Investigation of Continuous-Time Quantum Walk Via Modules of Bose-Mesner and Terwilliger Algebras

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 with a distance function, hence stratifies the graph into a (d+1) disjoint union of strata (associate classes), where the amplitudes of observing the continuous-time quantum walk on all sites belonging to a given stratum are the same. Using the similarity of all vertices of underlying graph of an association scheme, it is shown that the transition probabilities between the vertices depend only on the distance between the vertices (kind of relations or association classes). Hence for a continuous-time quantum walk over a graph associated with a given scheme with diameter d, we have exactly (d + 1) different transition probabilities (i.e., the number of strata or number of distinct eigenvalues of adjacency matrix). In graphs of association schemes with known spectrum, namely with relevant BoseMesner algebras of known eigenvalues and eigenstates, 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 with real and complex representations 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 Sn, Dihedral D2m and cyclic groups. Also, following Ref.[1], 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 strongly regular graphs such as cycle graph Cn and Johnson, and strongly regular graphs such as Petersen graphs and normal subgroup continuous-time Quantum walk 3 graphs have been evaluated. Likewise, using the method of spectral distribution, we have evaluated the amplitudes of continuous-time quantum walk on symmetric product of trivial association schemes such as Hamming graphs, where their amplitudes are proportional to the product of amplitudes of constituent sub-graphs, and walk does not generate any entanglement between constituent sub-graphs.