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Journal of Quantum Information Science

A Fixed-Phase Quantum Search Algorithm with More Flexible Behavior

Xin Li, Panchi Li

Published Jan 1, 2012• 9 citations
Publisher

Abstract

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.

DOI
10.4236/jqis.2012.22006

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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.
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.
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%.
When the Grover's algorithm is applied to search an unordered database, the probability of getting correct results usually decreases with the increase of marked items. The reason for this phenomenon is analyzed in this Letter, the Grover iteration is studied, and a new phase matching is proposed. With application of the new phase matching, when the fraction of marked items is greater than 1/3, the probability of getting correct results is greater than 25/27 with only one Grover iteration. The validity of the new phase matching is verified by a search example.
When the Grover's algorithm is applied to search an unordered database, the success probability usually decreases with the increase of marked items. The reason for this phenomenon was analyzed, the Grover iteration was studied, and a new adaptive phase rotation was proposed. With application of the new phase rotation, when the fraction of marked items is a rational number in range (3-5~(1/2)) / 8to 1/4, the success probability is equal to 1 after two Grover iterations; and when the fraction of marked items is greater than1/4, the success probability is equal to 1 after the only one Grover iteration. The effectiveness of the new phase rotation is verified by a search example.
We demonstrate that the rotation phase can be suitably chosen to increase the efficiency of the phase-based quantum search algorithm, leading to a dynamic balance between iterations and success probabilities of the fixed-phase quantum Grover search algorithm with Q-learning for a given number of solutions. In this search algorithm, the proposed Q-learning algorithm, which is a model-free reinforcement learning strategy in essence, is used for performing a matching algorithm based on the fraction of marked items λ and the rotation phase α. After establishing the policy function α = π(λ), we complete the fixed-phase Grover algorithm, where the phase parameter is selected via the learned policy. Simulation results show that the Q-learning-based Grover search algorithm (QLGA) enables fewer iterations and gives birth to higher success probabilities. Compared with the conventional Grover algorithms, it avoids the optimal local situations, thereby enabling success probabilities to approach one.