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).
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 \textscICGn({2,n/2b}∪Q) has perfect state transfer, where b∈{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).
We study perfect state transfer on quantum networks represented by weighted graphs. Our focus is on graphs constructed from the join and related graph operators. Some specific results we prove include: (1) The join of a weighted two-vertex graph with any regular graph has perfect state transfer. This generalizes a result of Casaccino et al. [clms09] where the regular graph is a complete graph or a complete graph with a missing link. In contrast, the half-join of a weighted two-vertex graph with any weighted regular graph has no perfect state transfer. This implies that adding weights in a complete bipartite graph do not help in achieving perfect state transfer. (2) A Hamming graph has perfect state transfer between each pair of its vertices. This is obtained using a closure property on weighted Cartesian products of perfect state transfer graphs. Moreover, on the hypercube, we show that perfect state transfer occurs between uniform superpositions on pairs of arbitrary subcubes. This generalizes results of Bernasconi et al. [bgs08] and Moore and Russell [mr02]. Our techniques rely heavily on the spectral properties of graphs built using the join and Cartesian product operators.
We prove new results on perfect state transfer of quantum walks on quotient graphs. Since a graph G has perfect state transfer if and only if its quotient G/\pi, under any equitable partition \pi, has perfect state transfer, we exhibit graphs with perfect state transfer between two vertices but which lack automorphism swapping them. This answers a question of Godsil (Discrete Mathematics 312(1):129-147, 2011). We also show that the Cartesian product of quotient graphs \Box_{k} G_{k}/\pi_{k} is isomorphic to the quotient graph \Box_{k} G_{k}/\pi, for some equitable partition \pi. This provides an algebraic description of a construction due to Feder (Physical Review Letters 97, 180502, 2006) which is based on many-boson quantum walk.
This paper investigates perfect state transfer (PST) in discrete-time quantum walks constructed via the shunt decomposition method. The walks are defined on a graph G and its associated quotient graph G/π, induced by an equitable partition π. Through the shunt decomposition of G, we derive an explicit relation between the shift operator of the parent graph G and that of its quotient graph G/π. We construct a reflection operator based on the characteristic matrix, which establishes a connection between the transition operator of the parent graph and that of its lower-dimensional quotient graph. We then prove that PST occurs on G if and only if it occurs on G/π. Furthermore, we express the unitary evolution operator of the quotient graph in terms of Chebyshev polynomials of the first kind, from which we derive explicit criteria for PST. As an application, we establish PST on the cycle graph Cn at time k=n/2, and lift the result to the parent graph C2n via the equitable partition π. We further show that if an equitable partition π of G induces a quotient isomorphic to Kn↺, the complete digraph on n vertices with a loop at every vertex, then PST occurs at step k=n, and the walk is periodic at k=2n. This framework is applied to two families of graphs, which are the complete bipartite digraph Kn,n⇌ and the circulant graph Circ(2n,S), where S consists of all odd residues modulo 2n and n=2s for some s≥1, establishing PST in their respective line digraphs. Collectively, these results also answer the question posed by Godsil and Zhan concerning which shunt decompositions or embeddings of a graph admit PST.
The continuous-time quantum walk on an oriented circulant graph is determined by the Fourier eigenvalues of its Hermitian adjacency matrix. We classify perfect state transfer (PST) between distinct vertices in every nonempty oriented circulant graph. We show that each such graph is described by an odd primitive quadratic Dirichlet character of conductor Δ, a set of gcd-classes, and a choice between the two orientations of each selected class. For a graph of order n, we derive an explicit formula for every Fourier eigenvalue without assuming that n/Δ is coprime to Δ. We prove that PST occurs only for Δ∈{3,4,8} and give necessary and sufficient conditions on the connection set for each conductor. Equivalently, the square-free radicands of oriented circulant graphs with PST are exactly 1, 2, and 3. More generally, when λj=Dηj with ηj∈Z, congruences satisfied by the integers ηj determine all PST pairs and times, the minimum period, and the largest vertex sets supporting multiple state transfer (MST). In this class, pretty good state transfer is equivalent to PST. We also determine the connected orders and enumerate the resulting graphs.
We study the existence of quantum state transfer in Q-graphs in this paper. The Q-graph of a graph G, denoted by Q(G), is the graph derived from G by plugging a new vertex to each edge of G and joining two new vertices which lie on adjacent edges of G by an edge. We show that, if all eigenvalues of a regular graph G are integers, then its Q-graph Q(G) has no perfect state transfer. In contrast, we also prove that the Q-graph of a regular graph has pretty good state transfer under some mild conditions. Finally, applying the obtained results, we also exhibit many new families of Q-graphs having no perfect state transfer, but admitting pretty good state transfer.
In this paper, we define directed partial join graphs with signed couplings and construct discrete-time quantum-walk transition operators for these graphs using the shunt-decomposition framework. The resulting transition operators apply to several important graph families, including complete graphs with loops, circulant partial joins, complete bipartite graphs, tensor powers of complete bipartite graphs, all with signed couplings. For each family, we identify the corresponding structure of the transition operator and derive necessary and sufficient conditions for periodicity and perfect state transfer (PST), when one of the two directed regular graphs admits PST. Based on these results, we identify two types of state transfer; internal PST, which occurs between vertices within the same graph, and coupling PST, which occurs between two components of join graphs. We further develop a double-cover construction for directed partial join graphs and derive conditions for periodicity and PST when the associated transition operators do not necessarily commute. Using this construction, we establish PST results for double covers of complete graphs with loops. In particular, we provide an example in which the complete graph Kn does not exhibit PST for n≥4, whereas a suitable partial join of Kn exhibits PST when n=2m, m≥2. Hence, these results extend the class of graph families admitting PST in shunt-decomposition-based quantum walks and provide a unified framework for studying quantum state transfer in graph joins, products, and covers.