State transfer in strongly regular graphs with an edge perturbation

Abstract

Quantum walks, an important tool in quantum computing, have been very successfully investigated using techniques in algebraic graph theory. We are motivated by the study of state transfer in continuous-time quantum walks, which is understood to be a rare and interesting phenomenon. We consider a perturbation on an edge uv of a graph where we add a weight β to the edge and a loop of weight γ to each of u and v. We characterize when for this perturbation results in strongly cospectral vertices u and v. Applying this to strongly regular graphs, we give infinite families of strongly regular graphs where some perturbation results in perfect state transfer. Further, we show that, for every strongly regular graph, there is some perturbation which results in pretty good state transfer. We also show for any strongly regular graph X and edge e ∈ E(X), that φ(X \ e) does not depend on the choice of e.

Characterizing cospectral vertices via isospectral reduction

“…In addition, G\a and G\b need not be isomorphic in order for a and b to be cospectral, it is only necessary for their adjacency matrices to have the same spectrum. It is known that in strongly regular graphs, any pair of vertices is cospectral [13], and most examples of strongly regular graphs have no automorphisms and will not have G\a isomorphic to G\b. In addition, graphs with cospectral pairs can be constructed from cospectral graphs involved in Godsil-McKay switching (see [15]).…”

Pretty Good State Transfer and Minimal Polynomials

“…Godsil, Guo, Kempton, and Lippner [17] considered the problem of achieving perfect and pretty good state transfer in strongly regular graphs. A graph G is strongly regular if it is neither empty nor complete, it is k-regular for some integer k, every pair of adjacent vertices has a common neighbours, and every pair of nonadjacent vertices has c common neighbours.…”

“…Theorem 1.5. [17] Let G be a strongly regular graph with eigenvalues k, θ, τ . Then if k ≡ θ ≡ τ (mod 4) is odd (resp.…”

“…Theorem 1.6. [17] Let G be a strongly regular graph. Then for any pair of vertices x, y of G, there is a choice of β, γ such that there is pretty good state transfer between x and y in G β,γ .…”

Pretty good quantum state transfer in asymmetric graphs via potential

“…Previous work in [15] showed that in graphs with an involutional symmetry, one can often induce pretty good state transfer between a pair of nodes by appropriately choosing a potential on the vertex set. In [10], it is shown that a potential can induce pretty good state transfer in strongly regular graphs as well. The contribution of this paper is to show how to construct asymmetric, non-regular graphs that admit PGST between a pair of nodes u, v if a suitably chosen potential is added to the adjacency matrix at u and v. The novelty of our constructions is that we do not require any symmetry or regularity in the graph.…”