StoqMA meets distribution testing

• URL: http://arxiv.org/abs/2011.05733v3
• Date: Tue, 22 Jun 2021 07:58:46 GMT
• Title: StoqMA meets distribution testing
• Authors: Yupan Liu
• Abstract summary: We provide a novel connection between $mathsfStoqMA$ and distribution testing via reversible circuits. We show that both variants of $mathsfStoqMA$ that without any ancillary random bit and with perfect soundness are contained in $mathsfNP$. Our results make a step towards collapsing the hierarchy $mathsfMA subseteq mathsfStoqMA subseteq mathsfSBP$ [BBT06], in which all classes are contained in $
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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