Perfect quantum state transfer on the Johnson scheme

Abstract

For any graph X with the adjacency matrix A, the transition matrix of the continuous-time quantum walk at time t is given by the matrix-valued function H X (t) = e itA . We say that there is perfect state transfer in X from the vertex u to the vertex v at time τ if |H X (τ ) u,v | = 1. It is an important problem to determine whether perfect state transfers can happen on a given family of graphs. In this paper we characterize all the graphs in the Johnson scheme which have this property. Indeed, we show that the Kneser graph K(2k, k) is the only class in the scheme which admits perfect state transfers. We also show that, under some conditions, some of the unions of the graphs in the Johnson scheme admit perfect state transfer.

Perfect state transfer on abelian Cayley graphs

“…x ∈ G 0 } ≥ 2. On the other hand, (χ 6 (S), · · · , χ 9 (S)) = (−2, 2, 2, −2) and (d − χ 6 (S), · · · , d − χ 9 (S)) = (6,2,2,6). Thus for each y ∈ G 1 , v 2 (d − α y ) = v 2 (d − χ j (S)) = 1, j ∈ {6, 7, 8, 9}.…”

Perfect State Transfer on Cayley Graphs over Dihedral Groups: The Non-Normal Case

“…The phenomenon of perfect state transfer (PST) in quantum communication networks was originally introduced by Bose in [14]. This work has attracted much research interest since many applications have been found in quantum information processing and cryptography (see [1,2,3,5,11,15,17,18,34,38,37] and the references therein).…”

Perfect state transfer in two dimensions and the bivariate dual-Hahn polynomials

“…The dual-Hahn polynomials are attached to the Johnson scheme [17]. It should be noted however that while PST occurs in a spin chain based on the dual Hahn polynomials, no lifts are possible to unweighted graphs of the Johnson scheme [1,19] owing to parameter incompatibility. Moreover to our knowledge no analogue of the ordered Hamming scheme to which multivariate dual-Hahn would be connected has been designed.…”