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StoqMA meets distribution testing
arXiv - CS - Computational Complexity Pub Date : 2020-11-11 , DOI: arxiv-2011.05733
Yupan Liu

$\mathsf{StoqMA}$ 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 the 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. Second, by showing distinguishing reversible circuits with random ancillary bits is $\mathsf{StoqMA}$-complete (as a comparison, distinguishing quantum circuits is $\mathsf{QMA}$-complete [JWB05]), we construct soundness error reduction of $\mathsf{StoqMA}$. This new $\mathsf{StoqMA}$-complete problem further signifies that $\mathsf{StoqMA}$ with perfect completeness ($\mathsf{StoqMA}_1$) is contained in $\mathsf{eStoqMA}$, which leads us to an alternating proof for $\mathsf{StoqMA}_1 \subseteq \mathsf{MA}$ previously proved in [BBT06, BT10]. Additionally, we show that both variants of $\mathsf{StoqMA}$ that without any random ancillary 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.

中文翻译:

StoqMA 满足分布测试

$\mathsf{StoqMA}$ 捕获了近似不遭受所谓符号问题的局部哈密顿量的地面能量的计算难度。我们通过可逆电路提供了 $\mathsf{StoqMA}$ 和分布测试之间的新联系。首先,我们证明容易见证的 $\mathsf{StoqMA}$(即 $\mathsf{eStoqMA}$,$\mathsf{StoqMA}$ 的一个子类)包含在 $\mathsf{MA}$ 中。简单见证是子集状态的概括,这样关联集合的成员资格可以有效地验证,并且所有非零坐标不一定是统一的。其次,通过显示区分具有随机辅助位的可逆电路是 $\mathsf{StoqMA}$-complete(作为比较,区分量子电路是 $\mathsf{QMA}$-complete [JWB05]),我们构建了稳健性误差减少$\mathsf{StoqMA}$。这个新的 $\mathsf{StoqMA}$-complete 问题进一步表明具有完美完备性 ($\mathsf{StoqMA}_1$) 的 $\mathsf{StoqMA}$ 包含在 $\mathsf{eStoqMA}$ 中,这导致我们$\mathsf{StoqMA}_1 \subseteq \mathsf{MA}$ 的交替证明先前在 [BBT06, BT10] 中证明。此外,我们表明 $\mathsf{StoqMA}$ 的两个变体都包含在 $\mathsf{NP}$ 中,它们没有任何随机辅助位且具有完美的稳健性。我们的结果朝着折叠层次结构迈出了一步 $\mathsf{MA} \subseteq \mathsf{StoqMA} \subseteq \mathsf{SBP}$ [BBT06],其中所有类都包含在 $\mathsf{AM}$ 中并折叠到 $\mathsf{NP}$ 在去随机化假设下。我们表明 $\mathsf{StoqMA}$ 的两个变体都包含在 $\mathsf{NP}$ 中,它们没有任何随机辅助位并且具有完美的稳健性。我们的结果朝着折叠层次结构迈出了一步 $\mathsf{MA} \subseteq \mathsf{StoqMA} \subseteq \mathsf{SBP}$ [BBT06],其中所有类都包含在 $\mathsf{AM}$ 中并折叠到 $\mathsf{NP}$ 在去随机化假设下。我们表明 $\mathsf{StoqMA}$ 的两个变体都包含在 $\mathsf{NP}$ 中,它们没有任何随机辅助位并且具有完美的稳健性。我们的结果朝着折叠层次结构迈出了一步 $\mathsf{MA} \subseteq \mathsf{StoqMA} \subseteq \mathsf{SBP}$ [BBT06],其中所有类都包含在 $\mathsf{AM}$ 中并折叠到 $\mathsf{NP}$ 在去随机化假设下。
更新日期:2020-11-12
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