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Lecture notes in computer science

Generalized Grover Search Algorithm for Arbitrary Initial Amplitude Distribution

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

Published Jan 1, 1999• 87 citations
Publisher
DOI
10.1007/3-540-49208-9_10

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Grover's algorithm for quantum searching is generalized to deal with arbitrary initial complex amplitude distributions. First-order linear difference equations are found for the time evolution of the amplitudes of the marked and unmarked states. These equations are solved exactly. New expressions are derived for the optimal time of measurement and the maximal probability of success. They are found to depend on the averages and variances of the initial amplitude distributions of the marked and unmarked states, but not on higher moments. Our results imply that Grover's algorithm is robust against modest noise in the amplitude initialization procedure.
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 \sim O(\sqrt{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.
For searching any item in an unsorted database with N items, Grover's quantum searching algorithm takes only O(√N) steps, which is much faster than O(N) steps required for a classical computer. However, it is known that Grover's algorithm is effective only in the case where the initial amplitude distribution of dataset is uniform. On the other hand, Ventura has also proposed a quantum searching algorithm. The algorithm is effective only in the special case for the initial amplitude distribution. Therefore, we have proposed an effective quantum searching algorithm in another case in the previous paper. In this paper, we generalize the previous algorithm and propose several effective algorithms in several cases where initial amplitude distributions of dataset are non-uniform. Further, in order to show the effectiveness of the algorithms, we analyze their dynamics.