Laplacian versus adjacency matrix in quantum walk search
Abstract
A quantum particle evolving by Schrödinger's equation contains, from the kinetic energy of the particle, a term in its Hamiltonian proportional to Laplace's operator. In discrete space, this is replaced by the discrete or graph Laplacian, which gives rise to a continuoustime quantum walk. Besides this natural definition, some quantum walk algorithms instead use the adjacency matrix to effect the walk. While this is equivalent to the Laplacian for regular graphs, it is different for nonregular graphs and is thus an inequivalent quantum walk. We algorithmically explore this distinction by analyzing search on the complete bipartite graph with multiple marked vertices, using both the Laplacian and adjacency matrix. The two walks differ qualitatively and quantitatively in their required jumping rate, runtime, sampling of marked vertices, and in what constitutes a natural initial state. Thus the choice of the Laplacian or adjacency matrix to effect the walk has important algorithmic consequences.
 Publication:

Quantum Information Processing
 Pub Date:
 October 2016
 DOI:
 10.1007/s1112801613731
 arXiv:
 arXiv:1512.05554
 Bibcode:
 2016QuIP...15.4029W
 Keywords:

 Quantum walk;
 Continuous time;
 Spatial search;
 Laplacian;
 Adjacency matrix;
 Quantum Physics
 EPrint:
 21 pages, 8 figures