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.
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References
Dorit Aharonov, Andris Ambainis, Julia Kempe, and Umesh Vazirani. Quantum walks on graphs. In ACM [ACM01].
Andris Ambainis, Eric Bach, Ashwin Nayak, Ashvin Vishwanath, and John Watrous. One-dimensional quantum walks. In ACM [ACM01].
Proc. 33rd Annual ACM Symposium on Theory of Computing (STOC) 2001.
Noga Alon and Joel H. Spencer. The Probabilistic Method. Wiley & Sons, 1992.
Norman Bleistein and Richard Handelsman. Asymptotic expansions of integrals. Holt, Rinehart and Winston, 1975.
Andrew Childs, Edward Farhi, and Sam Gutmann. An example of the difference between quantum and classical random walks. Preprint, quant-ph/0103020.
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.
Persi Diaconis. Group Representations in Probability and Statistics. Lecture notes-Monograph series. Institute of Mathematical Statistics, 1988.
Persi Diaconis and Mehrdad Shahshahani. Generating a random permutation with random transpositions. Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 57:159–179, 1981.
Edward Farhi and Sam Gutmann. Quantum computation and decision trees. Phys. Rev. A, 58:915-, 1998.
Ofer Gabber and Zvi Galil. Explicit constructions of linear-sized superconcentrators. Journal of Computer and System Sciences, 22(3):407–420, June 1981.
Lov K. Grover. A fast quantum mechanical algorithm for database search. Proc. 28th Annual ACM Symposium on the Theory of Computing (STOC) 1996.
Mark Jerrum and Alistair Sinclair. Approximating the permanent. SIAM Journal on Computing, 18(6):1149–1178, December 1989.
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.
László Lovász and Ravi Kannan. Faster mixing via average conductance. Proc. 31st Annual ACM Symposium on Theory of Computing (STOC) 1999.
Alexander Lubotzky. Discrete Groups, Expanding Graphs, and Invariant Measures, volume 125 of Progress in Mathematics. Birkhäuser Verlag, 1994.
Cristopher Moore and Alexander Russell. Quantum Walks on the Hypercube. Preprint, quant-ph/0104137.
Rajeev Motwani and Prabhakar Raghavan. Randomized Algorithms. Cambridge University Press, 1995.
Noam Nisan. Pseudorandom generators for space-bounded computation. Proc. 22nd Annual ACM Symposium on Theory of Computing (STOC) 1990.
Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM Journal on Computing, 22(4):838–856, August 1993.
Ashwin Nayak and Ashvin Vishawanath. Quantum walk on the line. Los Alamos preprint archive, quant-ph/0010117, 2000.
Rene Peralta. On the distribution of quadratic residues and nonresidues modulo a prime number. Mathematics of Computation, 58(197):433–440, 1992.
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.
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.
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.
John Watrous. Quantum simulations of classical random walks and undirected graph connectivity. Journal of Computer and System Sciences, 62:376–391, 2001.
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Moore, C., Russell, A. (2002). Quantum Walks on the Hypercube. In: Rolim, J.D.P., Vadhan, S. (eds) Randomization and Approximation Techniques in Computer Science. RANDOM 2002. Lecture Notes in Computer Science, vol 2483. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45726-7_14
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DOI: https://doi.org/10.1007/3-540-45726-7_14
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