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 (K_n) does not exhibit PST for (n\geq4), whereas a suitable partial join of (K_n) exhibits PST when (n=2^m), (m\geq2). 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.
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.
We study perfect state transfer of quantum walks on signed graphs. Our aim is to show that negative edges are useful for perfect state transfer. First, we show that the signed join of a negative
2-clique with any positive
(n,3)-regular graph has perfect state transfer even if the unsigned join does not. Curiously, the perfect state transfer time improves as
n increases. Next, we prove that a signed complete graph has perfect state transfer if its positive subgraph is a regular graph with perfect state transfer and its negative subgraph is periodic. This shows that signing is useful for creating perfect state transfer since no complete graph (except for the
2-clique) has perfect state transfer. Also, we show that the double-cover of a signed graph has perfect state transfer if the positive subgraph has perfect state transfer and the negative subgraph is periodic.Here, signing is useful for constructing unsigned graphs with perfect state transfer. Finally, we study perfect state transfer on a family of signed graphs called the exterior powers which is derived from a many-fermion quantum walk on graphs.
Perfect state transfer (PST) is a key requirement for high-fidelity quantum communication. In discrete-time quantum walks, the shift matrix governs the system dynamics and plays a central role in enabling PST. This work presents a shift-matrix construction based on shunt decompositions of d-regular directed graphs, under which path, cycle, and complete graphs exhibit both perfect state transfer and periodic dynamics. Unlike existing permutation-based approaches that yield periodicity without PST, the proposed method achieves both properties. A SHAP-like attribution analysis is performed to identify the arcs essential for PST. In addition, quantum-circuit implementations of the shift matrices and benchmark results under ideal and noisy conditions are reported. The framework is further extended to a unitary construction of a reset channel, whose reduced Kraus representation leverages the underlying graph structure to minimize information loss.
Abstract Continuous-time quantum walks (CTQW) becoming more significant for quantum information and quantum computation. The perfect state transfer (PST) on graphs is a specific phenomena of CTQW which is widely used in quantum communication of small system, and be found critical for quantum algorithm design recently, such as quantum approximate optimization algorithms and quantum search. If a network
G contains
2N vertices, and PST occurs between
N pairs of vertices at the same time
T, then we say
G admits symmetric PST (SPST). In this work, we will study the SPST on bipartite graphs via spectra. And then, we will present how to set couplings of special bipartite graphs such as cycles, grids and chains to admit SPST is given. For cycles with 6, 8, 10, 12 vertices, we give the analytical method to obtain the couplings to admit SPST. Analogously, the couplings for chains and grids to admit SPST are analytical obtained. We also present a typical approach by using the graph tensor product to construct new quantum networks that admit SPST, in which the new quantum networks are derived from special kinds of integral graphs, such as complete graphs, strongly regular graphs and distance regular graphs.
The join X ∨ Y of two graphs X and Y is the graph obtained by joining each vertex of X to each vertex of Y . We explore the behaviour of a continuous quantum walk on a join graph with positive edge weights having the adjacency matrix or Laplacian matrix as its associated Hamiltonian, where the underlying graphs are assumed to be regular when dealing with the adjacency matrix. We characterize strong cospectrality, periodicity and perfect state transfer (PST) in a join graph. We also determine conditions in which strong cospectrality, periodicity and PST are preserved in the join. Under certain conditions, we show that there are graphs with no PST that exhibit PST when joined with another graph. We also show that | | U M ( X ∨ Y , t ) u , v | − | U M ( X , t ) u , v | | ≤ 2 | V ( X ) | for all vertices u and v of X , where U M ( X ∨ Y , t ) and U M ( X , t ) denote the transition matrices of X ∨ Y and X respectively relative to the adjacency or Laplacian matrix. We demonstrate that the bound 2 | V ( X ) | is tight for infinite families of graphs.
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∈{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).