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

Patel, Apoorva D and Raghunathan, KS and Rahaman, Md Aminoor (2010) Search on a hypercubic lattice using a quantum random walk. II. d = 2. In: Physical Review A, 82 (3).

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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(root N ln N) algorithm, which is not optimal. The scaling behavior can be improved to O(root N ln N) 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.

Item Type:	Journal Article
Publication:	Physical Review A
Publisher:	The American Physical Society
Additional Information:	Copyright of this article belongs to The American Physical Society.
Department/Centre:	Division of Interdisciplinary Sciences > Supercomputer Education & Research Centre Division of Physical & Mathematical Sciences > Centre for High Energy Physics
Date Deposited:	18 Oct 2010 10:08
Last Modified:	18 Oct 2010 10:08
URI:	http://eprints.iisc.ac.in/id/eprint/33267

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