Quantum walk search on the complete bipartite graph

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 an 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 time step 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.

Figure 1

A complete bipartite graph with $N_1=4$ and $N_2=3$ vertices in each partite set.

Figure 2

The generalized complete bipartite graph containing $N_1$ vertices in set $X$ and $N_2$ vertices in set $Y$. There are $k$ marked vertices in set $X$, indicated by double circles. The vertices that evolve identically share the same color and label.

Figure 3

Success probability for search on the complete bipartite graph from initial state $\left|s\right.〉$ with $k=3$ marked vertices and (a) $N_1=N_2=400$ vertices, (b) $N_1=400$ vertices and $N_2=200$ vertices, (c) $N_1=400$ vertices and $N_2=1$ vertex, (d) $N_1=200$ vertices and $N_2=400$ vertices, and (e) $N_1=3$ vertices and $N_2=400$ vertices.

Figure 4

Success probability for search on the complete bipartite graph from initial state $\left|\sigma \right.〉$ with $k=3$ marked vertices and (a) $N_1=N_2=400$ vertices, (b) $N_1=400$ vertices and $N_2=200$ vertices, (c) $N_1=400$ vertices and $N_2=1$ vertex, (d) $N_1=200$ vertices and $N_2=400$ vertices, and (e) $N_1=3$ vertices and $N_2=400$ vertices.

Figure 5

The generalized complete bipartite graph containing $N_1$ vertices in set $X$ and $N_2$ vertices in set $Y$. There are $k_1$ marked vertices in set $X$ and $k_2$ marked vertices in set $Y$, indicated by double circles. The vertices that evolve identically share the same color and label.

Figure 6

Success probability for search on the complete bipartite graph from initial state $\left|s\right.〉$ with $k_1=3$ marked vertices in set $X$, $k_2=2$ marked vertices in set $Y$, and (a) $N_1=400$ and $N_2=600$ vertices, (b) $N_1=600$ and $N_2=400$ vertices, and (c) $N_1=3$ and $N_2=997$ vertices. The solid black curve is the total probability at all the marked vertices, the dashed red curve is the probability at marked vertices in set $X$, and the dotted green curve is the probability at the marked vertices in set $Y$.

Figure 7

Success probability for search on the complete bipartite graph from initial state $\left|\sigma \right.〉$ with $k_1=3$ marked vertices in set $X$, $k_2=2$ marked vertices in set $Y$, and (a) $N_1=400$ and $N_2=600$ vertices, (b) $N_1=600$ and $N_2=400$ vertices, and (c) $N_1=3$ and $N_2=997$ vertices. The solid black curve is the total probability at all the marked vertices, the dashed red curve is the probability at marked vertices in set $X$, and the dotted green curve is the probability at the marked vertices in set $Y$.