Universality of the fully connected vertex in Laplacian continuous-time quantum walk problems

TL;DR

This paper proves that the behavior of continuous-time quantum walks with Laplacian Hamiltonians on graphs with a fully connected vertex is universal, independent of the specific graph structure, and applies this to quantum search and transport.

Contribution

It establishes the universality of CTQW dynamics at fully connected vertices across different graphs with Laplacian Hamiltonians, extending known results to a broader class of graphs.

Findings

Quantum walk probabilities at fully connected vertices are graph-independent.

Spatial search on any graph with a fully connected vertex is optimal, like on the complete graph.

Results unify and extend previous findings on quantum transport and search algorithms.

Abstract

A fully connected vertex $w$ in a simple graph $G$ of order $N$ is a vertex connected to all the other $N-1$ vertices. Upon denoting by $L$ the Laplacian matrix of the graph, we prove that the continuous-time quantum walk (CTQW) -- with Hamiltonian $H=\gamma L$ -- of a walker initially localized at $\vert w \rangle$ does not depend on the graph $G$. We also prove that for any Grover-like CTQW -- with Hamiltonian $H=\gamma L +\sum_w \lambda_w \vert w \rangle\langle w \vert$ -- the probability amplitude at the fully connected marked vertices $w$ does not depend on $G$. The result does not hold for CTQW with Hamiltonian $H=\gamma A$ (adjacency matrix). We apply our results to spatial search and quantum transport for single and multiple fully connected marked vertices, proving that CTQWs on any graph $G$ inherit the properties already known for the complete graph of the same order, including the optimality of the spatial search. Our results provide a unified framework for several partial results already reported in literature for fully connected vertices, such as the equivalence of CTQW and of spatial search for the central vertex of the star and wheel graph, and any vertex of the complete graph.