# StoqMA meets distribution testing

π²ππππ¬π 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 π²ππππ¬π and the distribution testing via reversible circuits. First, we prove that easy-witness π²ππππ¬π (viz. πΎπ²ππππ¬π , a sub-class of π²ππππ¬π ) is contained in π¬π . 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 π²ππππ¬π -complete (as a comparison, distinguishing quantum circuits is π°π¬π -complete [JWB05]), we construct soundness error reduction of π²ππππ¬π . This new π²ππππ¬π -complete problem further signifies that π²ππππ¬π with perfect completeness (π²ππππ¬π _1) is contained in πΎπ²ππππ¬π , which leads us to an alternating proof for π²ππππ¬π _1 βπ¬π previously proved in [BBT06, BT10]. Additionally, we show that both variants of π²ππππ¬π that without any random ancillary bit and with perfect soundness are contained in ππ―. Our results make a step towards collapsing the hierarchy π¬π βπ²ππππ¬π βπ²π‘π― [BBT06], in which all classes are contained in π π¬ and collapse to ππ― under derandomization assumptions.

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