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Physical Review Letters

Quantum-Merlin-Arthur Problems Have Perfect Completeness with an Infinite Counter

Stacey Jeffery, Freek Witteveen

Published Apr 2, 2026• 0 citations
Publisher

Abstract

A longstanding open problem in quantum complexity theory is whether Quantum Merlin-Arthur (QMA), the quantum analog of nondeterministic polynomial time, is equal to QMA_{1}, its one-sided error variant. We show that QMA=QMA^{∞}=QMA_{1}^{∞}, where QMA_{1}^{∞} is like QMA_{1}, but the verifier has an infinite register, as part of their witness system, in which they can efficiently perform a shift (increment) operation. We call this register an "infinite counter," and compare it to a program counter in a Las Vegas algorithm. The result, QMA=QMA^{∞} means such an infinite register does not increase the power of QMA, but does imply perfect completeness. By truncating our construction to finite dimensions, we get a QMA-amplifier that only amplifies completeness, not soundness, but does so in significantly less time than previous QMA amplifiers. Our new construction achieves completeness 1-2^{-q} using O(1) calls to each of the original verifier and its inverse, and O(logq) other gates, proving that QMA has completeness doubly exponentially close to 1, i.e., QMA=QMA(1-2^{-2^{r}},2^{-r}) for any polynomial r.

DOI
10.1103/pwdd-htbf

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A longstanding open problem in quantum complexity theory is whether Quantum Merlin-Arthur (QMA), the quantum analog of nondeterministic polynomial time, is equal to QMA_{1}, its one-sided error variant. We show that QMA=QMA^{∞}=QMA_{1}^{∞}, where QMA_{1}^{∞} is like QMA_{1}, but the verifier has an infinite register, as part of their witness system, in which they can efficiently perform a shift (increment) operation. We call this register an "infinite counter," and compare it to a program counter in a Las Vegas algorithm. The result, QMA=QMA^{∞} means such an infinite register does not increase the power of QMA, but does imply perfect completeness. By truncating our construction to finite dimensions, we get a QMA-amplifier that only amplifies completeness, not soundness, but does so in significantly less time than previous QMA amplifiers. Our new construction achieves completeness 1-2^{-q} using O(1) calls to each of the original verifier and its inverse, and O(logq) other gates, proving that QMA has completeness doubly exponentially close to 1, i.e., QMA=QMA(1-2^{-2^{r}},2^{-r}) for any polynomial r.
QMA=QMA1{\sf QMA}={\sf QMA}_1QMA=QMA1​ with an infinite counter
A long-standing open problem in quantum complexity theory is whether QMA{\sf QMA}QMA, the quantum analogue of NP{\sf NP}NP, is equal to QMA1{\sf QMA}_1QMA1​, its one-sided error variant. We show that QMA=QMA∞=QMA1∞{\sf QMA}={\sf QMA}^{\infty}= {\sf QMA}_1^{\infty}QMA=QMA∞=QMA1∞​, where QMA1∞{\sf QMA}_1^\inftyQMA1∞​ is like QMA1{\sf QMA}_1QMA1​, but the verifier has an infinite register, as part of their witness system, in which they can efficiently perform a shift (increment) operation. We call this register an ``infinite counter'', and compare it to a program counter in a Las Vegas algorithm. The result QMA=QMA∞{\sf QMA}={\sf QMA}^\inftyQMA=QMA∞ means such an infinite register does not increase the power of QMA{\sf QMA}QMA, but does imply perfect completeness. By truncating our construction to finite dimensions, we get a QMA{\sf QMA}QMA-amplifier that only amplifies completeness, not soundness, but does so in significantly less time than previous QMA{\sf QMA}QMA amplifiers. Our new construction achieves completeness 1−2−q1-2^{-q}1−2−q using O(1)O(1)O(1) calls to each of the original verifier and its inverse, and O(log⁡q)O(\log q)O(logq) other gates, proving that QMA{\sf QMA}QMA has completeness doubly exponentially close to 1, i.e. QMA=QMA(1−2−2r,2−r){\sf QMA}={\sf QMA}(1-2^{-2^r},2^{-r})QMA=QMA(1−2−2r,2−r) for any polynomial rrr.
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.
This paper studies quantum Arthur-Merlin games, which are a restricted form of quantum interactive proof system in which the verifier's messages are given by unbiased coin-flips. The following results are proved. For one-message quantum Arthur-Merlin games, which correspond to the complexity class QMA, completeness and soundness errors can be reduced exponentially without increasing the length of Merlin's message. Previous constructions for reducing error required a polynomial increase in the length of Merlin's message. Applications of this fact include a proof that logarithmic length quantum certificates yield no increase in power over BQP and a simple proof that QMA /spl sube/ PP. In the case of three or more messages, quantum Arthur-Merlin games are equivalent in power to ordinary quantum interactive proof systems. In fact, for any language having a quantum interactive proof system there exists a three-message quantum Arthur-Merlin game in which Arthur's only message consists of just a single coin-flip that achieves perfect completeness and soundness error exponentially close to 1/2. Any language having a two-message quantum Arthur-Merlin game is contained in BP /spl middot/ PP. This gives some suggestion that three messages are stronger than two in the quantum Arthur-Merlin setting.
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.
This paper studies quantum Arthur-Merlin games, which are Arthur-Merlin games in which Arthur and Merlin can perform quantum computations and Merlin can send Arthur quantum information. As in the classical case, messages from Arthur to Merlin are restricted to be strings of uniformly generated random bits. It is proved that for one-message quantum Arthur-Merlin games, which correspond to the complexity class QMA, completeness and soundness errors can be reduced exponentially without increasing the length of Merlin's message. Previous constructions for reducing error required a polynomial increase in the length of Merlin's message. Applications of this fact include a proof that logarithmic length quantum certificates yield no increase in power over BQP and a simple proof that QMA is contained in PP. Other facts that are proved include the equivalence of three (or more) message quantum Arthur-Merlin games with ordinary quantum interactive proof systems and some basic properties concerning two-message quantum Arthur-Merlin games.
This paper studies quantum Arthur–Merlin games, which are Arthur–Merlin games in which Arthur and Merlin can perform quantum computations and Merlin can send Arthur quantum information. As in the classical case, messages from Arthur to Merlin are restricted to be strings of uniformly generated random bits. It is proved that for one-message quantum Arthur–Merlin games, which correspond to the complexity class QMA, completeness and soundness errors can be reduced exponentially without increasing the length of Merlin’s message. Previous constructions for reducing error required a polynomial increase in the length of Merlin’s message. Applications of this fact include a proof that logarithmic length quantum certificates yield no increase in power over BQP and a simple proof that QMA⊆PP. {\text{QMA}} \subseteq {\text{PP}}. QMA⊆PP. Other facts that are proved include the equivalence of three (or more) message quantum Arthur–Merlin games with ordinary quantum interactive proof systems and some basic properties concerning two-message quantum Arthur–Merlin games.