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Canadian Journal of Physics

Quantum search with an ensemble of single-query search processes

F. Masafumi Toyama, W. van Dijk

Published Oct 8, 2020• 0 citations
Publisher

Abstract

Recently we introduced an additive composition formulation of an iterative quantum-search algorithm of Grover type. In this paper, an ensemble-processing scheme suggested by the additive composition is presented. The ensemble-processing consists of single-query search processes and yields the quantum search with certainty. Furthermore, we propose a possible quantum circuit for the ensemble processing.

DOI
10.1139/cjp-2020-0245

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Recently we introduced an additive composition formulation of an iterative quantum-search algorithm of Grover type. In this paper, an ensemble-processing scheme suggested by the additive composition is presented. The ensemble-processing consists of single-query search processes and yields the quantum search with certainty. Furthermore, we propose a possible quantum circuit for the ensemble processing.
In the Grover-type quantum search process a search operator is iteratively applied, say, k times, on the initial uniform superposition database state. We present an additive decomposition scheme such that the iteration process is expressed, in the computational space, as a linear combination of k operators, each of which consists of a single Grover-search followed by an overall phase-rotation. The value of k and the rotation phase are the same as those determined in the framework of the search with certainty. We discuss how the additive form can be effectively utilized and we propose an alternative gate that realizes the same outcome as the iterative search.
In the Grover-type quantum search process a search operator is iteratively applied, say, k times, on the initial database state. We present an additive decomposition scheme such that the iteration process is expressed, in the computational space, as a linear combination of k operators, each of which consists of a single Grover-search followed by an overall phase-rotation. The value of k and the rotation phase are the same as those determined in the framework of the search with certainty. We further show that the final state can be expressed in terms of a single oracle operator of the Grover-search and phase-rotation factors. We discuss how the additive form can be utilized so that it effectively reduces the computational load of the iterative search, and we propose an effective shortcut gate that realizes the same outcome as the iterative search.
IntroductionFor a search problem associated with a unsorted database, the remarkable Grover's quantum algorithm (1) provides a quadratic speedup over its classical counterpart.The search problem can be described as follows: for a given function f , there exists one unknown element in the set {0, 1, ..., N -1} that satisfies f (x)=-1, say x = τ, whereas the other N -1o n e sg i v e f (x)=1.How many times of evaluations of f are required to determine the element τ for f (τ)=-1?Through a conventional algorithm, one needs O(N) trials to achieve this aim.How about the utility of quantum algorithm for the search?For the scenario in the quantum world, the search problem can be rephrased in the quantum mechanical language: for a given unitary operator I τ , that is sometimes called the oracle operator, and a set of state vectors (orthonormal basis): s = {|0 , |1 , ..., |N -1 }, I τ |x = |x for all states in the set except I τ |x = -|x for x = τ.H owma n yqueriesofI τ are required to determine |τ ?By Grover's algorithm, one needs only O( √ N) quantum mechanical steps to find the marked state |τ out.It has been shown that Grover's algorithm is optimal since it needs minimal oracle calls to perform a quantum search (2).The quantum searching process will be briefly reviewed as follows.The first step of Grover's algorithm is to prepare a superposition state of all elements with uniform probability amplitude:Then apply the Grover kernel G = -I η I τ to |s ,w h er eI η is a unitary operator and contains no bias against the marked state.For large N,a f t e ra b o u tm = π √ N/4 repetitions of G operations, the probability to observe |τ is close to one, i.e., G m |s ∼ |τ .( 2 ) Since every single G involves one query of I τ ,onlyO( √ N) searching steps are required for a quantum search task.In what follows, we will first investigate on the general SU(2) formulation for the kernel of Grover's searching operator G.The discussions of quantum searching certainty, robustness, and the analog analogue version of the Grover's algorithm will be given afterwards.
Grover's quantum search algorithm has recently been implemented without entanglement, by replacing multiple particles with a single particle having exponentially many states. We recall that all physical resources must be accounted for to quantify algorithm complexity, and that this scheme typically incurs exponential costs in some other resource(s). In particular, we demonstrate that a recent experimental realization requires exponentially increasing precision. There is, however, a quantum algorithm which searches a "sophisticated" database (not unlike a Web search engine) with a single query, but which we show does not require entanglement even for multiparticle implementations.