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Controlled quantum search

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Abstract

Quantum searching for one of N marked items in an unsorted database of n items is solved in \(\mathcal {O}(\sqrt{n/N})\) steps using Grover’s algorithm. Using nonlinear quantum dynamics with a Gross–Pitaevskii-type quadratic nonlinearity, Childs and Young (Phys Rev A 93:022314, 2016, https://doi.org/10.1103/PhysRevA.93.022314) discovered an unstructured quantum search algorithm with a complexity \(\mathcal {O}( \min \{ 1/g \, \log (g n), \sqrt{n} \}) \), which can be used to find a marked item after \(o(\log (n))\) repetitions, where g is the nonlinearity strength. In this work we develop an quantum search on a complete graph using a time-dependent nonlinearity which obtains one of the N marked items with certainty. The protocol has runtime \(\mathcal {O}(n /(g \sqrt{N(n-N)}))\), where g is related to the time-dependent nonlinearity. We also extend the analysis to a quantum search on general symmetric graphs and can greatly simplify the resulting equations when the graph diameter is less than 4.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of Western Australia, Perth, WA, 6009, Australia

    K. de Lacy & L. Noakes

  2. Department of Physics and Astronomy, ARC Centre for Engineered Quantum Systems, Macquarie University, Sydney, NSW, 2109, Australia

    J. Twamley

  3. School of Physics and Astrophysics, University of Western Australia, Perth, WA, 6009, Australia

    J. B. Wang

Authors
  1. K. de Lacy
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  2. L. Noakes
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  3. J. Twamley
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  4. J. B. Wang
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Correspondence to K. de Lacy.

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de Lacy, K., Noakes, L., Twamley, J. et al. Controlled quantum search. Quantum Inf Process 17, 266 (2018). https://doi.org/10.1007/s11128-018-2031-6

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  • DOI: https://doi.org/10.1007/s11128-018-2031-6

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