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Chinese Physics Letters

Research on Quantum Searching Algorithms Based on Phase Shifts

Zhong Pu-Cha, Wan‐Su Bao

Published Jul 29, 2008• 7 citations
Publisher

Abstract

One iterative in Grover's original quantum search algorithm consists of two Hadamard–Walsh transformations, a selective amplitude inversion and a diffusion amplitude inversion. We concentrate on the relation among the probability of success of the algorithm, the phase shifts, the number of target items and the number of iterations via replacing the two amplitude inversions by phase shifts of an arbitrary ϕ = ϕ(0 ⩽ ϕ, ϕ ⩽ 2π). Then, according to the relation we find out the optimal phase shifts when the number of iterations is given. We present a new quantum search algorithm based on the optimal phase shifts of 1.018 after 0.5π/ iterations. The new algorithm can obtain either a single target item or multiple target items in the search space with the probability of success at least 93.43%.

DOI
10.1088/0256-307x/25/8/011

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One iterative in Grover's original quantum search algorithm consists of two Hadamard–Walsh transformations, a selective amplitude inversion and a diffusion amplitude inversion. We concentrate on the relation among the probability of success of the algorithm, the phase shifts, the number of target items and the number of iterations via replacing the two amplitude inversions by phase shifts of an arbitrary ϕ = ϕ(0 ⩽ ϕ, ϕ ⩽ 2π). Then, according to the relation we find out the optimal phase shifts when the number of iterations is given. We present a new quantum search algorithm based on the optimal phase shifts of 1.018 after 0.5π/ iterations. The new algorithm can obtain either a single target item or multiple target items in the search space with the probability of success at least 93.43%.
The success probability of the Grover quantum search algorithm decreases quickly when the fraction of target items exceeds 1/4, where the phase plays a significant role. Therefore, we use multiple phases to complement each other. We obtain three useful properties and an important theorem of the success probability and design a systematic solution of the optimal phases for an arbitrary number of phases. Based on these results, we finally propose a multi-phase quantum search algorithm whose success probability rises with the increase of the number of phases with just a single iteration, and it tends to be 100% when the fraction of target items is over a lower limit.
Each iteration in Grover’s original quantum search algorithm contains 4 steps: two Hadamard-Walsh transformations and two amplitudes inversions. When the inversion of the marked state is replaced by arbitrary phase rotation θ and the inversion for the prepared state |γ 〉 is replaced by rotation through φ, we found that these phase rotations must satisfy a matching condition θ = φ. Approximate formula for the amplitude of the marked state after an arbitrary number of iterations are also derived. When phase matching is obtained, we found that the generalized quantum search is still a rotation in 2-dimensional space, but with a small angle β ′ ≈ (2sin θ 2)β, where θ is the angle of phase rotation of the marked state and 2β is the angle of U(2) rotation when the phase rotations are the inversions in the original version. We give also a simple explanation of the phase matching requirement.
We investigate the phase matching problem in the fixed-point quantum search algorithm proposed by Grover [Phys. Rev. Lett.95 (2005) 150501]. We show that the optimal phase shift is π/3.61, which replaces the π/3 phase shift in fixed-point quantum search algorithm. The π/3.61-phase algorithm can be achieved in [Formula: see text] with the success probability of at least 94.11%, which offsets disadvantage that the success probability of getting correct results usually decreases with the increase of marked items when original Grover quantum search algorithm is applied to search an unordered database. In the meantime, this work also indicates that Grover quantum search algorithm is considerably robust to certain kinds of perturbations and is robust against modest noise in the initialization procedure.
The quantum search algorithm consists of an iterative sequence of selective inversions and diffusion type operations, as a result of which it is able to find a state with desired properties (target state) in an unsorted database of size N in only sqrt[N] queries. This is achieved by designing the iterative transformations in a way that each iteration results in a small rotation of the state vector in a two-dimensional Hilbert space that includes the target state; if we choose the right number of iterative steps, we will stop just at the target state. This Letter shows that by replacing the selective inversions by selective phase shifts of pi/3, the algorithm preferentially converges to the target state irrespective of the step size or number of iterations. This feature leads to robust search algorithms and also to new schemes for quantum control and error correction.
When the Grover’s algorithm is applied to search an unordered database, the probability of success usually decreases with the increase of marked items. To address this phenomenon, a fixed-phase quantum search algorithm with more flexible behavior is proposed. In proposed algorithm, the phase shifts can be fixed at the different values to meet the needs of different practical problems. If research requires a relatively rapid speed, the value of the phase shifts should be appropriately increased, if search requires a higher success probability, the value of the phase shifts should be appropriately decreased. When the phase shifts are fixed at , the success probability of at least 99.38% can be obtained in iterations.
Building quantum devices using fixed operators is a must to simplify the hardware construction. Quantum search engine is not an exception. In this paper, a fixed phase quantum search algorithm that searches for M matches in an unstructured search space of size N will be presented. Selecting phase shifts of 1.91684πin the standard amplitude amplification will make the technique perform better so as to get probability of success at least 99.58% in O(sqrt(N/M)) better than any know fixed operator quantum search algorithms. The algorithm will be able to handle either a single match or multiple matches in the search space. The algorithm will find a match in O(sqrt(N/M)) whether the number of matches is known or not in advance.