Document Open Access Logo

On the Power of Nonstandard Quantum Oracles

Authors Roozbeh Bassirian, Bill Fefferman, Kunal Marwaha

PDF
Thumbnail PDF

File

LIPIcs.TQC.2023.11.pdf
  • Filesize: 0.9 MB
  • 25 pages

Document Identifiers

Author Details

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

Funding

  • Bassirian, Roozbeh: AFOSR (YIP number FA9550-18-1-0148 and FA9550-21-1-0008); NSF under Grant CCF-2044923 (CAREER); U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers; DOE QuantISED grant DE-SC0020360.
  • Fefferman, Bill: AFOSR (YIP number FA9550-18-1-0148 and FA9550-21-1-0008); NSF under Grant CCF-2044923 (CAREER); U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers; DOE QuantISED grant DE-SC0020360.
  • Marwaha, Kunal: NSF Graduate Research Fellowship Program under Grant No. DGE-1746045

Acknowledgements

Thanks to Casey Duckering, Juspreet Singh Sandhu, Peter Shor, and Justin Yirka for collaborating on early stages of this project. KM thanks many others for engaging discussions, including Adam Bouland, Antares Chen, Aram Harrow, Matt Hastings, Eric Hester, Neng Huang, Robin Kothari, Brian Lawrence, Yi-Kai Liu, Patrick Lutz, Tushant Mittal, Abhijit Mudigonda, Chinmay Nirkhe, and Aaron Potechin. Thanks to Chinmay Nirkhe for feedback on a previous version of this manuscript.

Cite AsGet BibTex

Roozbeh Bassirian, Bill Fefferman, and Kunal Marwaha. On the Power of Nonstandard Quantum Oracles. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 11:1-11:25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.TQC.2023.11

Abstract

We study how the choices made when designing an oracle affect the complexity of quantum property testing problems defined relative to this oracle. We encode a regular graph of even degree as an invertible function f, and present f in different oracle models. We first give a one-query QMA protocol to test if a graph encoded in f has a small disconnected subset. We then use representation theory to show that no classical witness can help a quantum verifier efficiently decide this problem relative to an in-place oracle. Perhaps surprisingly, a simple modification to the standard oracle prevents a quantum verifier from efficiently deciding this problem, even with access to an unbounded witness.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • quantum complexity
  • QCMA
  • expander graphs
  • representation theory

Metrics

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

Related Versions

References

  1. Scott Aaronson. Open problems related to quantum query complexity, 2021. URL: http://arxiv.org/abs/2109.06917.
  2. Scott Aaronson and Greg Kuperberg. Quantum versus classical proofs and advice. Theory Comput., 3(1):129-157, 2007. URL: http://arxiv.org/abs/quant-ph/0604056.
  3. Dorit Aharonov and Tomer Naveh. Quantum np - a survey, 2002. URL: http://arxiv.org/abs/quant-ph/0210077.
  4. Andris Ambainis. Quantum lower bounds by quantum arguments. In Proceedings of the thirty-second annual ACM symposium on Theory of computing - STOC '00. ACM Press, 2000. URL: http://arxiv.org/abs/quant-ph/0002066.
  5. Andris Ambainis, Andrew M. Childs, and Yi-Kai Liu. Quantum property testing for bounded-degree graphs. Lecture Notes in Computer Science, pages 365-376, 2011. URL: https://doi.org/10.1007/978-3-642-22935-0_31.
  6. Atul Singh Arora, Alexandru Gheorghiu, and Uttam Singh. Oracle separations of hybrid quantum-classical circuits, 2022. URL: http://arxiv.org/abs/2201.01904.
  7. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM (JACM), 45(3):501-555, 1998. URL: https://doi.org/10.1145/1236457.1236459.
  8. Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani. Strengths and weaknesses of quantum computing. SIAM Journal on Computing, 26(5):1510-1523, October 1997. URL: https://doi.org/10.1137/s0097539796300933.
  9. Andrew Childs. Lecture Notes on Quantum Algorithms, 2022. URL: https://www.cs.umd.edu/~amchilds/qa/qa.pdf.
  10. Paul Erdos. On the central limit theorem for samples from a finite population. Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 4:49-61, 1959. URL: https://old.renyi.hu/~p_erdos/1959-13.pdf.
  11. Bill Fefferman and Shelby Kimmel. Quantum vs classical proofs and subset verification, 2018. URL: http://arxiv.org/abs/1510.06750.
  12. Joel Friedman. A proof of Alon’s second eigenvalue conjecture and related problems. CoRR, 2004. URL: http://arxiv.org/abs/cs/0405020.
  13. Chris Godsil and Hanmeng Zhan. Discrete-time quantum walks and graph structures. Journal of Combinatorial Theory, Series A, 167:181-212, 2019. URL: https://doi.org/10.1016/j.jcta.2019.05.003.
  14. Alex Bredariol Grilo, Iordanis Kerenidis, and Jamie Sikora. QMA with subset state witnesses. In Mathematical Foundations of Computer Science 2015 - 40th International Symposium. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-48054-0_14.
  15. Aram W. Harrow. The church of the symmetric subspace, 2013. URL: http://arxiv.org/abs/1308.6595.
  16. Aram W. Harrow and David J. Rosenbaum. Uselessness for an oracle model with internal randomness, 2011. URL: http://arxiv.org/abs/1111.1462.
  17. Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen. Mip*=re. Communications of the ACM, 64(11):131-138, 2021. URL: http://arxiv.org/abs/2001.04383.
  18. Elham Kashefi, Adrian Kent, Vlatko Vedral, and Konrad Banaszek. Comparison of quantum oracles. Physical Review A, 65(5), May 2002. URL: https://doi.org/10.1103/physreva.65.050304.
  19. Jonathan Katz and Yehuda Lindell. Introduction to modern cryptography. CRC press, 2020. URL: http://www.cs.umd.edu/~jkatz/imc.html.
  20. Eyal Kushilevitz and Yishay Mansour. Learning decision trees using the fourier spectrum. SIAM J. Comput., 22(6):1331-1348, 1993. URL: https://doi.org/10.1137/0222080.
  21. Andrew Lutomirski. Component mixers and a hardness result for counterfeiting quantum money, 2011. URL: http://arxiv.org/abs/1107.0321.
  22. Anand Natarajan and Chinmay Nirkhe. A classical oracle separation between qma and qcma, 2022. URL: http://arxiv.org/abs/2210.15380.
  23. Julius Petersen. Die Theorie der regulären graphs. Acta Mathematica, 15:193-220, 1900. URL: https://doi.org/10.1007/BF02392606.
  24. Jason D. M. Rennie. Relating the trace and frobenius matrix norms, August 2005. URL: http://people.csail.mit.edu/jrennie/writing/traceFrobenius.pdf.
  25. Adi Shamir. IP=PSPACE. Journal of the ACM (JACM), 39(4):869-877, 1992. URL: https://doi.org/10.1145/146585.146609.
  26. Luca Trevisan. Max cut and the smallest eigenvalue, 2008. URL: http://arxiv.org/abs/0806.1978.
  27. John Watrous. Quantum simulations of classical random walks and undirected graph connectivity, 1998. URL: http://arxiv.org/abs/cs/9812012.
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 Schloss Dagstuhl - LZI GmbH Imprint Privacy Contact