Robustness of Quantum Random Walk Search Algorithm in Hypercube when only first or both first and second neighbors are measured

TL;DR

This paper investigates the robustness of modified quantum random walk search algorithms on hypercubes, focusing on phase deviations and neighbor measurements, and finds that certain modifications enhance stability especially in large dimensions.

Contribution

It introduces and analyzes the robustness of two modifications of quantum random walk search algorithms, highlighting how neighbor measurements improve stability in high-dimensional hypercubes.

Findings

Unmodified algorithm's robustness improves with specific phase relations.

First modification does not affect robustness.

Neighbor measurements increase stability, especially in large dimensions.

Abstract

In this work we study the robustness of two modifications of quantum random walk search algorithm on hypercube. In the first previously suggested modification, on each even iteration only quantum walk is applied. And in the second, the closest neighbors of the solution are measured classically. In our approach the traversing coin is constructed by both generalized Householder reflection and an additional phase multiplier and we investigate the stability of the algorithm to deviations in those phases. We have shown that the unmodified algorithm becomes more robust when a certain relation between those phases is preserved. The first modification we study here does not lead to any change in the robustness of quantum random walk search algorithm. However, when a measurement of the first and second neighbors is included, there are some differences. The most important one, in view of our study of the robustness, is an increase in the stability of the algorithm, especially for large coin dimensions.