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On Perfect Completeness for QMA

Scott Aaronson

Published Jun 3, 2008• 3 citations
OpenAlex

Abstract

Whether the class QMA (Quantum Merlin Arthur) is equal to QMA1, or QMA with onesided 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 QMA ≠ QMA1. 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 ⊆ ZQEXP.

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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.

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Non-Standard Oracles for Bounded-Error Complexity Classes

Avantika Agarwal, Srijita Kundu

In recent years, the quantum oracle model introduced by Aaronson and Kuperberg (2007) has found a lot of use in showing oracle separations between complexity classes and cryptographic primitives. It is generally assumed that proof techniques that do not relativize with respect to quantum oracles will also not relativize with respect to classical oracles. Aaronson (2009) showed that this is not the case by showing a complexity class containment that relativizes with respect to classical oracles but not with quantum oracles. However, their result only works for zero-error quantum complexity classes and they leave open the problem for bounded-error complexity classes. We show that there is a quantum oracle problem that is contained in the class QMA, but not in a class we call polyQCPH. However, with respect to classical oracles, QMA is contained in polyQCPH, because polyQCPH is equal to PSPACE with respect to classical oracles. Our result works for polyQCPH, which is a bounded-error complexity class, thus it resolves the open problem from Aaronson (2009). We also show that the same separation holds relative to a distributional oracle, which is a model introduced by Natarajan and Nirkhe (2024). We believe our findings show the need for some caution when using these non-standard oracle models, particularly when showing separations between quantum and classical resources.

arXiv:2504.19470•Apr 28, 2025•0 citations
Whether the class QMA (Quantum Merlin Arthur) is equal to QMA1, or QMA with onesided 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 QMA ≠ QMA1. 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 ⊆ ZQEXP.
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.
We prove QMA=QMA1\mathsf{QMA} = \mathsf{QMA_1}QMA=QMA1​, i.e., every quantum Merlin-Arthur proof system can be made perfectly complete. Our construction uses only Hadamard, Toffoli, and XXX gates, yielding a universal gate set for QMA1\mathsf{QMA_1}QMA1​. As a consequence, quantum 333-SAT is QMA\mathsf{QMA}QMA-complete. The construction relativizes to classical oracles, so known classical-oracle separations of QMA\mathsf{QMA}QMA from QCMA\mathsf{QCMA}QCMA extend to QMA1\mathsf{QMA_1}QMA1​.
In recent years, the quantum oracle model introduced by Aaronson and Kuperberg (2007) has found a lot of use in showing oracle separations between complexity classes and cryptographic primitives. It is generally assumed that proof techniques that do not relativize with respect to quantum oracles will also not relativize with respect to classical oracles. Aaronson (2009) showed that this is not the case by showing a complexity class containment that relativizes with respect to classical oracles but not with quantum oracles. However, their result only works for zero-error quantum complexity classes and they leave open the problem for bounded-error complexity classes. We show that there is a quantum oracle problem that is contained in the class QMA, but not in a class we call polyQCPH. However, with respect to classical oracles, QMA is contained in polyQCPH, because polyQCPH is equal to PSPACE with respect to classical oracles. Our result works for polyQCPH, which is a bounded-error complexity class, thus it resolves the open problem from Aaronson (2009). We also show that the same separation holds relative to a distributional oracle, which is a model introduced by Natarajan and Nirkhe (2024). We believe our findings show the need for some caution when using these non-standard oracle models, particularly when showing separations between quantum and classical resources.