Perfect state transfer, integral circulants, and join of graphs

Abstract

We propose new families of graphs which exhibit quantum perfect state transfer. Our constructions are based on the join operator on graphs, its circulant generalizations, and the Cartesian product of graphs. We build upon the results of Ba\v{s}i\'{c} and Petkovi\'{c} ({\em Applied Mathematics Letters} {\bf 22}(10):1609-1615, 2009) and construct new integral circulants and regular graphs with perfect state transfer. More specifically, we show that the integral circulant $\textsc{ICG}_{n}(\{2,n/2^{b}\} \cup Q)$ has perfect state transfer, where $b \in \{1,2\}$, $n$ is a multiple of $16$ and $Q$ is a subset of the odd divisors of $n$. Using the standard join of graphs, we also show a family of double-cone graphs which are non-periodic but exhibit perfect state transfer. This class of graphs is constructed by simply taking the join of the empty two-vertex graph with a specific class of regular graphs. This answers a question posed by Godsil (arxiv.org math/08062074).

Perfect State Transfer in Weighted Cubelike Graphs

“…In [5], it is shown that the join of a weighted two-vertex graph with any regular graph has perfect state transfer. For more reference on weighted graphs allowing PST, see [4,19,31].…”

Perfect quantum state transfer of hard-core bosons on weighted path graphs

“…The standard definition of an equitable partition for an unweighted graph via the normalized partition matrix (12) must be extended to the case of weighted graphs in order to prove the main results of the present work. This extension has not previously been made, to the best knowledge of the authors, and so it is outlined here with full definitions to be found in Appendix A.…”

“…A complete set of coupling constants for 1D chains has been subsequently characterized [3]. The singleexcitation subspace of a spin network is equivalent to a single particle undergoing a continuous-time quantum walk on a graph with the same connectivity [4], and in this context PST has been shown to occur on variety of graphs including graph quotients and joins, circulants, double cones, cubelike graphs, and signed graphs, among others [5][6][7][8][9][10][11][12][13][14][15]. If local control is permitted at the endpoints, then PST is possible on a wider assortment of networks [16,17].…”

Perfect quantum state transfer on diamond fractal graphs

“…In addition to its fundamental interest, this means that perfect quantum state transfer also has potential applications to the design of sub-protocols for quantum information and quantum computation [3][4][5]. A number of one dimensional cases, when perfect transmission can be achieved, have been found in some XX chains with inhomogeneous couplings, see [2,3,[6][7][8][9][10][11][12][13][14][15], and references therein]. These models have the advantage that the perfect transfer can be done without the need for active control.…”