  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.
