StoqMA meets distribution testing

TL;DR

This paper establishes new connections between the complexity class StoqMA and distribution testing, showing that certain subclasses are contained in MA and NP, and exploring the completeness of reversible circuit distinguishing.

Contribution

It introduces a novel link between StoqMA and distribution testing, proves containment results for subclasses, and demonstrates StoqMA's completeness for distinguishing reversible circuits.

Findings

Easy-witness StoqMA is contained in MA.

Distinguishing reversible circuits with ancillary randomness is StoqMA-complete.

Variants of StoqMA without random bits are in NP.

Abstract

$\mathsf{StoqMA}$ captures the computational hardness of approximating the ground energy of local Hamiltonians that do not suffer the so-called sign problem. We provide a novel connection between $\mathsf{StoqMA}$ and distribution testing via reversible circuits. First, we prove that easy-witness $\mathsf{StoqMA}$ (viz. $\mathsf{eStoqMA}$, a sub-class of $\mathsf{StoqMA}$) is contained in $\mathsf{MA}$. Easy witness is a generalization of a subset state such that the associated set's membership can be efficiently verifiable, and all non-zero coordinates are not necessarily uniform. This sub-class $\mathsf{eStoqMA}$ contains $\mathsf{StoqMA}$ with perfect completeness ($\mathsf{StoqMA}_1$), which further signifies a simplified proof for $\mathsf{StoqMA}_1 \subseteq \mathsf{MA}$ [BBT06, BT10]. Second, by showing distinguishing reversible circuits with ancillary random bits is $\mathsf{StoqMA}$-complete (as a comparison, distinguishing quantum circuits is $\mathsf{QMA}$-complete [JWB05]), we construct soundness error reduction of $\mathsf{StoqMA}$. Additionally, we show that both variants of $\mathsf{StoqMA}$ that without any ancillary random bit and with perfect soundness are contained in $\mathsf{NP}$. Our results make a step towards collapsing the hierarchy $\mathsf{MA} \subseteq \mathsf{StoqMA} \subseteq \mathsf{SBP}$ [BBT06], in which all classes are contained in $\mathsf{AM}$ and collapse to $\mathsf{NP}$ under derandomization assumptions.