Generalized grover search algorithm for arbitrary initial amplitude distribution

David Biron; Ofer Biham; Eli Biham; Markus Grassl; Daniel A. Lidar

doi:10.1007/3-540-49208-9_10

Generalized grover search algorithm for arbitrary initial amplitude distribution

David Biron, Ofer Biham, Eli Biham, Markus Grassl, Daniel A. Lidar

Racah Institute of Physics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

45 Scopus citations

Abstract

Grover’s algorithm for quantum searching of a database is generalized to deal with arbitrary initial amplitude distributions. First order linear difference equations are found for the time evolution of the amplitudes of the r marked and N − r unmarked states. These equations are solved exactly. An expression for the optimal measurement time T ∼ O(√N/r) is derived which is shown to depend only on the initial average amplitudes of the marked and unmarked states. A bound on the probability of measuring a marked state is derived, which depends only on the standard deviation of the initial amplitude distributions of the marked or unmarked states.

Original language	English
Title of host publication	Quantum Computing and Quantum Communications - 1st NASA International Conference, QCQC 1998, Selected Papers
Editors	Colin P. Williams
Publisher	Springer Verlag
Pages	140-147
Number of pages	8
ISBN (Print)	354065514X, 9783540655145
DOIs	https://doi.org/10.1007/3-540-49208-9_10
State	Published - 1999
Event	1st NASA International Conference on Quantum Computing and Quantum Communications, QCQC 1998 - Palm Springs, United States Duration: 17 Feb 1998 → 20 Feb 1998

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	1509
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	1st NASA International Conference on Quantum Computing and Quantum Communications, QCQC 1998
Country/Territory	United States
City	Palm Springs
Period	17/02/98 → 20/02/98

Bibliographical note

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1999.

Keywords

Exact solution
Grover’s algorithm
Quantum searching

Access to Document

10.1007/3-540-49208-9_10

Cite this

Biron, D., Biham, O., Biham, E., Grassl, M., & Lidar, D. A. (1999). Generalized grover search algorithm for arbitrary initial amplitude distribution. In C. P. Williams (Ed.), Quantum Computing and Quantum Communications - 1st NASA International Conference, QCQC 1998, Selected Papers (pp. 140-147). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1509). Springer Verlag. https://doi.org/10.1007/3-540-49208-9_10

Biron, David ; Biham, Ofer ; Biham, Eli et al. / Generalized grover search algorithm for arbitrary initial amplitude distribution. Quantum Computing and Quantum Communications - 1st NASA International Conference, QCQC 1998, Selected Papers. editor / Colin P. Williams. Springer Verlag, 1999. pp. 140-147 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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title = "Generalized grover search algorithm for arbitrary initial amplitude distribution",

abstract = "Grover{\textquoteright}s algorithm for quantum searching of a database is generalized to deal with arbitrary initial amplitude distributions. First order linear difference equations are found for the time evolution of the amplitudes of the r marked and N − r unmarked states. These equations are solved exactly. An expression for the optimal measurement time T ∼ O(√N/r) is derived which is shown to depend only on the initial average amplitudes of the marked and unmarked states. A bound on the probability of measuring a marked state is derived, which depends only on the standard deviation of the initial amplitude distributions of the marked or unmarked states.",

keywords = "Exact solution, Grover{\textquoteright}s algorithm, Quantum searching",

author = "David Biron and Ofer Biham and Eli Biham and Markus Grassl and Lidar, \{Daniel A.\}",

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year = "1999",

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Biron, D, Biham, O, Biham, E, Grassl, M & Lidar, DA 1999, Generalized grover search algorithm for arbitrary initial amplitude distribution. in CP Williams (ed.), Quantum Computing and Quantum Communications - 1st NASA International Conference, QCQC 1998, Selected Papers. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1509, Springer Verlag, pp. 140-147, 1st NASA International Conference on Quantum Computing and Quantum Communications, QCQC 1998, Palm Springs, United States, 17/02/98. https://doi.org/10.1007/3-540-49208-9_10

Generalized grover search algorithm for arbitrary initial amplitude distribution. / Biron, David; Biham, Ofer; Biham, Eli et al.
Quantum Computing and Quantum Communications - 1st NASA International Conference, QCQC 1998, Selected Papers. ed. / Colin P. Williams. Springer Verlag, 1999. p. 140-147 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1509).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Biron D, Biham O, Biham E, Grassl M, Lidar DA. Generalized grover search algorithm for arbitrary initial amplitude distribution. In Williams CP, editor, Quantum Computing and Quantum Communications - 1st NASA International Conference, QCQC 1998, Selected Papers. Springer Verlag. 1999. p. 140-147. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/3-540-49208-9_10

Generalized grover search algorithm for arbitrary initial amplitude distribution

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