Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

On QMA Protocols with Two Short Quantum Proofs (New Trends in Algorithms and Theory of Computation)

François Le Gall, Shota Nakagawa, Harumichi Nishimura

Published Jun 1, 2012• 1 citation
OpenAlex

Abstract

This paper gives $a$ .QMA (Quantum Merlin-Arthur) protocol for 3-SAT with two logarith- mic-size quantum proofs (that are not entangled with each other) such that the gap between the completeness and the soundness is $\Omega(\frac{1}{npoly\log(n)})$ .This improves the best completeness/soundness gaps known for NP-complete problems in this setting.

Related papers

On QMA protocols with two short quantum proofs

Francois Le Gall, Shota Nakagawa, Harumichi Nishimura

This paper gives a QMA (Quantum Merlin-Arthur) protocol for 3-SAT with two logarithmic-size quantum proofs (that are not entangled with each other) such that the gap between the completeness and the soundness is $\Omega(\frac{1}{n\polylog{n}})$. This improves the best completeness/soundness gaps known for NP-complete problems in this setting.

arXiv:1108.4306•Quantum Information and Computation•Jul 1, 2012•25 citations

NP vs QMA_\log(2)

Salman Beigi

Although it is believed unlikely that $\NP$-hard problems admit efficient quantum algorithms, it has been shown that a quantum verifier can solve NP-complete problems given a "short" quantum proof; more precisely, NP\subseteq QMA_{\log}(2) where QMA_{\log}(2) denotes the class of quantum Merlin-Arthur games in which there are two unentangled provers who send two logarithmic size quantum witnesses to the verifier. The inclusion NP\subseteq QMA_{\log}(2) has been proved by Blier and Tapp by stating a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap 1/24n^6. Moreover, Aaronson et al. have shown the above inclusion with a constant gap by considering $\widetilde{O}(\sqrt{n})$ witnesses of logarithmic size. However, we still do not know if QMA_{\log}(2) with a constant gap contains NP. In this paper, we show that 3-SAT admits a QMA_{\log}(2) protocol with the gap 1/n^{3+\epsilon}} for every constant \epsilon>0.

Quantum Information and Computation•Jan 1, 2010•26 citations

NP vs QMA_log(2)

Salman Beigi

Although it is believed unlikely that $\NP$-hard problems admit efficient quantum algorithms, it has been shown that a quantum verifier can solve $\NP$-complete problems given a short quantum proof; more precisely, $\NP\subseteq \QMA_{\log}(2)$ where $\QMA_{\log}(2)$ denotes the class of quantum Merlin-Arthur games in which there are two unentangled provers who send two logarithmic size quantum witnesses to the verifier. The inclusion $\NP\subseteq \QMA_{\log}(2)$ has been proved by Blier and Tapp by stating a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap $\frac{1}{24n^6}$. Moreover, Aaronson {\it et al.} have shown the above inclusion with a constant gap by considering $\widetilde{O}(\sqrt{n})$ witnesses of logarithmic size. However, we still do not know if $\QMA_{\log}(2)$ with a constant gap contains $\NP$. In this paper, we show that 3-SAT admits a $\QMA_{\log}(2)$ protocol with the gap $\frac{1}{n^{3+\epsilon}}$ for every constant $\epsilon>0$.

•Oct 28, 2008•9 citations

NP vs QMA_log(2)

Salman Beigi

Although it is believed unlikely that $\NP$-hard problems admit efficient quantum algorithms, it has been shown that a quantum verifier can solve $\NP$-complete problems given a "short" quantum proof; more precisely, $\NP\subseteq \QMA_{\log}(2)$ where $\QMA_{\log}(2)$ denotes the class of quantum Merlin-Arthur games in which there are two unentangled provers who send two logarithmic size quantum witnesses to the verifier. The inclusion $\NP\subseteq \QMA_{\log}(2)$ has been proved by Blier and Tapp by stating a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap $\frac{1}{24n^6}$. Moreover, Aaronson {\it et al.} have shown the above inclusion with a constant gap by considering $\widetilde{O}(\sqrt{n})$ witnesses of logarithmic size. However, we still do not know if $\QMA_{\log}(2)$ with a constant gap contains $\NP$. In this paper, we show that 3-SAT admits a $\QMA_{\log}(2)$ protocol with the gap $\frac{1}{n^{3+ε}}$ for every constant $ε>0$.

arXiv:0810.5109•Oct 28, 2008•0 citations

QMA has perfect completeness

Sabee Grewal, Dorian Rudolph

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 $\mathsf{QMA_1}$.

quant-pharXiv:2609.13032•Sep 14, 2026•0 citations

Short Multi-Prover Quantum Proofs for SAT without Entangled Measurements

Jing Chen, Andrew Drucker

BellQMA protocols are a subclass of multi-prover quantum Merlin-Arthur protocols in which the verifier is restricted to perform nonadaptive,unentangled measurements on the quantum states received from each Merlin. In this paper, we prove that m-clause 3-SAT instances have BellQMA proofs of satisfiability with constant soundness gap, in which O(sqrt(m)polylog(m)) Merlins each send O(log m) qubits to Arthur. Our result answers a question of Aaronson et al., who gave a protocol with similar parameters that used entangled measurements; the analysis of our protocol is significantly simpler than that of Aaronson et al. Our result also complements recent work of Brandao, Christandl, and Yard, who showed upper bounds on the power of multiprover quantum proofs with unentangled but adaptive (LOCC) measurements.

arXiv:1011.0716•Nov 2, 2010•20 citations
This paper gives aaa .QMA (Quantum Merlin-Arthur) protocol for 3-SAT with two logarith- mic-size quantum proofs (that are not entangled with each other) such that the gap between the completeness and the soundness is Ω(1npolylog⁡(n))\Omega(\frac{1}{npoly\log(n)})Ω(npolylog(n)1​) .This improves the best completeness/soundness gaps known for NP-complete problems in this setting.
Although it is believed unlikely that $\NP$-hard problems admit efficient quantum algorithms, it has been shown that a quantum verifier can solve NP-complete problems given a "short" quantum proof; more precisely, NP\subseteq QMA_{\log}(2) where QMA_{\log}(2) denotes the class of quantum Merlin-Arthur games in which there are two unentangled provers who send two logarithmic size quantum witnesses to the verifier. The inclusion NP\subseteq QMA_{\log}(2) has been proved by Blier and Tapp by stating a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap 1/24n^6. Moreover, Aaronson et al. have shown the above inclusion with a constant gap by considering O~(n)\widetilde{O}(\sqrt{n})O(n​) witnesses of logarithmic size. However, we still do not know if QMA_{\log}(2) with a constant gap contains NP. In this paper, we show that 3-SAT admits a QMA_{\log}(2) protocol with the gap 1/n^{3+\epsilon}} for every constant \epsilon>0.
Although it is believed unlikely that $\NP$-hard problems admit efficient quantum algorithms, it has been shown that a quantum verifier can solve $\NP$-complete problems given a short quantum proof; more precisely, $\NP\subseteq \QMA_{\log}(2)$ where $\QMA_{\log}(2)$ denotes the class of quantum Merlin-Arthur games in which there are two unentangled provers who send two logarithmic size quantum witnesses to the verifier. The inclusion $\NP\subseteq \QMA_{\log}(2)$ has been proved by Blier and Tapp by stating a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap 124n6\frac{1}{24n^6}24n61​. Moreover, Aaronson {\it et al.} have shown the above inclusion with a constant gap by considering O~(n)\widetilde{O}(\sqrt{n})O(n​) witnesses of logarithmic size. However, we still do not know if $\QMA_{\log}(2)$ with a constant gap contains $\NP$. In this paper, we show that 3-SAT admits a $\QMA_{\log}(2)$ protocol with the gap 1n3+ϵ\frac{1}{n^{3+\epsilon}}n3+ϵ1​ for every constant ϵ>0\epsilon>0ϵ>0.
Although it is believed unlikely that $\NP$-hard problems admit efficient quantum algorithms, it has been shown that a quantum verifier can solve $\NP$-complete problems given a "short" quantum proof; more precisely, $\NP\subseteq \QMA_{\log}(2)$ where $\QMA_{\log}(2)$ denotes the class of quantum Merlin-Arthur games in which there are two unentangled provers who send two logarithmic size quantum witnesses to the verifier. The inclusion $\NP\subseteq \QMA_{\log}(2)$ has been proved by Blier and Tapp by stating a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap 124n6\frac{1}{24n^6}24n61​. Moreover, Aaronson {\it et al.} have shown the above inclusion with a constant gap by considering O~(n)\widetilde{O}(\sqrt{n})O(n​) witnesses of logarithmic size. However, we still do not know if $\QMA_{\log}(2)$ with a constant gap contains $\NP$. In this paper, we show that 3-SAT admits a $\QMA_{\log}(2)$ protocol with the gap 1n3+ε\frac{1}{n^{3+ε}}n3+ε1​ for every constant ε>0ε>0ε>0.
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​.
BellQMA protocols are a subclass of multi-prover quantum Merlin-Arthur protocols in which the verifier is restricted to perform nonadaptive,unentangled measurements on the quantum states received from each Merlin. In this paper, we prove that m-clause 3-SAT instances have BellQMA proofs of satisfiability with constant soundness gap, in which O(sqrt(m)polylog(m)) Merlins each send O(log m) qubits to Arthur. Our result answers a question of Aaronson et al., who gave a protocol with similar parameters that used entangled measurements; the analysis of our protocol is significantly simpler than that of Aaronson et al. Our result also complements recent work of Brandao, Christandl, and Yard, who showed upper bounds on the power of multiprover quantum proofs with unentangled but adaptive (LOCC) measurements.