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Quantum Walks on the Hypercube

  • Cristopher Moore
  • Alexander Russell
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2483)

Abstract

Recently, it has been shown that one-dimensional quantum walks can mix more quickly than classical random walks, suggesting that quantum Monte Carlo algorithms can outperform their classical counterparts. We study two quantum walks on the n-dimensional hypercube, one in discrete time and one in continuous time. In both cases we show that the instantaneous mixing time is (π/4)n steps, faster than the Θ(n log n) steps required by the classical walk. In the continuous-time case, the probability distribution is exactly uniform at this time. On the other hand, we show that the average mixing time as defined by Aharonov et al. [AAKV01] is Ω(n 3/2) in the discrete-time case, slower than the classical walk, and nonexistent in the continuous-time case. This suggests that the instantaneous mixing time is a more relevant notion than the average mixing time for quantum walks on large, well-connected graphs. Our analysis treats interference between terms of different phase more carefully than is necessary for the walk on the cycle; previous general bounds predict an exponential average mixing time when applied to the hypercube.

Keywords

Cayley Graph Quantum Walk Total Variation Distance Undirected Graph Connectivity Classical Random Walk 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AAKV01]
    Dorit Aharonov, Andris Ambainis, Julia Kempe, and Umesh Vazirani. Quantum walks on graphs. In ACM [ACM01].Google Scholar
  2. [ABN+01]
    Andris Ambainis, Eric Bach, Ashwin Nayak, Ashvin Vishwanath, and John Watrous. One-dimensional quantum walks. In ACM [ACM01].Google Scholar
  3. [ACM01]
    Proc. 33rd Annual ACM Symposium on Theory of Computing (STOC) 2001.Google Scholar
  4. [AS92]
    Noga Alon and Joel H. Spencer. The Probabilistic Method. Wiley & Sons, 1992.Google Scholar
  5. [BH75]
    Norman Bleistein and Richard Handelsman. Asymptotic expansions of integrals. Holt, Rinehart and Winston, 1975.Google Scholar
  6. [CFG01]
    Andrew Childs, Edward Farhi, and Sam Gutmann. An example of the difference between quantum and classical random walks. Preprint, quant-ph/0103020. Google Scholar
  7. [DFK91]
    Martin Dyer, Alan Frieze, and Ravi Kannan. A random polynomial-time algorithm for approximating the volume of convex bodies. Journal of the ACM, 38(1):1–17, January 1991.Google Scholar
  8. [Dia88]
    Persi Diaconis. Group Representations in Probability and Statistics. Lecture notes-Monograph series. Institute of Mathematical Statistics, 1988.Google Scholar
  9. [DS81]
    Persi Diaconis and Mehrdad Shahshahani. Generating a random permutation with random transpositions. Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 57:159–179, 1981.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [FG98]
    Edward Farhi and Sam Gutmann. Quantum computation and decision trees. Phys. Rev. A, 58:915-, 1998.CrossRefMathSciNetGoogle Scholar
  11. [GG81]
    Ofer Gabber and Zvi Galil. Explicit constructions of linear-sized superconcentrators. Journal of Computer and System Sciences, 22(3):407–420, June 1981.Google Scholar
  12. [Gro96]
    Lov K. Grover. A fast quantum mechanical algorithm for database search. Proc. 28th Annual ACM Symposium on the Theory of Computing (STOC) 1996.Google Scholar
  13. [JS89]
    Mark Jerrum and Alistair Sinclair. Approximating the permanent. SIAM Journal on Computing, 18(6):1149–1178, December 1989.Google Scholar
  14. [JSV00]
    Mark Jerrum, Alistair Sinclair, and Eric Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. Technical Report TR00-079, Electronic Colloquium on Computational Complexity, 2000.Google Scholar
  15. [LK99]
    László Lovász and Ravi Kannan. Faster mixing via average conductance. Proc. 31st Annual ACM Symposium on Theory of Computing (STOC) 1999.Google Scholar
  16. [Lub94]
    Alexander Lubotzky. Discrete Groups, Expanding Graphs, and Invariant Measures, volume 125 of Progress in Mathematics. Birkhäuser Verlag, 1994.Google Scholar
  17. [MR01]
    Cristopher Moore and Alexander Russell. Quantum Walks on the Hypercube. Preprint, quant-ph/0104137.Google Scholar
  18. [MR95]
    Rajeev Motwani and Prabhakar Raghavan. Randomized Algorithms. Cambridge University Press, 1995.Google Scholar
  19. [Nis90]
    Noam Nisan. Pseudorandom generators for space-bounded computation. Proc. 22nd Annual ACM Symposium on Theory of Computing (STOC) 1990.Google Scholar
  20. [NN93]
    Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM Journal on Computing, 22(4):838–856, August 1993.Google Scholar
  21. [NV00]
    Ashwin Nayak and Ashvin Vishawanath. Quantum walk on the line. Los Alamos preprint archive, quant-ph/0010117, 2000.Google Scholar
  22. [Per92]
    Rene Peralta. On the distribution of quadratic residues and nonresidues modulo a prime number. Mathematics of Computation, 58(197):433–440, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [Sch99]
    Uwe Schöning. A probabilistic algorithm for k-SAT and constraint satisfaction problems. In 40th Annual Symposium on Foundations of Computer Science, pages 17–19. IEEE, 1999.Google Scholar
  24. [Sho97]
    Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5):1484–1509, October 1997.Google Scholar
  25. [Vaz92]
    Umesh Vazirani. Rapidly mixing markov chains. In Béla Bollobás, editor, Probabilistic Combinatorics and Its Applications, volume 44 of Proceedings of Symposia in Applied Mathematics. American Mathematical Society, 1992.Google Scholar
  26. [Wat01]
    John Watrous. Quantum simulations of classical random walks and undirected graph connectivity. Journal of Computer and System Sciences, 62:376–391, 2001.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Cristopher Moore
    • 1
  • Alexander Russell
    • 2
  1. 1.Computer Science DepartmentUniversity of New Mexico, Albuquerque and the Santa Fe Institute
  2. 2.Department of Computer Science and EngineeringUniversity of ConnecticutStorrs

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