse Hamiltonians problem and the StoqMA(2)-completeness of its separable variant.

Findings

Stoquastic Sparse Hamiltonians problem is in StoqMA.

Stoquastic Local Hamiltonians are StoqMA-hard.

Separable version of the problem is StoqMA(2)-complete.

Abstract

Despite having an unnatural definition, $\mathsf{StoqMA}$ plays a central role in Hamiltonian complexity, e.g., in the classification theorem of the complexity of Hamiltonians by Cubitt and Montanaro (SICOMP 2016). Moreover, it lies between the two randomized extensions of $\mathsf{NP}$, $\mathsf{MA}$ and $\mathsf{AM}$. Therefore, understanding the exact power of $\mathsf{StoqMA}$ (and hopefully collapsing it with more natural complexity classes) is of great interest for different reasons. In this work, we take a step further in understanding this complexity class by showing that the Stoquastic Sparse Hamiltonians problem ($\mathsf{StoqSH}$) is in $\mathsf{StoqMA}$. Since Stoquastic Local Hamiltonians are $\mathsf{StoqMA}$-hard, this implies that $\mathsf{StoqSH}$ is $\mathsf{StoqMA}$-complete. We complement this result by showing that the separable version of $\mathsf{StoqSH}$ is $\mathsf{StoqMA}(2)$-complete, where $\mathsf{StoqMA}(2)$ is the version of $\mathsf{StoqMA}$ that receives two unentangled proofs.