Quantum Probability and Spectral Analysis of Graphs

M. A. Jafarizadeh, Rahime Sufiani

Continuous-time quantum walk (CTQW) on a given graph is investigated using the techniques of the spectral analysis and inverse Laplace transform of the Stieltjes function (Stieltjes transform of the spectral distribution) associated with the graph. It is shown that the probability amplitude of observing the CTQW at a given site at time t is related to the inverse Laplace transformation of the Stieltjes function, namely, one can calculate the probability amplitudes only by taking the inverse laplace transform of the function iGμ(is), where Gμ(x) is the Stieltjes function of the graph. The preference of this procedure is that there is no need to know the spectrum of the graph.

Z. Pluhař, Hans A. Weidenmüller

Gamal Mograby, Benjamin Maloy, Anastasiia Minenkova

We study Grover's quantum walk on a new class of graphs, termed segmented complete graphs, which combine high symmetry with detailed spectral properties. Using these graphs, we implement Grover's search algorithm and investigate its performance, focussing on the relationship between graph volume, optimal search time and success probabilities. Our results generalize classical findings for directed weighted graphs and provide new insights into enhancing quantum search algorithms on complex graph structures.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.

Ide Yusuke, Norio Konno, Tomoyuki, Terada

In this paper, we consider a spectral analysis of the Szegedy walk on path graph. Here, we have an explicit formula for the probability measure of the walk by using eigenvalues and eigenvectors of the Jacobi matrix related to the birth and death chain.