Table of Contents
Journal of Computational Methods in Physics
Volume 2013, Article ID 235624, 5 pages
http://dx.doi.org/10.1155/2013/235624
Research Article

Preparation of Approximate Eigenvector by Unitary Operations on Eigenstate in Abrams-Lloyd Quantum Algorithm

Homi Bhabha Centre for Science Education, TIFR, Mumbai 400088, India

Received 6 May 2013; Revised 8 July 2013; Accepted 7 August 2013

Academic Editor: Gian Marco Rignanese

Copyright © 2013 Latha S. Warrier. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The Abrams-Lloyd quantum algorithm computes an eigenvalue and the corresponding eigenstate of a unitary matrix from an approximate eigenvector . The eigenstate is a basis vector in the orthonormal eigenspace. Finding another eigenvalue, using a random approximate eigenvector, may require many trials as the trial may repeatedly result in the eigenvalue measured earlier. We present a method involving orthogonalization of the eigenstate obtained in a trial. It is used as the for the next trial. Because of the orthogonal construction, Abrams-Lloyd algorithm will not repeat the eigenvalue measured earlier. Thus, all the eigenvalues are obtained in sequence without repetitions. An operator that anticommutes with a unitary operator orthogonalizes the eigenvectors of the unitary. We implemented the method on the programming language model of quantum computation and tested it on a unitary matrix representing the time evolution operator of a small spin chain. All the eigenvalues of the operator were obtained sequentially. Another use of the first eigenvector from Abrams-Lloyd algorithm is preparing a state that is the uniform superposition of all the eigenvectors. This is possible by nonorthogonalizing the first eigenvector in all dimensions and then applying the Abrams-Lloyd algorithm steps stopping short of the last measurement.