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Interactive Proofs with Polynomial-Time Quantum Prover for Computing the Order of Solvable Groups

Authors François Le Gall, Tomoyuki Morimae, Harumichi Nishimura, Yuki Takeuchi

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Author Details

François Le Gall
  • Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan
Tomoyuki Morimae
  • Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
Harumichi Nishimura
  • Graduate School of Informatics, Nagoya University, Chikusa-ku, Nagoya, Aichi 464-8601, Japan
Yuki Takeuchi
  • NTT Communication Science Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan
  • Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan

Funding

  • Le Gall, François: Partially supported by the JSPS KAKENHI grants No. 15H01677, No. 16H01705 and No. 16H05853.
  • Morimae, Tomoyuki: Supported by JST PRESTO No. JPMJPR176A, and the Grant-in-Aid for Young Scientists (B) No. JP17K12637 of JSPS.
  • Nishimura, Harumichi: Partially supported by the JSPS KAKENHI grants No. 26247016, No. 16H01705 and No. 16K00015.
  • Takeuchi, Yuki: Supported by the Program for Leading Graduate Schools: Interactive Materials Science Cadet Program.

Cite As Get BibTex

François Le Gall, Tomoyuki Morimae, Harumichi Nishimura, and Yuki Takeuchi. Interactive Proofs with Polynomial-Time Quantum Prover for Computing the Order of Solvable Groups. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 26:1-26:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.MFCS.2018.26

Abstract

In this paper we consider what can be computed by a user interacting with a potentially malicious server, when the server performs polynomial-time quantum computation but the user can only perform polynomial-time classical (i.e., non-quantum) computation. Understanding the computational power of this model, which corresponds to polynomial-time quantum computation that can be efficiently verified classically, is a well-known open problem in quantum computing. Our result shows that computing the order of a solvable group, which is one of the most general problems for which quantum computing exhibits an exponential speed-up with respect to classical computing, can be realized in this model.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
Keywords
  • Quantum computing
  • interactive proofs
  • group-theoretic problems

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References

  1. Scott Aaronson and Greg Kuperberg. Quantum versus classical proofs and advice. Theory of Computing, 3:129-157, 2007. URL: http://dx.doi.org/10.4086/toc.2007.v003a007.
  2. Dorit Aharonov, Michael Ben-Or, Elad Eban, and Urmila Mahadev. Interactive proofs for quantum computations. arXiv:1704.04487, 2017. Google Scholar
  3. Dorit Aharonov and Ayal Green. A quantum inspired proof of P^#P ⊆ IP. arXiv:1710.09078, 2017. Google Scholar
  4. Dorit Aharonov and Umesh Vazirani. Is quantum mechanics falsifiable? A computational perspective on the foundations of quantum mechanics. arXiv:1206.3686, 2012. Google Scholar
  5. László Babai. Local expansion of vertex-transitive graphs and random generation in finite groups. In Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, pages 164-174, 1991. URL: http://dx.doi.org/10.1145/103418.103440.
  6. László Babai. Bounded round interactive proofs in finite groups. SIAM Journal on Discrete Mathematics, 5(1):88-111, 1992. URL: http://dx.doi.org/10.1137/0405008.
  7. László Babai, Robert Beals, and Ákos Seress. Polynomial-time theory of matrix groups. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, pages 55-64, 2009. URL: http://dx.doi.org/10.1145/1536414.1536425.
  8. László Babai, Gene Cooperman, Larry Finkelstein, Eugene M. Luks, and Ákos Seress. Fast monte carlo algorithms for permutation groups. Journal of Computer and System Sciences, 50(2):296-308, 1995. URL: http://dx.doi.org/10.1006/jcss.1995.1024.
  9. László Babai and Endre Szemerédi. On the complexity of matrix group problems I. In Proceedings of the 25th Annual Symposium on Foundations of Computer Science, pages 229-240, 1984. URL: http://dx.doi.org/10.1109/SFCS.1984.715919.
  10. Ethan Bernstein and Umesh Vazirani. Quantum complexity theory. SIAM Journal on Computing, 26:1411-1473, 1997. URL: http://dx.doi.org/10.1137/S0097539796300921.
  11. Simon R. Blackburn, Peter M. Neumann, and Geetha Venkataraman. Enumeration of Finite Groups. Cambridge University Press, 2017. URL: http://dx.doi.org/10.1017/CBO9780511542756.
  12. Tommaso F. Demarie, Yungkai Ouyang, and Joseph F. Fitzsimons. Classical verification of quantum circuits containing few basis changes. arXiv:1612.04914, 2016. Google Scholar
  13. Joseph F. Fitzsimons, Michael Hajdušek, and Tomoyuki Morimae. Post hoc verification of quantum computation. Physical Review Letters, 120:040501, 2018. URL: http://dx.doi.org/10.1103/PhysRevLett.120.040501.
  14. Joseph F. Fitzsimons and Elham Kashefi. Unconditionally verifiable blind computation. Physical Review A, 96:012303, 2017. URL: http://dx.doi.org/10.1103/PhysRevA.96.012303.
  15. Shafi Goldwasser, Ofer Grossman, and Dhiraj Holden. Pseudo-deterministic proofs. In Proceedings of the 9th Innovations in Theoretical Computer Science Conference, pages 17:1-17:18, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ITCS.2018.17.
  16. Masahito Hayashi and Tomoyuki Morimae. Verifiable measurement-only blind quantum computing with stabilizer testing. Physical Review Letters, 115:220502, 2015. URL: http://dx.doi.org/10.1103/PhysRevLett.115.220502.
  17. Derek F. Holt, Bettina Eick, and Eamonn A. O'Brien. Handbook of computational group theory. Chapman &Hall/CRC, 2005. URL: http://dx.doi.org/10.1201/9781420035216.
  18. I. Martin Isaacs. Finite group theory. American Mathematical Society, 2008. Google Scholar
  19. Gábor Ivanyos, Frédéric Magniez, and Miklos Santha. Efficient quantum algorithms for some instances of the non-abelian hidden subgroup problem. International Journal of Foundations of Computer Science, 14(5):723-740, 2003. URL: http://dx.doi.org/10.1142/S0129054103001996.
  20. Zhengfeng Ji. Classical verification of quantum proofs. In Proceedings of the 48th Annual ACM symposium on Theory of Computing, pages 885-898, 2016. URL: http://dx.doi.org/10.1145/2897518.2897634.
  21. Carsten Lund, Lance Fortnow, Howard J. Karloff, and Noam Nisan. Algebraic methods for interactive proof systems. Journal of the ACM, 39(4):859-868, 1992. URL: http://dx.doi.org/10.1145/146585.146605.
  22. Urmila Mahadev. Classical verification of quantum computations. arXiv:1804.01082, 2018. Google Scholar
  23. Mathew McKague. Interactive proofs with efficient quantum prover for recursive fourier sampling. Chicago Journal of Theoretical Computer Science, 2012(6), 2012. URL: http://dx.doi.org/10.4086/cjtcs.2012.006.
  24. Mathew McKague. Interactive proofs for BQP via self-tested graph states. Theory of Computing, 12(3):1-42, 2016. URL: http://dx.doi.org/10.4086/toc.2016.v012a003.
  25. Tomoyuki Morimae, Daniel Nagaj, and Norbert Schuch. Quantum proofs can be verified using only single-qubit measurements. Physical Review A, 93:022326, 2016. URL: http://dx.doi.org/10.1103/PhysRevA.93.022326.
  26. Tomoyuki Morimae, Yuki Takeuchi, and Harumichi Nishimura. Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the fourier hierarchy. arXiv:1711.10605, 2017. Google Scholar
  27. Ben W. Reichardt, Falk Unger, and Umesh Vazirani. Classical command of quantum systems. Nature, 496:456-460, 2013. URL: http://dx.doi.org/10.1038/nature12035.
  28. Adi Shamir. IP = PSPACE. Journal of the ACM, 39(4):869-877, 1992. URL: http://dx.doi.org/10.1145/146585.146609.
  29. Yaoyun Shi. Quantum and classical tradeoffs. Theoretical Computer Science, 344:335-343, 2005. URL: http://dx.doi.org/10.1016/j.tcs.2005.03.053.
  30. Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5):1484-1509, 1997. URL: http://dx.doi.org/10.1137/S0097539795293172.
  31. Daniel R. Simon. On the power of quantum computation. SIAM Journal on Computing, 26(5):1474-1483, 1997. URL: http://dx.doi.org/10.1137/S0097539796298637.
  32. John Watrous. Succinct quantum proofs for properties of finite groups. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pages 537-546, 2000. URL: http://dx.doi.org/10.1109/SFCS.2000.892141.
  33. John Watrous. Quantum algorithms for solvable groups. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pages 60-67, 2001. URL: http://dx.doi.org/10.1145/380752.380759.
  34. URL: https://www-03.ibm.com/press/us/en/pressrelease/52403.wss.
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