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Encyclopedia of Algorithms pp 1683–1691Cite as

Quantum Analogues of Markov Chains

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Years and Authors of Summarized Original Work

  • 2004; Szegedy

Problem Definition

Spatial Search and Walk Processes

Spatial search by quantum walk is database search with the additional constraint that one is required to move through the search space that obeys some locality structure. For example, the data items may be stored at the vertices of a two-dimensional grid. The requirement of moves along the edges of the grid captures the cost of accessing different items starting from some fixed position in the database.

One of possible ways of carrying out spatial search is by performing a random walk on the search space or its quantum analog, a quantum walk. The complexity of spatial search by quantum walk is strongly tied to the quantum hitting time [19] of the walk.

Let S, with | S | = n, be a finite set of states. Assume that a subset \(M \subseteq S\) of states are marked. We are given a procedure \(\mathcal{C}\) that, on input x ∈ S and an associated data structure d(x), checks...

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Author information

Authors and Affiliations

  1. Department of Combinatorics and Optimization, and Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada

    Ashwin Nayak & Peter C. Richter

  2. Department of Computer Science, Rutgers, The State University of New Jersey, New Brunswick, NJ, USA

    Mario Szegedy

Authors
  1. Ashwin Nayak
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  2. Peter C. Richter
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  3. Mario Szegedy
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Corresponding author

Correspondence to Ashwin Nayak .

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Nayak, A., Richter, P.C., Szegedy, M. (2016). Quantum Analogues of Markov Chains. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_302

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