Faster Search of Clustered Marked States with Lackadaisical Quantum Walks

TL;DR

This paper demonstrates that lackadaisical quantum walks significantly improve the success probability and reduce the runtime for locating clustered marked states in a grid, outperforming previous quantum walk methods.

Contribution

It introduces a novel application of lackadaisical quantum walks to efficiently find clustered marked states, optimizing self-loop weights for this specific configuration.

Findings

Success probability nearly 1 with smaller runtime using lackadaisical quantum walks.

Existing self-loop weights are suboptimal for clustered marked states.

Proposed weight range improves search efficiency for this configuration.

Abstract

The nature of discrete-time quantum walk in the presence of multiple marked states has been studied by Nahimovs and Rivosh. They introduced an exceptional configuration of clustered marked states $i.e.,$ if the marked states are arranged in a $\sqrt{k} \times \sqrt{k}$ cluster within a $\sqrt{N} \times \sqrt{N}$ grid, where $k=n^{2}$ and $n$ an odd integer. They showed that finding a single marked state among the multiple ones using quantum walk with AKR (Ambainis, Kempe and Rivosh) coin requires $\Omega(\sqrt{N} - \sqrt{k})$ time. Furthermore, Nahimov and Rivosh also showed that the Grover's coin can find the same configuration of marked state both faster and with higher probability compared to that with the AKR coin. In this article, we show that using lackadaisical quantum walk, a variant of a three-state discrete-time quantum walk on a line, the success probability of finding all the clustered marked states of this exceptional configuration is nearly 1 with smaller run-time. We also show that the weights of the self-loop suggested for multiple marked states in the state-of-the-art works are not optimal for this exceptional configuration of clustered mark states. We propose a range of weights of the self-loop from which only one can give the desired result for this configuration.