Abstract
The lackadaisical quantum walk, which is a quantum walk with a weighted self-loop at each vertex, has been shown to speed up dispersion on the line and improve spatial search on the complete graph and periodic square lattice. In these investigations, each self-loop had the same weight, owing to each graph's vertex transitivity. In this paper, we propose lackadaisical quantum walks where the self-loops have different weights. We investigate spatial search on the complete bipartite graph, which can be irregular with N1 and N2 vertices in each partite set, and this naturally leads to self-loops in each partite set having different weights l1 and l2, respectively. We analytically prove that for large N1 and N2, if the k marked vertices are confined to, say, the first partite set, then with the typical initial uniform state over the vertices, the success probability is improved from its nonlackadaisical value when l1=kN2/2N1 and N2>(3-22)N1, regardless of l2. When the initial state is stationary under the quantum walk, however, then the success probability is improved when l1=kN2/2N1, now without a constraint on the ratio of N1 and N2, and again independent of l2. Next, when marked vertices lie in both partite sets, then for either initial state there are many configurations for which the self-loops yield no improvement in quantum search, no matter what weights they take.