The coined quantum walk is a discretization of the Dirac equation of relativistic quantum mechanics, and it is the basis of many quantum algorithms. We investigate how it searches the complete bipartite graph of N vertices for one of k marked vertices with different initial states. We prove intriguing dependence on the number of marked and unmarked vertices in each partite set. For example, when the graph is irregular and the initial state is the typical uniform superposition over the vertices, then the success probability can vary greatly from one timestep to the next, even alternating between 0 and 1, so the precise time at which measurement occurs is crucial. When the initial state is a uniform superposition over the edges, however, the success probability evolves smoothly. As another example, if the complete bipartite graph is regular, then the two initial states are equivalent. Then if two marked vertices are in the same partite set, the success probability reaches 1/2, but if they are in different partite sets, it instead reaches 1. This differs from the complete graph, which is the quantum walk formulation of Grover's algorithm, where the success probability with two marked vertices is 8/9. This reveals a contrast to the continuous-time quantum walk, whose evolution is governed by Schrödinger's equation, which asymptotically searches the regular complete bipartite graph with any arrangement of marked vertices in the same manner as the complete graph.

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 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, 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.

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