Grover Search with Lackadaisical Quantum Walks

TL;DR

This paper explores how adding self-loops to vertices in quantum walks affects Grover's search algorithm, revealing that the impact varies with the type of quantum walk and the number of marked vertices.

Contribution

It introduces the concept of lackadaisical quantum walks and analyzes their effects on Grover's algorithm across different quantum walk models.

Findings

Self-loops can increase success probability to 1 with phase flip coin.

Additional self-loops can decrease success probability depending on the coin used.

Continuous-time quantum walks are unaffected by self-loops.

Abstract

The lazy random walk, where the walker has some probability of staying put, is a useful tool in classical algorithms. We propose a quantum analogue, the lackadaisical quantum walk, where each vertex is given $l$ self-loops, and we investigate its effects on Grover's algorithm when formulated as search for a marked vertex on the complete graph of $N$ vertices. For the discrete-time quantum walk using the phase flip coin, adding a self-loop to each vertex boosts the success probability from 1/2 to 1. Additional self-loops, however, decrease the success probability. Using instead the Ambainis, Kempe, and Rivosh (2005) coin, adding self-loops simply slows down the search. These coins also differ in that the first is faster than classical when $l$ scales less than $N$, while the second requires that $l$ scale less than $N^2$. Finally, continuous-time quantum walks differ from both of these discrete-time examples---the self-loops make no difference at all. These behaviors generalize to multiple marked vertices.