Preparation of Approximate Eigenvector by Unitary Operations on Eigenstate in Abrams-Lloyd Quantum Algorithm. (11th September 2013)

• Record Type: Journal Article
• Authors: Warrier, Latha S.
• Other Names: Rignanese Gian Marco Academic Editor.
• Is Part Of: Journal of computational methods in physics. Volume 2013(2013)
• Publication Date: 2013-09-11
• Subjects: Physics -- Data processing -- Periodicals; Mathematical physics -- Data processing -- Periodicals; Mathematical physics -- Data processing; Physics -- Data processing; Electronic journals; Periodicals; Electronic journals; 530.0285
• DOI: 10.1155/2013/235624
• Languages: English
• ISSNs: 2356-7287

Abstract

Abstract : The Abrams-Lloyd quantum algorithm computes an eigenvalue and the corresponding eigenstate of a unitary matrix from an approximate eigenvectorV a . 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 theV a 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.