Abstract
Joining independent quantum searches provides novel collective modes of quantum search (merging) by utilizing the algorithm’s underlying algebraic structure. If n quantum searches, each targeting a single item, join the domains of their classical oracle functions and sum their Hilbert spaces (merging), instead of acting independently (concatenation), then they achieve a reduction of the search complexity by factor O(√n).
Highlights
The quantum search algorithm, from its initial conception [1,2,3], to the subsequent manifold of ongoing developments, see e.g., the various open research projects addressing the association of quantum search with e.g., quantum entanglement [4], quantum programming [5], error faultiness [6], fixed-point quantum search [7], and quantum walks [8], constitutes one of the pillars of the research area of quantum computing
Despite its simplicity and the manifested versatility in applications the algorithm remains a challenge to meet, especially when it is considered as a computational primitive that could be synthesized in non trivial ways with itself. This point of view is put forward in this work, where utilizing the underlying algebraic structure of the search algorithm and its matrix representation theory [9], the algorithm is treated as a computational unit
An important follow up of this work concerns the fact that the collective quantum search can be cast in the language of cooperative game theory, and so wider problems of search complexity reduction can be addressed
Summary
The quantum search algorithm, from its initial conception [1,2,3], to the subsequent manifold of ongoing developments, see e.g., the various open research projects addressing the association of quantum search with e.g., quantum entanglement [4], quantum programming [5], error faultiness [6], fixed-point quantum search [7], and quantum walks [8], constitutes one of the pillars of the research area of quantum computing. Despite its simplicity and the manifested versatility in applications the algorithm remains a challenge to meet, especially when it is considered as a computational primitive that could be synthesized in non trivial ways with itself This point of view is put forward in this work, where utilizing the underlying algebraic structure of the search algorithm and its matrix representation theory [9], the algorithm is treated as a computational unit. It is shown that the merging of n single searches with database dimensions Nk = 2k , k = 1, ..., n, causes a complexity reduction proportional of square root of n This main result of collective search is scrutinized in all intermediated joining schemes, where among n searches k are merged and the rest are left concatenated, via partitioning databases into distinct groups of merged algorithms and concatenating the resulting groups. The logistics of joining schemes is carried out via Young diagrams and tableaux of partitions, as well as majorization theory [10]. (Proofs and examples are placed in the second part of the paper)
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