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The Entangled Quantum Polynomial Hierarchy Collapses

Authors Sabee Grewal , Justin Yirka

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Author Details

Sabee Grewal
  • The University of Texas at Austin, TX, USA
Justin Yirka
  • The University of Texas at Austin, TX, USA

Funding

Supported via Scott Aaronson by a Vannevar Bush Fellowship from the US Department of Defense, the NSF QLCI program (Grant No. OMA-2016245), and a Simons Investigator Award, the Simons "It from Qubit" collaboration. This material is based upon work supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator.

Acknowledgements

We thank Khang Le, Daniel Liang, William Kretschmer, Siddhartha Jain, and Scott Aaronson for helpful conversations. Joshua Cook was especially helpful at early stages of this project. We thank John Watrous for identifying an error in an earlier draft of this work.

Cite AsGet BibTex

Sabee Grewal and Justin Yirka. The Entangled Quantum Polynomial Hierarchy Collapses. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 6:1-6:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CCC.2024.6

Abstract

We introduce the entangled quantum polynomial hierarchy, QEPH, as the class of problems that are efficiently verifiable given alternating quantum proofs that may be entangled with each other. We prove QEPH collapses to its second level. In fact, we show that a polynomial number of alternations collapses to just two. As a consequence, QEPH = QRG(1), the class of problems having one-turn quantum refereed games, which is known to be contained in PSPACE. This is in contrast to the unentangled quantum polynomial hierarchy, QPH, which contains QMA(2). We also introduce DistributionQCPH, a generalization of the quantum-classical polynomial hierarchy QCPH where the provers send probability distributions over strings (instead of strings). We prove DistributionQCPH = QCPH, suggesting that only quantum superposition (not classical probability) increases the computational power of these hierarchies. To prove this equality, we generalize a game-theoretic result of Lipton and Young (1994) which says that, without loss of generality, the provers can send uniform distributions over a polynomial-size support. We also prove the analogous result for the polynomial hierarchy, i.e., DistributionPH = PH. Finally, we show that PH and QCPH are contained in QPH, resolving an open question of Gharibian et al. (2022).

Subject Classification

ACM Subject Classification
  • Theory of computation → Interactive proof systems
  • Theory of computation → Complexity classes
  • Theory of computation → Quantum complexity theory
Keywords
  • Polynomial hierarchy
  • Entangled proofs
  • Correlated proofs
  • Minimax

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