Search on a hypercubic lattice using a quantum random walk. II. d=2

Apoorva Patel, K. S. Raghunathan, and Md. Aminoor Rahaman
Phys. Rev. A 82, 032331 – Published 28 September 2010

Abstract

We investigate the spatial search problem on the two-dimensional square lattice, using the Dirac evolution operator discretized according to the staggered lattice fermion formalism. d=2 is the critical dimension for the spatial search problem, where infrared divergence of the evolution operator leads to logarithmic factors in the scaling behavior. As a result, the construction used in our accompanying article [A. Patel and M. A. Rahaman, Phys. Rev. A 82, 032330 (2010)] provides an O(NlnN) algorithm, which is not optimal. The scaling behavior can be improved to O(NlnN) by cleverly controlling the massless Dirac evolution operator by an ancilla qubit, as proposed by Tulsi [Phys. Rev. A 78, 012310 (2008)]. We reinterpret the ancilla control as introduction of an effective mass at the marked vertex, and optimize the proportionality constants of the scaling behavior of the algorithm by numerically tuning the parameters.

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  • Received 6 April 2010

DOI:https://doi.org/10.1103/PhysRevA.82.032331

©2010 American Physical Society

Authors & Affiliations

Apoorva Patel1,2,*, K. S. Raghunathan1,†, and Md. Aminoor Rahaman2,‡

  • 1Centre for High Energy Physics, Indian Institute of Science, Bangalore-560012, India
  • 2Supercomputer Education and Research Centre, Indian Institute of Science, Bangalore-560012, India

  • *adpatel@cts.iisc.ernet.in
  • ksraghu@yahoo.com
  • aminoorrahaman@yahoo.com

See Also

Search on a hypercubic lattice using a quantum random walk. I. d>2

Apoorva Patel and Md. Aminoor Rahaman
Phys. Rev. A 82, 032330 (2010)

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Vol. 82, Iss. 3 — September 2010

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