# Perfect state transfer of a qudit over underlying networks of group association schemes

M. A. Jafarizadeh, R. Sufiania, S. F. Taghavia and E. Barati

Arxiv ID: 0901.4504•Last updated: 12/29/2021

As generalizations of results of Christandl et al.\cite{8,9""} and Facer et
al.\cite{Facer}, Bernasconi et al.\cite{godsil,godsil1} studied perfect state
transfer (PST) between two particles in quantum networks modeled by a large
class of cubelike graphs (e.g., the hypercube) which are the Cayley graphs of
the elementary abelian group $Z_2^n$. In Refs. \cite{PST,psd}, respectively,
PST of a qubit over distance regular spin networks and optimal state transfer
(ST) of a $d$-level quantum state (qudit) over pseudo distance regular networks
were discussed, where the networks considered there were not in general related
with a certain finite group. In this paper, PST of a qudit over antipodes of
more general networks called underlying networks of association schemes, is
investigated. In particular, we consider the underlying networks of group
association schemes in order to employ the group properties (such as
irreducible characters) and use the algebraic structure of these networks (such
as Bose-Mesner algebra) in order to give an explicit analytical formula for
coupling constants in the Hamiltonians so that the state of a particular qudit
initially encoded on one site will perfectly evolve to the opposite site
without any dynamical control. It is shown that the only necessary condition in
order to PST over these networks be achieved is that the centers of the
corresponding groups be non-trivial. Therefore, PST over the underlying
networks of the group association schemes over all the groups with non-trivial
centers such as the abelian groups, the dihedral group $D_{2n}$ with even $n$,
the Clifford group CL(n) and all of the $p$-groups can be achieved.

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