The existence of perfect state transfer (PST) on quantum spin networks is a fundamental problem in mathematics and physics. Various works in the literature have explored PST in graphs with arithmetic origins, such as gcd-graphs over [Formula: see text] and cubelike graphs. In this paper, building on our recent work on gcd-graphs over an arbitrary finite Frobenius ring, we investigate the existence of PST on these graphs. Our approach is algebraic in nature, enabling us to unify various existing results in the literature.
The existence of perfect state transfer (PST) on quantum spin networks is a fundamental problem in mathematics and physics. Various works in the literature have explored PST in graphs with arithmetic origins, such as gcd-graphs over [Formula: see text] and cubelike graphs. In this paper, building on our recent work on gcd-graphs over an arbitrary finite Frobenius ring, we investigate the existence of PST on these graphs. Our approach is algebraic in nature, enabling us to unify various existing results in the literature.
Abstract. In this work, using eigenvalues and eigenvectors of unitary Cayley graphs over finite local rings and elementary linear algebra, we characterize which local rings allow a PST occurring in its unitary Cayley graph. Moreover, we have some developments when R is a product of local rings. 1. Perfect State Transfer and Unitary Cayley Graphs Let G be an undirected graph whose vertex set V (G) = {v1,..., vn}. The adjacency matrix of G, written AG, is the n×n matrix in which entry ajk is the number of edges in G with endpoint {vj, vk}. Define the matrix-valued function H(t) = exp(itAG) for all t ≥ 0. We say there is a perfect state transfer (PST) from vertex vj to vertex vk if there is a time t such that |H(t)jk | = 1. We note that our matrix H(t) determines what is known in graph theory as a continuous quantum walk. For background on quantum walks, we refer the reader to [9] and [10]. A perfect state transfer in continuous-time quantum walk on graphs has received considerable attention in quantum information and computations in Physics (e.g., [2, 4]). An excellent survey of perfect state transfer graphs and related questions are given by Godsil [8]. Observe that H(t) has the following properties: (i) H(t) is symmetric, (ii) H(t) = H(t)−1, where ¯ is the complex conjugate, (iii) H(t) is unitary, i.e., (H(t))T = H(t)−1. Thus, we have the next proposition.
In this paper we answer the question of when circulant quantum spin networks with nearest-neighbor couplings can give perfect state transfer. The network is described by a circulant graph G, which is characterized by its circulant adjacency matrix A. Formally, we say that there exists a {\it perfect state transfer} (PST) between vertices a,b∈V(G) if ∣F(τ)ab∣=1, for some positive real number τ, where F(t)=exp(ıAt). Saxena, Severini and Shparlinski ({\it International Journal of Quantum Information} 5 (2007), 417--430) proved that ∣F(τ)aa∣=1 for some a∈V(G) and τ∈R+ if and only if all eigenvalues of G are integer (that is, the graph is integral). The integral circulant graph $\ICG_n (D)$ has the vertex set Zn={0,1,2,...,n−1} and vertices a and b are adjacent if gcd(a−b,n)∈D, where $D \subseteq \{d : d \mid n,\ 1\leq d
Abstract. Pure states correspond to one-dimensional subspaces of [Formula: see text] represented by unit vectors. In this paper, we develop the theory of perfect state transfer (PST) between real pure states with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a quantum spin network. We characterize PST between real pure states based on the spectral information of a graph and prove three fundamental results: (i) every periodic real pure state [Formula: see text] admits PST with another real pure state [Formula: see text], (ii) every connected graph admits PST between real pure states, and (iii) for any pair of real pure states [Formula: see text] and [Formula: see text] and for any time [Formula: see text], there exists a real symmetric matrix [Formula: see text] such that [Formula: see text] and [Formula: see text] admit PST relative to [Formula: see text] at time [Formula: see text]. We also determine all real pure states that admit PST in complete graphs, complete bipartite graphs, paths, and cycles. This leads to a complete characterization of pair and plus state transfer in paths and complete bipartite graphs. We give constructions of graphs that admit PST between real pure states. Finally, using results on the spread of graphs, we prove that amongst all [Formula: see text]-vertex simple unweighted graphs, the least minimum PST time between real pure states is attained by any join graph for the Laplacian case, while it is attained by the join of an empty graph and a complete graph of appropriate sizes for the adjacency case. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at none . [Formula: see text]
Recently, perfect state transfer (PST for short) on graphs has attracted great attention due to their applications in quantum information processing and quantum computations. Many constructions and results have been established through various graphs. However, most of the graphs previously investigated are abelian Cayley graphs. Necessary and sufficient conditions for Cayley graphs over dihedral groups having perfect state transfer were studied recently. The key idea in that paper is the assumption of the normality of the connection set. In those cases, viewed as an element in a group algebra, the connection set is in the center of the group algebra, which makes the situations just like in the abelian case. In this paper, we study the non-normal case. In this case, the discussion becomes more complicated. Using the representations of the dihedral group Dn, we show that Cay(Dn,S) cannot have PST if n is odd. For even integers n, it is proved that if Cay(Dn,S) has PST, then S is normal.
Cubelike graphs are the Cayley graphs of the elementary Abelian group Z2n (e.g., the hypercube is a cubelike graph). We study perfect state transfer between two particles in quantum networks modeled by a large class of cubelike graphs. This generalizes the results of Christandl et al. [Phys. Rev. Lett. 92, 187902 (2004)] and Facer et al. [Phys. Rev. A 92, 187902 (2008)].
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.