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Science Technology and Engineering

Analysis of Quantum Search Algorithms for Arbitrary Amplitude Distribution

Huifang Li

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

The problem of Grover's and Ventura's algorithms are not effective in any arbitrary amplitude distribution are studied.First,combining with the characteristics of quantum computing,a new algorithm is proposed and its amplitude dynamics with recursive function are analyzed.Further,a simulation with Matlab to compare the performance of three algorithms are made.Finally,the conclusion that the proposed algorithm can make up for Grover's and Ventura's deficiencies.If the value of m in advance is known,the appropriate algorithm to increase the probability of success can be seected.

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The problem of Grover's and Ventura's algorithms are not effective in any arbitrary amplitude distribution are studied.First,combining with the characteristics of quantum computing,a new algorithm is proposed and its amplitude dynamics with recursive function are analyzed.Further,a simulation with Matlab to compare the performance of three algorithms are made.Finally,the conclusion that the proposed algorithm can make up for Grover's and Ventura's deficiencies.If the value of m in advance is known,the appropriate algorithm to increase the probability of success can be seected.
For searching any item in an unsorted database with N items, Grover's quantum searching algorithm takes only O(√N) steps, which is much faster than O(N) steps required for a classical computer. However, it is known that Grover's algorithm is effective only in the case where the initial amplitude distribution of dataset is uniform. On the other hand, Ventura has also proposed a quantum searching algorithm. The algorithm is effective only in the special case for the initial amplitude distribution. Therefore, we have proposed an effective quantum searching algorithm in another case in the previous paper. In this paper, we generalize the previous algorithm and propose several effective algorithms in several cases where initial amplitude distributions of dataset are non-uniform. Further, in order to show the effectiveness of the algorithms, we analyze their dynamics.
For searching an unsorted database with N items, a classical computer requires O(N) steps but Grover's quantum searching algorithm requires only O(¿N) steps. However, it is also known that Grover's algorithm is effective in the case where the initial amplitude distribution of dataset is uniform, but is not always effective in the non-uniform case. In this paper, we will consider the influence of initial amplitude distribution of dataset for Grover's and Ventura's algorithms. Further, in order to improve the performance of quantum searching, we propose a new algorithm and analyze its dynamics.
Grover's algorithm for quantum searching is generalized to deal with arbitrary initial complex amplitude distributions. First-order linear difference equations are found for the time evolution of the amplitudes of the marked and unmarked states. These equations are solved exactly. New expressions are derived for the optimal time of measurement and the maximal probability of success. They are found to depend on the averages and variances of the initial amplitude distributions of the marked and unmarked states, but not on higher moments. Our results imply that Grover's algorithm is robust against modest noise in the amplitude initialization procedure.
A quantum computation problem of quantum search is discussed in this paper.Grover's search algorithm, the most commonly used quantum search algorithm, is introduced in details.The flow of quantum search algorithm and the quantum circuit model are shown.And the error of the search value, as well as the probability of measurement, is analyzed.For further research we simulated the Grover's search algorithm in classical computer.By quantum simulator the probability distribution of the measuring result of search value is presented and the computational efficiency is discussed.The simulation result demonstrates the effectiveness of the proposed algorithm.
The recursion equation analysis of Grover's quantum search algorithm presented by Biham et al. [Phys. Rev. A 60, 2742 (1999)] is generalized. It is applied to the large class of Grover-type algorithms in which the Hadamard transform is replaced by any other unitary transformation and the phase inversion is replaced by a rotation by an arbitrary angle. The time evolution of the amplitudes of the marked and unmarked states, for any initial complex amplitude distribution, is expressed using first-order linear difference equations. These equations are solved exactly. The solution provides the number of iterations T after which the probability of finding a marked state upon measurement is the highest, as well as the value of this probability, Pmax.{P}_{\mathrm{max}}.Pmax​. Both T and Pmax{P}_{\mathrm{max}}Pmax​ are found to depend on the averages and variances of the initial amplitude distributions of the marked and unmarked states, but not on higher moments.
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.