Quantum Merlin-Arthur and Proofs Without Relative Phase

Authors Roozbeh Bassirian, Bill Fefferman, Kunal Marwaha

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Roozbeh Bassirian
  • University of Chicago, IL, USA
Bill Fefferman
  • University of Chicago, IL, USA
Kunal Marwaha
  • University of Chicago, IL, USA


Thanks to Zachary Remscrim for collaborating on early stages of this project. Thanks to Noam Lifshitz, Dor Minzer, and Kevin Pratt for answering questions about algebraic constructions of expanders. Thanks to Srinivasan Arunachalam, Fernando Granha Jeronimo, Supartha Podder, and Pei Wu for comments on a draft of this manuscript.

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Roozbeh Bassirian, Bill Fefferman, and Kunal Marwaha. Quantum Merlin-Arthur and Proofs Without Relative Phase. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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 [Grilo et al., 2014]; but if both completeness and soundness are modified, the class (named QMA+ by Jeronimo and Wu [Jeronimo and Wu, 2023]) 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) ⊆ NEXP.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • quantum complexity
  • QMA(2)
  • PCPs


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