Performance Analysis and Parameter Optimization of the Optimal Fixed-Point Quantum Search

Tianyi Bao,Duan Huang

doi:10.3390/app9173584

Abstract

The optimal fixed-point quantum search (OFPQS) algorithm [Phys. Rev. Lett. 113, 210501 (2014)] achieves both the fixed-point property and quadratic speedup over classical algorithms, which gives a sufficient condition on the number of iterations to ensure the success probability is no less than a given lower bound (denoted by 1 − δ 2 ). However, this condition is approximate and not exact. In this paper, we derive the sufficient and necessary condition on the number of feasible iterations, based on which the exact least number of iterations can be obtained. For example, when δ = 0 . 8 , iterations can be saved by almost 25%. Moreover, to find a target item certainly, setting directly 1 − δ 2 = 100 % , the quadratic advantage of the OFPQS algorithm will be lost, then, applying the OFPQS algorithm with 1 − δ 2 < 100 % requires multiple executions, which leads to a natural problem of choosing the optimal parameter δ . For this, we analyze the extreme and minimum properties of the success probability and further analytically derive the optimal δ which minimizes the query complexity. Our study can be a guideline for both the theoretical and application research on the fixed-point quantum search algorithms.

Highlights

Grover search [1,2] provides a quadratic speedup over classical search algorithms, and has been proven optimal [3,4,5,6]
Inspired by [4], which achieves about 12% reduction of the expected queries through stopping the Grover algorithm short of the optimal number of iterations and restarting again in case of failure, to find a target item as soon as possible, a natural strategy is to set the lower bound of success probability 1 − δ2 < 100% and repeat the optimal fixed-point quantum search (OFPQS) algorithm until it succeeds
Different from our optimal number of iterations lopt of Equation (13), to achieve a success probability no less than 1 − δ2 for any λ ≥ λ0 in the OFPQS algorithm [20], another number of iterations can be obtained from Equation (12), denoted by

Summary

Introduction

Grover search [1,2] provides a quadratic speedup over classical search algorithms, and has been proven optimal [3,4,5,6]. Based on the quantum amplitude amplification [9,10,11,12] and phase-matching methods [13,14,15,16], a fixed-point quantum search algorithm [17] has been proposed, where the final state of the algorithm converges to the target states and the success probability increases as the number of iterations grows This algorithm applies to the case where the lower bound (denoted by λ0 ) of the fraction of target items (denoted by λ) is known. We expect to give the minimum feasible number of iterations, analyze the extreme and minimum properties of the success probability, and further derive the optimal parameter of the OFPQS algorithm to find a target item with the minimum query complexity.

OFPQS Algorithm Revisited

Performance Analysis of the OFPQS Algorithm

Optimization of Parameters of the OFPQS Algorithm

Findings

Effects of lopt and δopt

Conclusions