Quantum Walk Sampling by Growing Seed Sets

Author Simon Apers



PDF
Thumbnail PDF

File

LIPIcs.ESA.2019.9.pdf
  • Filesize: 474 kB
  • 12 pages

Document Identifiers

Author Details

Simon Apers
  • Inria, Paris, France
  • CWI, Amsterdam, The Netherlands

Acknowledgements

This work benefited from discussions with Alain Sarlette, Stacey Jeffery, Anthony Leverrier, Ronald de Wolf, André Chailloux and Frédéric Magniez. Part of this work was supported by the CWI-Inria International Lab.

Cite AsGet BibTex

Simon Apers. Quantum Walk Sampling by Growing Seed Sets. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 9:1-9:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ESA.2019.9

Abstract

This work describes a new algorithm for creating a superposition over the edge set of a graph, encoding a quantum sample of the random walk stationary distribution. The algorithm requires a number of quantum walk steps scaling as O~(m^(1/3) delta^(-1/3)), with m the number of edges and delta the random walk spectral gap. This improves on existing strategies by initially growing a classical seed set in the graph, from which a quantum walk is then run. The algorithm leads to a number of improvements: (i) it provides a new bound on the setup cost of quantum walk search algorithms, (ii) it yields a new algorithm for st-connectivity, and (iii) it allows to create a superposition over the isomorphisms of an n-node graph in time O~(2^(n/3)), surpassing the Omega(2^(n/2)) barrier set by index erasure.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Graph algorithms analysis
Keywords
  • Quantum algorithms
  • Quantum walks
  • Connectivity
  • Graph theory

Metrics

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

References

  1. Dorit Aharonov, Andris Ambainis, Julia Kempe, and Umesh Vazirani. Quantum walks on graphs. In \stoc33rd, pages 50-59. ACM, 2001. URL: https://arxiv.org/abs/quant-ph/0012090.
  2. Dorit Aharonov and Amnon Ta-Shma. Adiabatic quantum state generation and statistical zero knowledge. In \stoc35th, pages 20-29. ACM, 2003. URL: https://arxiv.org/abs/quant-ph/0301023.
  3. David Aldous and Jim Fill. Reversible Markov chains and random walks on graphs. Unfinished monograph, 2002. URL: https://www.stat.berkeley.edu/~aldous/RWG/book.pdf.
  4. Andris Ambainis. Quantum walk algorithm for element distinctness. \siamjc, 37(1):210-239, 2007. URL: https://arxiv.org/abs/quant-ph/0311001.
  5. Andris Ambainis, Andrew M Childs, and Yi-Kai Liu. Quantum property testing for bounded-degree graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 365-376. Springer, 2011. URL: https://arxiv.org/abs/1012.3174.
  6. Andris Ambainis, Loïck Magnin, Martin Roetteler, and Jérémie Roland. Symmetry-assisted adversaries for quantum state generation. In çc26th, pages 167-177. IEEE, 2011. URL: https://arxiv.org/abs/1012.2112.
  7. Reid Andersen and Yuval Peres. Finding sparse cuts locally using evolving sets. In \stoc41st, pages 235-244. ACM, 2009. URL: https://arxiv.org/abs/0811.3779.
  8. Simon Apers and Alain Sarlette. Quantum Fast-Forwarding Markov Chains and Property Testing. \qic, 19(3&4):181-213, 2019. URL: https://arxiv.org/abs/1804.02321.
  9. László Babai. Local expansion of vertex-transitive graphs and random generation in finite groups. In \stoc23rd, volume 91, pages 164-174. ACM, 1991. URL: https://doi.org/10.1145/103418.103440.
  10. László Babai. Graph isomorphism in quasipolynomial time. In \stoc48th, pages 684-697. ACM, 2016. URL: https://arxiv.org/abs/1512.03547.
  11. Tuğkan Batu, Lance Fortnow, Ronitt Rubinfeld, Warren D Smith, and Patrick White. Testing closeness of discrete distributions. \jacm, 60(1):4, 2013. URL: https://arxiv.org/abs/1009.5397.
  12. Aleksandrs Belovs. Quantum walks and electric networks. https://arxiv.org/abs/1302.3143, 2013.
  13. Aleksandrs Belovs and Ben W Reichardt. Span programs and quantum algorithms for st-connectivity and claw detection. In \esa20th, pages 193-204. Springer, 2012. URL: https://arxiv.org/abs/1203.2603.
  14. Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp. Quantum amplitude amplification and estimation. Contemporary Mathematics, 305:53-74, 2002. URL: https://arxiv.org/abs/quant-ph/0005055.
  15. Gilles Brassard, Peter Høyer, and Alain Tapp. Quantum algorithm for the collision problem. ACM SIGACT News (Cryptology Column), 28:14-19, 1997. URL: https://arxiv.org/abs/quant-ph/9705002.
  16. Chris Cade, Ashley Montanaro, and Aleksandrs Belovs. Time and space efficient quantum algorithms for detecting cycles and testing bipartiteness. https://arxiv.org/abs/1610.00581, 2016.
  17. Flavio Chiericetti, Anirban Dasgupta, Ravi Kumar, Silvio Lattanzi, and Tamás Sarlós. On sampling nodes in a network. In Proceedings of the 25th International Conference on World Wide Web (WWW), pages 471-481. International WWW Conferences, 2016. URL: https://doi.org/10.1145/2872427.2883045.
  18. Artur Czumaj, Pan Peng, and Christian Sohler. Testing cluster structure of graphs. In \stoc47th, pages 723-732. ACM, 2015. URL: https://arxiv.org/abs/1504.03294.
  19. Persi Diaconis and Mehrdad Shahshahani. Generating a random permutation with random transpositions. Probability Theory and Related Fields, 57(2):159-179, 1981. URL: https://doi.org/10.1007/BF00535487.
  20. Christoph Dürr, Mark Heiligman, Peter Høyer, and Mehdi Mhalla. Quantum query complexity of some graph problems. \siamjc, 35(6):1310-1328, 2006. URL: https://arxiv.org/abs/quant-ph/0401091.
  21. Oded Goldreich and Dana Ron. Property testing in bounded degree graphs. \algor, 32(2):302-343, 2002. URL: https://doi.org/10.1007/s00453-001-0078-7.
  22. Oded Goldreich and Dana Ron. On testing expansion in bounded-degree graphs. In Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation, pages 68-75. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22670-0_9.
  23. Michael Jarret, Stacey Jeffery, Shelby Kimmel, and Alvaro Piedrafita. Quantum algorithms for connectivity and related problems. In \esa26th, pages 49:1-49:13. Springer, 2018. URL: https://arxiv.org/abs/1804.10591.
  24. Iordanis Kerenidis and Anupam Prakash. Quantum recommendation systems. In \itcs8th, pages 49:1-49:21. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017. URL: https://arxiv.org/abs/1603.08675.
  25. Hari Krovi, Frédéric Magniez, Maris Ozols, and Jérémie Roland. Quantum walks can find a marked element on any graph. \algor, 74(2):851-907, 2016. URL: https://arxiv.org/abs/1002.2419.
  26. David A Levin, Yuval Peres, and Elizabeth L Wilmer. Markov chains and mixing times. American Mathematical Society, 2017. URL: https://doi.org/10.1090/mbk/058.
  27. Andrew Lutomirski. Component mixers and a hardness result for counterfeiting quantum money. https://arxiv.org/abs/1107.0321, 2011.
  28. Frédéric Magniez, Ashwin Nayak, Jérémie Roland, and Miklos Santha. Search via quantum walk. \siamjc, 40(1):142-164, 2011. https://arxiv.org/abs/quant-ph/0608026. URL: https://doi.org/10.1137/090745854.
  29. Davide Orsucci, Hans J. Briegel, and Vedran Dunjko. Faster quantum mixing for slowly evolving sequences of Markov chains. \quantum, 2:105, 2018. URL: https://arxiv.org/abs/1503.01334.
  30. David Poulin and Pawel Wocjan. Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer. \prl, 103(22):220502, 2009. URL: https://arxiv.org/abs/0905.2199.
  31. Peter C Richter. Quantum speedup of classical mixing processes. \pra, 76(4):042306, 2007. URL: https://arxiv.org/abs/quant-ph/0609204.
  32. Alistair Sinclair. Algorithms for random generation and counting: a Markov chain approach. Springer Science & Business Media, 2012. URL: https://doi.org/10.1007/978-1-4612-0323-0.
  33. Rolando D Somma, Sergio Boixo, Howard Barnum, and Emanuel Knill. Quantum simulations of classical annealing processes. Physical Review Letters, 101(13):130504, 2008. URL: https://arxiv.org/abs/0804.1571.
  34. Daniel A Spielman and Shang-Hua Teng. A local clustering algorithm for massive graphs and its application to nearly linear time graph partitioning. \siamjc, 42(1):1-26, 2013. URL: https://arxiv.org/abs/0809.3232.
  35. Mario Szegedy. Quantum speed-up of Markov chain based algorithms. In \focs45th, pages 32-41. IEEE, 2004. URL: https://arxiv.org/abs/quant-ph/0401053.
  36. Joran Van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf. Quantum SDP-Solvers: better upper and lower bounds. In \focs58th, pages 403-414. IEEE, 2017. URL: https://arxiv.org/abs/1705.01843.
  37. John Watrous. Succinct quantum proofs for properties of finite groups. In \focs41st, pages 537-546. IEEE, 2000. URL: https://arxiv.org/abs/cs/0009002.
  38. John Watrous. Quantum simulations of classical random walks and undirected graph connectivity. \jcss, 62(2):376-391, 2001. URL: https://arxiv.org/abs/cs/9812012.
  39. Pawel Wocjan and Anura Abeyesinghe. Speedup via quantum sampling. \pra, 78(4):042336, 2008. URL: https://arxiv.org/abs/0804.4259.
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