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Quantum Walks on the Hypercube


  • Cristopher Moore
  • Alexander Russell


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 quantum walk mixes in (pi/4)n steps, faster than the O(n log n) steps required by the classical walk. In the continuous-time case, the probability distribution is exactly uniform at this time. More importantly, these walks expose several subtleties in the definition of mixing time for quantum walks. Even though the continuous-time walk has an O(n) instantaneous mixing time at which it is precisely uniform, it never approaches the uniform distribution when the stopping time is chosen randomly as in [AharonovAKV2001]. 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, rather than linear, mixing time for the hypercube.

Suggested Citation

  • Cristopher Moore & Alexander Russell, 2001. "Quantum Walks on the Hypercube," Working Papers 01-05-026, Santa Fe Institute.
  • Handle: RePEc:wop:safiwp:01-05-026

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    References listed on IDEAS

    1. Cohen, Joel E. & Hajnal, John & Newman, Charles M., 1986. "Approaching consensus can be delicate when positions harden," Stochastic Processes and their Applications, Elsevier, vol. 22(2), pages 315-322, July.
    2. Follmer, Hans, 1974. "Random economies with many interacting agents," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 51-62, March.
    3. Orlean, Andre, 1995. "Bayesian interactions and collective dynamics of opinion: Herd behavior and mimetic contagion," Journal of Economic Behavior & Organization, Elsevier, vol. 28(2), pages 257-274, October.
    4. H. Peyton Young & Mary A. Burke, 2001. "Competition and Custom in Economic Contracts: A Case Study of Illinois Agriculture," American Economic Review, American Economic Association, vol. 91(3), pages 559-573, June.
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