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Yuanzihe wuli pinglun

Phase Matching in Quantum Searching and the Improved Grover Algorithm

Yang Sun

Published Jan 1, 2004• 7 citations
OpenAlex

Abstract

We briefly introduced some of our recent work related to the phase matching condition in quantum searching algorithms and the improved Grover algorithm. When one replaces the two phase inversions in the Grover algorithm with arbitrary phase rotations, the modified algorithm usually fails in searching the marked state unless a phase matching condition is satisfied between the two phases. The Grover algorithm is not 100% in success rate, an improved Grover algorithm with zero-failure rate is given by replacing the phase inversions with angles that depends on the size of the database. Other aspects of the Grover algorithm such as the SO(3) picture of quantum searching, the dominant gate imperfections in the Grover algorithm are also mentioned.

Related papers

Phase matching in quantum searching

Gui‐Lu Long, Yan Li, Wei Lin Zhang +2 more

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.

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An improved phase error tolerance in Quantum search algorithm

Jin-Yuan Hsieh, Che‐Ming Li, Der-San Chuu

As the matching condition in Grover search algorithm is transgressed due to inevitable errors in phase inversions, it gives a reduction in maximum probability of success. With a given degree of maximum success, we have derive the generalized and imroved criterion for tolerated error and corresponding size of quantum database under the inevitable gate imperfections. The vanished inaccurancy to this condition has also been shown. Besides, a concise formula for evaluating minimum number of iterations is also presented in this work.

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When the current Grover algorithm is applied to search some objects in an unsorted quantum database,the probability of correct objects usually falls with the increase of the searched objects.The reason for this problem is analyzed in this paper,and an improved measure based on the new phase matching condition is proposed.In the new phase matching condition,the amplitudes of two phase rotations are the same and the directions of two phase rotations are contrary.When the objects are more than one third of the total items,with the new phase matching condition,all objects can be found by at least 25/27 of the probability and by the only one Grover iteration.The validity of the improved measure is proved by experiment.

Caai Transactions on Intelligent Systems•Jan 1, 2007•3 citations

Phase matching in quantum search algorithm

Saptarshi Roy Chowdhury, Sudarshan Baruah, Biswaranjan Dikshit

Abstract We systematically generalize the Grover algorithm in a density matrix formalism by exploiting the underlying two-dimensional subspace of the problem. Using this, we derive analytic expressions for the success probability after arbitrary iterations of the generalized Grover operator with two generic phase angles . We show for the phase matching condition with three iterations, success probability can be achieved only with knowledge about the lower bound , where λ is the ratio of marked to total number of states in the database. This result will improve the quantum search algorithm when applied to databases with unknown number of marked states in the specified regime of λ , at the cost of decreased efficiency in the smaller λ region.

Europhysics Letters (EPL)•Feb 8, 2023•2 citations

PHASE MATCHING IN FIXED-POINT QUANTUM SEARCH ALGORITHM

Hong‐Fu Wang, Shou Zhang, Yongfang Zhao

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.

International Journal of Quantum Information•Sep 1, 2009•2 citations
We briefly introduced some of our recent work related to the phase matching condition in quantum searching algorithms and the improved Grover algorithm. When one replaces the two phase inversions in the Grover algorithm with arbitrary phase rotations, the modified algorithm usually fails in searching the marked state unless a phase matching condition is satisfied between the two phases. The Grover algorithm is not 100% in success rate, an improved Grover algorithm with zero-failure rate is given by replacing the phase inversions with angles that depends on the size of the database. Other aspects of the Grover algorithm such as the SO(3) picture of quantum searching, the dominant gate imperfections in the Grover algorithm are also mentioned.
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.
As the matching condition in Grover search algorithm is transgressed due to inevitable errors in phase inversions, it gives a reduction in maximum probability of success. With a given degree of maximum success, we have derive the generalized and imroved criterion for tolerated error and corresponding size of quantum database under the inevitable gate imperfections. The vanished inaccurancy to this condition has also been shown. Besides, a concise formula for evaluating minimum number of iterations is also presented in this work.
In a standard Grover's algorithm for quantum searching, the probability of finding the marked item is not exactly 1. In this paper we present a modified version of Grover's algorithm that searches a marked state with full successful rate. The modification is done by replacing the phase inversion by phase rotation through angle $\ensuremath{\varphi}.$ The rotation angle is given analytically to be $\ensuremath{\varphi}=2\mathrm{arcsin}(\mathrm{sin}[\ensuremath{\pi}/(4J+6)]/\mathrm{sin}\ensuremath{\beta}),$ where $\mathrm{sin}\ensuremath{\beta}=1/\sqrt{N},$ N is the number of items in the database, and J is any integer equal to or greater than the integer part of $[(\ensuremath{\pi}/2)\ensuremath{-}\ensuremath{\beta}]/(2\ensuremath{\beta}).$ Upon measurement at the (J+1)th(J+1)\mathrm{th}(J+1)th iteration, the marked state is obtained with certainty.
When the current Grover algorithm is applied to search some objects in an unsorted quantum database,the probability of correct objects usually falls with the increase of the searched objects.The reason for this problem is analyzed in this paper,and an improved measure based on the new phase matching condition is proposed.In the new phase matching condition,the amplitudes of two phase rotations are the same and the directions of two phase rotations are contrary.When the objects are more than one third of the total items,with the new phase matching condition,all objects can be found by at least 25/27 of the probability and by the only one Grover iteration.The validity of the improved measure is proved by experiment.
Abstract We systematically generalize the Grover algorithm in a density matrix formalism by exploiting the underlying two-dimensional subspace of the problem. Using this, we derive analytic expressions for the success probability after arbitrary iterations of the generalized Grover operator with two generic phase angles . We show for the phase matching condition with three iterations, success probability can be achieved only with knowledge about the lower bound , where λ is the ratio of marked to total number of states in the database. This result will improve the quantum search algorithm when applied to databases with unknown number of marked states in the specified regime of λ , at the cost of decreased efficiency in the smaller λ region.
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.