Lackadaisical quantum walk for spatial search

Lackadaisical quantum walk (LQW) has been an efficient technique in searching for a target state in a database which is distributed in a two-dimensional lattice. We numerically study the quantum search algorithm based on the lackadaisical quantum walk in 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 a speedup of the algorithm. Searching for a target state in one-dimensional lattice with periodic boundary conditions is possible using lackadaisical quantum walk, which can find a target state with O(1) success probability after O(N) time steps. In two dimensions, our numerical simulation up to M = 6 for specific sets of target states suggests that the lackadaisical quantum walk can search one of the M target states in O(root N/M log N/M) time steps.