Robustness of different modifications of Grovers algorithm based on generalized Householder reflections with different phases

TL;DR

This paper evaluates five modifications of Grover's algorithm based on generalized Householder reflections, analyzing their robustness to phase inaccuracies and their success probabilities through semi-empirical methods.

Contribution

It introduces a comparative analysis of multiple Grover modifications' robustness to phase errors using semi-empirical techniques, providing insights into their stability and performance.

Findings

Two modifications use the same Grover operator each iteration.

Robustness varies significantly among the modifications.

Semi-empirical methods enable predictions for larger register sizes.

Abstract

In this work we study five Grovers algorithm modifications, where each iteration is constructed by two generalized Householder reflections, against inaccuracies in the phases. By using semi-empirical methods, we investigate various characteristics of the dependence between the probability to find solution and the phase errors. The first of them is the robustness of the probability to errors in the phase. The second one is how quickly the probability falls beyond the stability interval. And finally, the average success rate of the algorithm when the parameters are in the range of the highly robust interval. Two of the modifications require usage of the same Grover operator each iteration and in the other three it differs. Those semi-empirical methods give us the, tool to make prediction of the quantum algorithm modifications overall behavior and compare them for even larger register size