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On the Hitting Times of Quantum Versus Random Walks

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Abstract

The hitting time of a classical random walk (Markov chain) is the time required to detect the presence of—or equivalently, to find—a marked state. The hitting time of a quantum walk is subtler to define; in particular, it is unknown whether the detection and finding problems have the same time complexity. In this paper we define new Monte Carlo type classical and quantum hitting times, and we prove several relationships among these and the already existing Las Vegas type definitions. In particular, we show that for some marked state the two types of hitting time are of the same order in both the classical and the quantum case.

Then, we present new quantum algorithms for the detection and finding problems. The complexities of both algorithms are related to the new, potentially smaller, quantum hitting times. The detection algorithm is based on phase estimation and is particularly simple. The finding algorithm combines a similar phase estimation based procedure with ideas of Tulsi from his recent theorem (Tulsi A.: Phys. Rev. A 78:012310 2008) for the 2D grid. Extending his result, we show that we can find a unique marked element with constant probability and with the same complexity as detection for a large class of quantum walks—the quantum analogue of state-transitive reversible ergodic Markov chains.

Further, we prove that for any reversible ergodic Markov chain P, the quantum hitting time of the quantum analogue of P has the same order as the square root of the classical hitting time of P. We also investigate the (im)possibility of achieving a gap greater than quadratic using an alternative quantum walk. In doing so, we define a notion of reversibility for a broad class of quantum walks and show how to derive from any such quantum walk a classical analogue. For the special case of quantum walks built on reflections, we show that the hitting time of the classical analogue is exactly the square of the quantum walk.

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Authors and Affiliations

  1. LIAFA, Univ. Paris 7, CNRS, 75205, Paris cedex 13, France

    Frédéric Magniez & Miklos Santha

  2. Department of Combinatorics and Optimization and Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada

    Ashwin Nayak

  3. Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Canada

    Ashwin Nayak

  4. LRI, Univ. Paris-Sud, CNRS, bat. 490, 91405, Orsay cedex, France

    Peter C. Richter

  5. Centre for Quantum Technologies, National University of Singapore, Singapore, 117543, Singapore

    Miklos Santha

Authors
  1. Frédéric Magniez
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  2. Ashwin Nayak
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  3. Peter C. Richter
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  4. Miklos Santha
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Corresponding author

Correspondence to Frédéric Magniez.

Additional information

A preliminary version of this article appeared in Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 86–95, 2009.

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Magniez, F., Nayak, A., Richter, P.C. et al. On the Hitting Times of Quantum Versus Random Walks. Algorithmica 63, 91–116 (2012). https://doi.org/10.1007/s00453-011-9521-6

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  • DOI: https://doi.org/10.1007/s00453-011-9521-6

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