Inexact and exact quantum searches with a preparation state in a three-dimensional subspace

Inexact and exact quantum searches with a preparation state in a three-dimensional subspace It is well known that exact quantum searches can be performed by the quantum amplitude amplification algorithm with some phase matching condition. However, recently it was shown that for some preparation states in a three-dimensional subspace, an exact search is impossible to accomplish. We show this impossibility derives from two sources: a problem of state restriction to a cyclic subspace and the solution of a linear system of equations with a $$k$$ k -potent coefficient matrix. Furthermore, using said system of equations, we introduce a class of preparation states in a three-dimensional space that, even though the quantum amplitude amplification algorithm is unable to find the target state exactly, the same system of equations implies modifications to the quantum amplitude amplification algorithm under which exact solutions in three-dimensional subspaces can be found. We also prove that an inexact quantum search in the 3-potent case can find the target state with high probability if the Grover operator is iterated a number of times inversely proportional to the uncertainty of said 3-potent coefficient matrix as an observable operator. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

Inexact and exact quantum searches with a preparation state in a three-dimensional subspace

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Publisher
Springer Journals
Copyright
Copyright © 2014 by Springer Science+Business Media New York
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-014-0810-2
Publisher site
See Article on Publisher Site

Abstract

It is well known that exact quantum searches can be performed by the quantum amplitude amplification algorithm with some phase matching condition. However, recently it was shown that for some preparation states in a three-dimensional subspace, an exact search is impossible to accomplish. We show this impossibility derives from two sources: a problem of state restriction to a cyclic subspace and the solution of a linear system of equations with a $$k$$ k -potent coefficient matrix. Furthermore, using said system of equations, we introduce a class of preparation states in a three-dimensional space that, even though the quantum amplitude amplification algorithm is unable to find the target state exactly, the same system of equations implies modifications to the quantum amplitude amplification algorithm under which exact solutions in three-dimensional subspaces can be found. We also prove that an inexact quantum search in the 3-potent case can find the target state with high probability if the Grover operator is iterated a number of times inversely proportional to the uncertainty of said 3-potent coefficient matrix as an observable operator.

Journal

Quantum Information ProcessingSpringer Journals

Published: Aug 15, 2014

References

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