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Signal Processing

An Improved Grover Quantum Searching Algorithm

Lenan Wu

Published Jan 1, 2009• 2 citations
OpenAlex

Abstract

Grover quantum search algorithm access the unsorted database O (N~(1/2)) times,however,the probability of acquiring so- lutions usually falls with the increase of the number of solutions,the reason of which is analyzed and an improved algorithm based on en- larging searching space is proposed with a new parameter i to make the algorithm controllable in order to trade off between success proba- bility and iterative steps.Simulated experiments of the reciprocal problem showcase that this proposed algorithm can find solutions with higher probability than traditional Grover algorithm at the condition of equivalent iterative steps,and the success probability can even be increased if the iterative steps is unlimited.Thus,it can conclude that our proposed algorithm is more efficient and effective.The validity of the improved algorithm is manifested by experiment.

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Grover quantum search algorithm access the unsorted database O (N~(1/2)) times,however,the probability of acquiring so- lutions usually falls with the increase of the number of solutions,the reason of which is analyzed and an improved algorithm based on en- larging searching space is proposed with a new parameter i to make the algorithm controllable in order to trade off between success proba- bility and iterative steps.Simulated experiments of the reciprocal problem showcase that this proposed algorithm can find solutions with higher probability than traditional Grover algorithm at the condition of equivalent iterative steps,and the success probability can even be increased if the iterative steps is unlimited.Thus,it can conclude that our proposed algorithm is more efficient and effective.The validity of the improved algorithm is manifested by experiment.
The algorithm proposed by Grover solves the unsorted database search problem with computation complexity of O(2～(1/2)n/m),where the size of database is N=2n and there is m target solutions.However,the successful probability of searching decreases rapidly,when the target solutions m is greater than N/4,And the algorithm is disabled when m is equal to N/2.This paper presents an improved Grover's algorithm,which can find the object with the successful probability at least 98.01% with one step when m is greater than N/4.
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.
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.
Grover search algorithm is applied to the current number of objectives in quantum unsorted database, the meaning of each goal difference is not taken into account. When the target is more than a quarter of the total items, the probability of the target falls quickly identify and target increases, and when the target is more than half of the total items, the algorithm will fail. To solve these problems, first of all, based on a weighted index improvement is proposed in which each target in accordance with the right given to the importance of the weight coefficient. These different weight coefficients, we give all the target state, which may make the probability of each target equal weights of the quantum superposition. Secondly, an improved phase matching is proposed the phase rotation in both directions is the same, and the inner product of two states by the target amplitude superposition and initial state of the system is determined. When this inner product is 1/2 or more, when using the improved phase matching, the target may be 100% probability and by only one Grover iteration. Finally, the effectiveness of these measures by a simple search instance validation.