Investigation of continuous-time quantum walk via modules of Bose–Mesner and Terwilliger algebras

Abstract

The continuous-time quantum walk on the underlying graphs of association schemes have been studied, via the algebraic combinatorics structures of association schemes, namely semi-simple modules of their Bose-Mesner and (reference state dependent) Terwilliger algebras. By choosing the (walk) starting site as a reference state, the Terwilliger algebra connected with this choice turns the graph into the metric space, hence stratifies the graph into a (d+1) disjoint union of strata, where the amplitudes of observing the continuous-time quantum walk on all sites belonging to a given stratum are the same. In graphs of association schemes with known spectrum, the transition amplitudes and average probabilities are given in terms of dual eigenvalues of association schemes. As most of association schemes arise from finite groups, hence the continuous-time walk on generic group association schemes have been studied in great details, where the transition amplitudes are given in terms of characters of groups. Further investigated examples are the walk on graphs of association schemes of symmetric $S_n$, Dihedral $D_{2m}$ and cyclic groups. Also, following Ref.\cite{js}, the spectral distributions connected to the highest irreducible representations of Terwilliger algebras of some rather important graphs, namely distance regular graphs, have been presented. Then using spectral distribution, the amplitudes of continuous-time quantum walk on graphs such as cycle graph $C_n$, Johnson and normal subgroup graphs have been evaluated. {\bf Keywords: Continuous-time quantum walk, Association scheme, Bose-Mesner algebra, Terwilliger algebra, Spectral distribution, Distance regular graph.} {\bf PACs Index: 03.65.Ud}