Lackadaisical quantum walk for spatial search

Lackadaisical quantum walk(LQW) has been an efficient technique in searching a target state from a database which is distributed on a two-dimensional lattice. We numerically study the quantum search algorithm based on the lackadaisical quantum walk on one- and two-dimensions. It is observed that specific values of the self-loop weight at each vertex of the graph is responsible for such speedup of the algorithm. Searching for a target state on one-dimensional lattice with periodic boundary conditions is possible using lackadaisical quantum walk, which can find a target state with $\mathcal{O}(1)$ success probability after $\mathcal{O} \left( N \right)$ time steps. In two-dimensions, our numerical simulation upto $M=6$ suggests that lackadaisical quantum walk can search one of the $M$ target states in $\mathcal{O}\left(\sqrt{\frac{N}{M}\log \frac{N}{M}}\right)$ time steps.