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arXiv:2410.19152 · Leibniz international proceedings in informatics

Quantum Merlin-Arthur with an Internally Separable Proof

Roozbeh Bassirian, Bill Fefferman, Itai Leigh, Kunal Marwaha, Pei Wu

Published Oct 24, 2024• 0 citations
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Abstract

We find a modification to QMA where having one quantum proof is strictly less powerful than having two unentangled proofs, assuming EXP $\ne$ NEXP. This gives a new route to prove QMA(2) = NEXP that overcomes the primary drawback of a recent approach [arXiv:2402.18790 , arXiv:2306.13247] (QIP 2024). Our modification endows each proof with a form of *multipartite* unentanglement: after tracing out one register, a small number of qubits are separable from the rest of the state.

DOI
10.4230/lipics.tqc.2026.5

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Fernando Granha Jeronimo, Pei Wu

Quantum entanglement is a fundamental property of quantum mechanics and it serves as a basic resource in quantum computation and information. Despite its importance, the power and limitations of quantum entanglement are far from being fully understood. Here, we study entanglement via the lens of computational complexity. This is done by studying quantum generalizations of the class NP with multiple unentangled quantum proofs, the so-called QMA(2) and its variants. The complexity of QMA(2) is known to be closely connected to a variety of problems such as deciding if a state is entangled and several classical optimization problems. However, determining the complexity of QMA(2) is a longstanding open problem, and only the trivial complexity bounds ⊆ (2) ⊆ are known.

arXiv:2402.18790•ACM Symposium on Theory of Computing (STOC)•May 16, 2023•5 citations
We find a modification to QMA where having one quantum proof is strictly less powerful than having two unentangled proofs, assuming EXP ≠\ne= NEXP. This gives a new route to prove QMA(2) = NEXP that overcomes the primary drawback of a recent approach [arXiv:2402.18790 , arXiv:2306.13247] (QIP 2024). Our modification endows each proof with a form of multipartite unentanglement: after tracing out one register, a small number of qubits are separable from the rest of the state.
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​.
We study a variant of QMA where quantum proofs have no relative phase (i.e. non-negative amplitudes, up to a global phase). If only completeness is modified, this class is equal to QMA [arXiv:1410.2882]; but if both completeness and soundness are modified, the class (named QMA+ by Jeronimo and Wu) can be much more powerful. We show that QMA+ with some constant gap is equal to NEXP, yet QMA+ with some *other* constant gap is equal to QMA. One interpretation is that Merlin's ability to "deceive" originates from relative phase at least as much as from entanglement, since QMA(2) ⊆\subseteq⊆ NEXP.
We find a quantum oracle relative to which QMA≠QMA(2)\mathsf{QMA} \neq \mathsf{QMA}(2)QMA=QMA(2). As a consequence, we resolve the no-disentanglers conjecture of Watrous: for every ε+δ<1ε+δ<1ε+δ<1, any (ε,δ)(ε,δ)(ε,δ)-disentangler requires input size exponential in the number of output qubits. Our proof combines the unitarily invariant polynomial method of She and Yuen (ITCS '23) with a new construction based on the symmetric and antisymmetric subspace projectors, reducing the QMA\mathsf{QMA}QMA lower bound to the approximate degree of OR\mathrm{OR}OR.
In QMA(2)\mathsf{QMA}(2)QMA(2), two computationally unbounded provers submit unentangled quantum witnesses to a polynomial-time quantum verifier. We ask whether allowing limited shared entanglement between the two provers changes the power of QMA(2)\mathsf{QMA}(2)QMA(2). To this end, we formalize QMA(2;h)\mathsf{QMA}(2;h)QMA(2;h), a variant in which the provers initially share hhh EPR pairs, as suggested by Aaronson, Beigi, Drucker, Fefferman, and Shor (ToC~2009). We show that polynomially many shared EPR pairs give the provers no cheating advantage: QMA(2)⊆QMA(2;h)\mathsf{QMA}(2)\subseteq\mathsf{QMA}(2;h)QMA(2)⊆QMA(2;h) for h=poly⁡(n)h=\operatorname{poly}(n)h=poly(n). For logarithmically many pairs, the power of QMA(2)\mathsf{QMA}(2)QMA(2) remains unchanged, strengthening this inclusion to QMA(2)=QMA(2;h)\mathsf{QMA}(2)=\mathsf{QMA}(2;h)QMA(2)=QMA(2;h) for h=O(log⁡n)h=O(\log n)h=O(logn). Finally, we introduce a variant that also allows classical communication between the provers when preparing the witnesses. This setting is particularly revealing for logarithmic-size witnesses: assuming NP⊈BQP\mathsf{NP}\nsubseteq\mathsf{BQP}NP⊈BQP, we show that superlogarithmically many shared EPR pairs strictly reduce the model's computational power.
Since the introduction of the complexity class QMA as a quantum-verifier analogue of NP (Kitaev, 1997), many have wondered whether quantum proofs are necessary or classical proofs suffice - that is, whether QMA = QCMA or QCMA != QMA (Aharonov and Naveh, 2002; Aaronson and Kuperberg, CCC '07). This longstanding question was recently answered by works of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC '26) and Bostanci, Huang, and Vaikuntanathan (FOCS '26), which showed that quantum proofs are more powerful than classical ones in the classical-oracle setting. However, it remains unclear what exactly makes quantum proofs more powerful than classical ones. In the information-theoretic setting, a family of quantum states is not classicalizable if and only if it is unclonable. Indeed, these recent works also explicitly highlight the unclonability of their quantum proofs as a key mechanism behind their separations, and their arguments crucially rely on this property. This raises the question of whether unclonability is necessary for quantum proofs to be more powerful than classical ones. In this work, we show that even clonable quantum proofs can be more powerful than classical ones relative to a classical oracle by constructing a classical oracle O such that QCMA^O != ClonableQMA^O. This resolves the open question of Nehoran and Zhandry (ITCS '24), who established the analogous separation relative to a quantum oracle. We also show a classical-oracle separation between BQP/clonableqpoly and BQP/poly, and give applications of our results to quantum cryptography.
Quantum entanglement is a fundamental property of quantum mechanics and it serves as a basic resource in quantum computation and information. Despite its importance, the power and limitations of quantum entanglement are far from being fully understood. Here, we study entanglement via the lens of computational complexity. This is done by studying quantum generalizations of the class NP with multiple unentangled quantum proofs, the so-called QMA(2) and its variants. The complexity of QMA(2) is known to be closely connected to a variety of problems such as deciding if a state is entangled and several classical optimization problems. However, determining the complexity of QMA(2) is a longstanding open problem, and only the trivial complexity bounds ⊆ (2) ⊆ are known.