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The nature of discrete-time quantum walk in the presence of multiple markedstates has been studied by Nahimovs and Rivosh. They introduced an exceptionalconfiguration of clustered marked states $i.e.,$ if the marked states arearranged 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 thatfinding a single marked state among the multiple ones using quantum walk withAKR (Ambainis, Kempe and Rivosh) coin requires $\Omega(\sqrt{N} - \sqrt{k})$time. Furthermore, Nahimov and Rivosh also showed that the Grover's coin canfind the same configuration of marked state both faster and with higherprobability compared to that with the AKR coin. In this article, we show thatusing lackadaisical quantum walk, a variant of a three-state discrete-timequantum walk on a line, the success probability of finding all the clusteredmarked states of this exceptional configuration is nearly 1 with smallerrun-time. We also show that the weights of the self-loop suggested for multiplemarked states in the state-of-the-art works are not optimal for thisexceptional configuration of clustered mark states. We propose a range ofweights of the self-loop from which only one can give the desired result forthis configuration.

Faster Search of Clustered Marked States with Lackadaisical Quantum Walks