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arXiv:2609.13032

QMA has perfect completeness

Sabee Grewal, Dorian Rudolph

quant-phcs.CCSubmitted Sep 11, 2026• Announced Sep 14, 2026• 0 citations
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Abstract

We prove $\mathsf{QMA} = \mathsf{QMA_1}$, i.e., every quantum Merlin-Arthur proof system can be made perfectly complete. Our construction uses only Hadamard, Toffoli, and $X$ gates, yielding a universal gate set for $\mathsf{QMA_1}$. As a consequence, quantum $3$-SAT is $\mathsf{QMA}$-complete. The construction relativizes to classical oracles, so known classical-oracle separations of $\mathsf{QMA}$ from $\mathsf{QCMA}$ extend to $\mathsf{QMA_1}$.

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13 pages

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This paper proves that classical-witness quantum Merlin-Arthur proof systems can achieve perfect completeness. That is, ${\QCMA = \QCMA_1}$. This holds under any gate set with which the Hadamard and arbitrary classical reversible transformations can be exactly implemented, \emph{e.g.}, {Hadamard, Toffoli, NOT}{\{\textrm{Hadamard, Toffoli, NOT}\}}{Hadamard, Toffoli, NOT}. The proof is quantumly nonrelativizing, and uses a simple but novel quantum technique that \emph{additively} adjusts the success probability, which may be of independent interest.
Whether the class QMA (Quantum Merlin Arthur) is equal to QMA1, or QMA with one-sided error, has been an open problem for years. This note helps to explain why the problem is difficult, by using ideas from real analysis to give a "quantum oracle" relative to which they are different. As a byproduct, we find that there are facts about quantum complexity classes that are classically relativizing but not quantumly relativizing, among them such "trivial" containments as BQP in ZQEXP.
Whether the class QMA (Quantum Merlin Arthur)\ is equal to QMA_1, or QMA with one-sided error, has been an open problem for years. This note helps to explain why the problem is difficult,\ by using ideas from real analysis to give a "quantum" relative to which QMA \neq QMA_1. As a byproduct, we find that there are facts about quantum complexity classes that are classically relativizing but not quantumly relativizing, among them such "trivial" containments as BQP \subseteq ZQEXP.