Search of clustered marked states with lackadaisical quantum walks

TL;DR

This paper improves the efficiency of finding clustered marked states in a grid using lackadaisical quantum walks by increasing success probability and proposing a classical search method for all marked states.

Contribution

It introduces a specific weight for the self-loop in lackadaisical quantum walks that enhances success probability and suggests a classical search approach to find all marked states efficiently.

Findings

Probability of finding a marked state increases by ~0.2 with the proposed method.

Classical search can find all marked states in O(√k) time after initial quantum walk.

The method reduces the number of quantum walk applications needed to locate all marked states.

Abstract

Nature of quantum walk in presence of multiple marked state has been studied by Nahimovs and Rivosh \cite{10.1007/978-3-662-49192-8_31}. They have shown that if the marked states are arranged in a $\sqrt{k} \times \sqrt{k}$ cluster in a $\sqrt{N} \times \sqrt{N}$ grid, then to find a single marked state among the multiple ones, quantum walk requires $\Omega(\sqrt{N} - \sqrt{k})$ time. In this paper, we show that using lackadaisical quantum walk with the weight of the self-loop as $\frac{4}{N(k + \lfloor{\frac{\sqrt{{k}}}{2}}\rfloor)}$, where $k$ is odd, the probability of finding a marked state increases by $\sim 0.2$. Furthermore, we show that instead of applying the quantum walk $\mathcal{O}(k)$ times to find all the marked states, classical search in the vicinity of the marked state found after the first implementation of the quantum walk can find all the marked states in $\mathcal{O}(\sqrt{k})$ time on average.