Search by Lackadaisical Quantum Walk with Nonhomogeneous Weights

The lackadaisical quantum walk, a quantum walk with weighted self-loops, speeds up dispersion on a line and improves spatial search on the complete graph and periodic square lattice. In these investigations, each self-loop had the same weight, due to each graph's vertex-transitivity. In this paper, we propose lackadaisical quantum walks with self-loops of different weight. We investigate spatial search on the complete bipartite graph, which can be irregular with partitions of size N₁ and N₂, which naturally leads to self-loops having different weights l₁ and l₂, respectively. We prove that if the k marked vertices are confined to one partite set, then with the typical initial uniform state over the vertices, the success probability is improved from its non-lackadaisical value when l₁=kN₂/2N₁ and N₂>(3−2√2)N₁, regardless of l₂. When the initial state is stationary under the quantum walk, however, the success probability is improved when l₁=kN₂/2N₁ without a constraint on the ratio of N₁ and N₂. 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.