Quantum sampling for balanced allocations

Authors:
Kazuo Iwama;Akinori Kawachi;Shigeru Yamashita
Affiliations:
Imai Quantum Computation and Information Project, ERATO, Japan Sci. and Tech. Corp., Kyoto, Japan and Graduate School of Informatics, Kyoto University, Kyoto, Japan;Imai Quantum Computation and Information Project, ERATO, Japan Sci. and Tech. Corp., Kyoto, Japan and Graduate School of Informatics, Kyoto University, Kyoto, Japan;Graduate School of Information Science, Nara Institute of Science and Technology and Imai Quantum Computation and Information Project, ERATO, Japan Sci. and Tech. Corp., Kyoto, Japan
Venue:
COCOON'03 Proceedings of the 9th annual international conference on Computing and combinatorics
Year:
2003

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Abstract

It is known that the original Grover Search (GS) can be modified to use a general value for the phase θ of the diffusion transform. Then, if the number of answers is relatively large, this modified GS can find one of the answers with probability one in a single iteration. However, such a quick and error-free GS can only be possible if we can initially adjust the value of θ correctly against the number of answers, and this seems very hard in usual occasions. A natural question now arises: Can we enjoy a merit even if GS is used without such an adjustment? In this paper, we give a positive answer using the balls-and-bins game in which the random sampling of bins is replaced by the quantum sampling, i.e., a single round of modified GS. It is shown that by using the quantum sampling: (i) The maximum load can be improved quadratically for the static model of the game and this improvement is optimal. (ii) That is also improved to O(1) for the continuous model if we have a certain knowledge about the total number of balls in the bins after the system becomes stable.