Document Open Access Logo

The Entangled Quantum Polynomial Hierarchy Collapses

Authors Sabee Grewal , Justin Yirka

PDF
Thumbnail PDF

File

LIPIcs.CCC.2024.6.pdf
  • Filesize: 0.88 MB
  • 23 pages

Document Identifiers

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Scott Aaronson, DeVon Ingram, and William Kretschmer. The Acrobatics of BQP. In 37th Computational Complexity Conference (CCC 2022), Leibniz International Proceedings in Informatics (LIPIcs), pages 20:1-20:17, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.20.
  2. Avantika Agarwal, Sevag Gharibian, Venkata Koppula, and Dorian Rudolph. Quantum Polynomial Hierarchies: Karp-Lipton, error reduction, and lower bounds, 2024. URL: https://arxiv.org/abs/2401.01633.
  3. Ingo Althöfer. On sparse approximations to randomized strategies and convex combinations. Linear Algebra and its Applications, 199:339-355, 1994. Special Issue Honoring Ingram Olkin. URL: https://doi.org/10.1016/0024-3795(94)90357-3.
  4. Sanjeev Arora and Boaz Barak. Computational Complexity: A Modern Approach. Cambridge University Press, 2009. URL: https://doi.org/10.1017/CBO9780511804090.
  5. Roozbeh Bassirian, Bill Fefferman, and Kunal Marwaha. Quantum Merlin-Arthur and proofs without relative phase, 2023. URL: https://arxiv.org/abs/2306.13247.
  6. Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf. Quantum fingerprinting. Physical Review Letters, 87(16):167902, 2001. URL: https://doi.org/10.1103/PhysRevLett.87.167902.
  7. Chirag Falor, Shu Ge, and Anand Natarajan. A Collapsible Polynomial Hierarchy for Promise Problems, 2023. URL: https://arxiv.org/abs/2311.12228.
  8. Uriel Feige and Joe Kilian. Making Games Short (Extended Abstract). In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 506-516, New York, NY, USA, 1997. Association for Computing Machinery. URL: https://doi.org/10.1145/258533.258644.
  9. J. Feigenbaum, D. Koller, and P. Shor. A Game-Theoretic Classification of Interactive Complexity Classes. In Proceedings of Structure in Complexity Theory. Tenth Annual IEEE Conference, pages 227-237, 1995. URL: https://doi.org/10.1109/SCT.1995.514861.
  10. Sevag Gharibian and Julia Kempe. Hardness of approximation for quantum problems. In International Colloquium on Automata, Languages, and Programming, pages 387-398. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-31594-7_33.
  11. Sevag Gharibian, Miklos Santha, Jamie Sikora, Aarthi Sundaram, and Justin Yirka. Quantum generalizations of the Polynomial Hierarchy with applications to QMA(2). Computational Complexity, 31(2):13, 2022. URL: https://doi.org/10.1007/s00037-022-00231-8.
  12. Soumik Ghosh and John Watrous. Complexity limitations on one-turn quantum refereed games. Theory of Computing Systems, 67(2):383-412, 2023. URL: https://doi.org/10.1007/s00224-022-10105-9.
  13. Gus Gutoski and John Watrous. Quantum interactive proofs with competing provers. In STACS 2005: 22nd Annual Symposium on Theoretical Aspects of Computer Science, pages 605-616. Springer, 2005. URL: https://doi.org/10.1007/978-3-540-31856-9_50.
  14. Gus Gutoski and John Watrous. Toward a General Theory of Quantum Games. In Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, pages 565-574, New York, NY, USA, 2007. Association for Computing Machinery. URL: https://doi.org/10.1145/1250790.1250873.
  15. Gus Gutoski and Xiaodi Wu. Parallel Approximation of Min-Max Problems. Computational Complexity, 22:385-428, 2013. URL: https://doi.org/10.1007/s00037-013-0065-9.
  16. Aram W. Harrow and Ashley Montanaro. Testing Product States, Quantum Merlin-Arthur Games and Tensor Optimization. J. ACM, 60(1), 2013. URL: https://doi.org/10.1145/2432622.2432625.
  17. Rahul Jain, Zhengfeng Ji, Sarvagya Upadhyay, and John Watrous. QIP = PSPACE. Journal of the ACM (JACM), 58(6):1-27, 2011. URL: https://doi.org/10.1145/2049697.2049704.
  18. Rahul Jain and John Watrous. Parallel Approximation of Non-interactive Zero-sum Quantum Games. In 24th Annual IEEE Conference on Computational Complexity, pages 243-253, 2009. URL: https://doi.org/10.1109/CCC.2009.26.
  19. Fernando Granha Jeronimo and Pei Wu. The power of unentangled quantum proofs with non-negative amplitudes. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 1629-1642, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3564246.3585248.
  20. Alexei Kitaev and John Watrous. Parallelization, amplification, and exponential time simulation of quantum interactive proof systems. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pages 608-617, 2000. URL: https://doi.org/10.1145/335305.335387.
  21. Alexei Y. Kitaev, Alexander Shen, and Mikhail N. Vyalyi. Classical and Quantum Computation. American Mathematical Soc., 2002. URL: https://doi.org/10.1090/gsm/047.
  22. Robert Kleinberg, Oliver Korten, Daniel Mitropolsky, and Christos Papadimitriou. Total Functions in the Polynomial Hierarchy. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021), volume 185 of Leibniz International Proceedings in Informatics (LIPIcs), pages 44:1-44:18, 2021. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.44.
  23. Clemens Lautemann. BPP and the polynomial hierarchy. Information Processing Letters, 17(4):215-217, 1983. URL: https://doi.org/10.1016/0020-0190(83)90044-3.
  24. Richard J. Lipton and Neal E. Young. Simple Strategies for Large Zero-Sum Games with Applications to Complexity Theory. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 734-740, 1994. URL: https://doi.org/10.1145/195058.195447.
  25. Chris Marriott and John Watrous. Quantum Arthur-Merlin Games. Computational Complexity, 14(2):122-152, 2005. URL: https://doi.org/10.1007/s00037-005-0194-x.
  26. A. R. Meyer and L. J. Stockmeyer. The equivalence problem for regular expressions with squaring requires exponential space. In 13th Annual Symposium on Switching and Automata Theory (SWAT 1972), pages 125-129, 1972. URL: https://doi.org/10.1109/SWAT.1972.29.
  27. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010. URL: https://doi.org/10.1017/CBO9780511976667.
  28. Adi Shamir. IP = PSPACE. J. ACM, 39(4):869-877, 1992. URL: https://doi.org/10.1145/146585.146609.
  29. Maurice Sion. On General Minimax Theorems. Pacific Journal of Mathematics, 1958. URL: https://doi.org/10.2140/pjm.1958.8.171.
  30. Michael Sipser. A Complexity Theoretic Approach to Randomness. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 330-335. Association for Computing Machinery, 1983. URL: https://doi.org/10.1145/800061.808762.
  31. Larry J. Stockmeyer. The Polynomial-Time Hierarchy. Theoretical Computer Science, 3(1):1-22, 1976. URL: https://doi.org/10.1016/0304-3975(76)90061-X.
  32. Lieuwe Vinkhuijzen. A Quantum Polynomial Hierarchy and a Simple Proof of Vyalyi’s Theorem. Master’s thesis, Leiden University, 2018. URL: https://theses.liacs.nl/1505.
  33. John Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. URL: https://doi.org/10.1017/9781316848142.
  34. Tomoyuki Yamakami. Quantum NP and a Quantum Hierarchy. In Foundations of Information Technology in the Era of Networking and Mobile Computing, volume 96 of IFIP — The International Federation for Information Processing, pages 323-336, Boston, MA, 2002. Springer. URL: https://doi.org/10.1007/978-0-387-35608-2_27.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact