Abstract
An explicit quantum dual adversary for the S-isomorphism problem is constructed. As a consequence, this gives an alternative proof that the query complexity of the dihedral hidden subgroup problem is polynomial.
This research is partly supported by the ERDF grant number 1.1.1.2/VIAA/1/16/113.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ambainis, A., Belovs, A., Regev, O., de Wolf, R.: Efficient quantum algorithms for (gapped) group testing and junta testing. In: Proceedings of 27th ACM-SIAM SODA, pp. 903–922 (2016)
Babai, L., Chakraborty, S.: Property testing of equivalence under a permutation group action (2008)
Bacon, D., Childs, A.M., van Dam, W.: From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups. In: Proceedings of 46th IEEE FOCS, pp. 469–478 (2005)
Belovs, A., Childs, A.M., Jeffery, S., Kothari, R., Magniez, F.: Time-efficient quantum walks for 3-distinctness. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 105–122. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39206-1_10
Belovs, A., Reichardt, B.W.: Span programs and quantum algorithms for st-connectivity and claw detection. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 193–204. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-33090-2_18
Ettinger, M., Høyer, P., Knill, E.: The quantum query complexity of the hidden subgroup problem is polynomial. Inf. Process. Lett. 91(1), 43–48 (2004)
Friedl, K., Ivanyos, G., Magniez, F., Santha, M., Sen, P.: Hidden translation and orbit coset in quantum computing. In: Proceedings of 35th ACM STOC, pp. 1–9 (2003)
Grigni, M., Schulman, L., Vazirani, M., Vazirani, U.: Quantum mechanical algorithms for the nonabelian hidden subgroup problem. Combinatorica 24(1), 137–154 (2004)
Harrow, A.W., Lin, C.Y.Y., Montanaro, A.: Sequential measurements, disturbance and property testing. In: Proceedings of 28th ACM-SIAM SODA, pp. 1598–1611 (2017)
Hausladen, P., Wootters, W.K.: A “pretty good” measurement for distinguishing quantum states. J. Mod. Opt. 41(12), 2385–2390 (1994)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
Høyer, P., Lee, T., Špalek, R.: Negative weights make adversaries stronger. In: Proceedings of 39th ACM STOC, pp. 526–535 (2007)
Kitaev, A.: Quantum measurements and the Abelian stabilizer problem (1995)
Kobayashi, H., Le Gall, F.: Dihedral hidden subgroup problem: a survey. Inf. Media Technol. 1(1), 178–185 (2006)
Kuperberg, G.: A subexponential-time quantum algorithm for the dihedral hidden subgroup problem. SIAM J. Comput. 35, 170–188 (2005)
Kuperberg, G.: Another subexponential-time quantum algorithm for the dihedral hidden subgroup problem (2011)
Lee, T., Mittal, R., Reichardt, B.W., Špalek, R., Szegedy, M.: Quantum query complexity of state conversion. In: Proceedings of 52nd IEEE FOCS, pp. 344–353 (2011)
Moore, C., Rockmore, D., Russell, A., Schulman, L.J.: The power of basis selection in Fourier sampling: hidden subgroup problems in affine groups. In: Proceedings of 15th ACM-SIAM SODA, pp. 1113–1122 (2004)
Regev, O.: Quantum computation and lattice problems. SIAM J. Comput. 33(3), 738–760 (2004)
Regev, O.: A subexponential time algorithm for the dihedral hidden subgroup problem with polynomial space (2004)
Reichardt, B.W.: Span programs and quantum query complexity: the general adversary bound is nearly tight for every boolean function. In: Proceedings of 50th IEEE FOCS, pp. 544–551 (2009)
Reichardt, B.W.: Reflections for quantum query algorithms. In: Proceedings of 22nd ACM-SIAM SODA, pp. 560–569 (2011)
Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26, 1484–1509 (1997)
Simon, D.: On the power of quantum computation. SIAM J. Comput. 26, 1474–1483 (1997)
Acknowledgements
I am grateful to all the persons with whom I have discussed this problem. Especially, I would like to thank Martin Roetteler, Dmitry Gavinsky and Tsuyoshi Ito.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Belovs, A. (2019). Quantum Dual Adversary for Hidden Subgroups and Beyond. In: McQuillan, I., Seki, S. (eds) Unconventional Computation and Natural Computation. UCNC 2019. Lecture Notes in Computer Science(), vol 11493. Springer, Cham. https://doi.org/10.1007/978-3-030-19311-9_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-19311-9_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-19310-2
Online ISBN: 978-3-030-19311-9
eBook Packages: Computer ScienceComputer Science (R0)