Adjustable self-loop on discrete-time quantum walk and its application in spatial search

TL;DR

This paper introduces a novel adjustable self-loop model for discrete-time quantum walks, enabling continuous control of self-loop effects and significantly improving spatial search success rates.

Contribution

The authors propose a real-parameter controlled self-loop model for quantum walks, overcoming integer limitations and enhancing search algorithm performance.

Findings

Success rate on 20x20 lattice increased from 23.6% to 97.2%.

Improved search success scales better with lattice size.

First demonstration of such an improvement in quantum walk spatial search.

Abstract

How self-loops on vertices affect quantum walks is an interesting issue, and self-loops play important roles in quantum walk based algorithms. However, the original model that adjusting the effect of self-loops by changing their number has limitations. For example, the effect of self-loops cannot be adjusted continuously, for their number must be an integer. In this paper, we proposed a model of adjustable self-loop on discrete-time quantum walk, whose weight is controlled by a real parameter in the coin operator. The proposed method not only generalises the situations where the number of self-loops is an integer, but also provides a way to adjust the weight of the self-loop continuously. It enhances the potential of self-loops in applications. For instance, we improve the success rate of the quantum walk based search on a $20\times20$ two-dimension lattice from $23.6\%$ to $97.2\%$ by the proposed method. And the success rate of the improved search, which only scales as $O(1/\log{N})$ before being improved, even increases slightly with the size of the lattice. To the best of our knowledge, this is the first time that such an improvement is achieved on the quantum walk based spatial search.