The power of unentanglement without destructive interference

Abstract

Stoquasticity, originating in sign-problem-free physical systems, gives rise to $\sf StoqMA$, introduced by Bravyi, Bessen, and Terhal (2006), a quantum-inspired intermediate class between $\sf MA$ and $\sf AM$. Unentanglement similarly gives rise to ${\sf QMA}(2)$, introduced by Kobayashi, Matsumoto, and Yamakami (CJTCS 2009), which generalizes $\sf QMA$ to two unentangled proofs and still has only the trivial $\sf NEXP$ upper bound. In this work, we initiate a systematic study of the power of unentanglement without destructive interference via ${\sf StoqMA}(2)$, the class of unentangled stoquastic Merlin--Arthur proof systems. Beyond its complexity-theoretic interest, ${\sf StoqMA}(2)$ is connected to the optimality of non-negative tensor optimization algorithms. We highlight: 1. ${\sf NP} \subseteq {\sf StoqMA}(2)$ with $\widetilde{O}(\sqrt{n})$-qubit proofs and completenes $1-2^{-{\rm polylog}(n)}$. Conversely, the Sum-of-Squares algorithm of Barak, Kelner, and Steurer (STOC 2014) gives an exponential-time upper bound for ${\sf StoqMA}(2)$. Our tightened analysis shows the optimality of our protocol and the BKS algorithm under ETH. 2. For ${\sf StoqMA}(2)_1$, the parameter dependence in the general ETH-optimal time bound can be exponentially improved, or the bound achieved simultaneously with polynomial space. 3. For logarithmic-size proofs, ${\sf NP} \subseteq {\sf StoqMA}(2)_{\log}$ with completeness $1-O(n^{-2})$ and vanishing gap, while ${\sf StoqMA}(2)_{\log} \subseteq {\sf MA}$. Consequently, quantum-inspired randomness enables \emph{exponentially} shorter unentangled proofs even under the assumption ${\sf MA}={\sf NP}$. Our lower bounds are obtained by stoquastizing the short-proof ${\sf QMA}(2)$ protocols using distribution testing techniques. Our upper bounds for the nearly perfect completeness case are proved via our rectangular closure testing framework.