Abstract In this paper, within the framework of continuous-time quantum walks (CTQWs), we decompose graphs into disjoint strata by adopting the idea of equitable partitions. Based on the thought, we investigate the success probability of finding marked vertices on the N -fold star power graph G ⋆ N (where G is a cycle graph with three vertices), as well as the time required for perfect state transfer. Additionally, we explore the time for perfect state transfer on G ⋆ N under partition conditions from the perspective of quantum probability. Specifically, we surprisingly discover an interesting result: the times for perfect state transfer under quantum probability-based partitions and equitable partitions are completely consistent when the coupling strength J = 1 and the parameter λ = 1.
Abstract In this paper, within the framework of continuous-time quantum walks (CTQWs), we decompose graphs into disjoint strata by adopting the idea of equitable partitions. Based on the thought, we investigate the success probability of finding marked vertices on the N -fold star power graph G ⋆ N (where G is a cycle graph with three vertices), as well as the time required for perfect state transfer. Additionally, we explore the time for perfect state transfer on G ⋆ N under partition conditions from the perspective of quantum probability. Specifically, we surprisingly discover an interesting result: the times for perfect state transfer under quantum probability-based partitions and equitable partitions are completely consistent when the coupling strength J = 1 and the parameter λ = 1.
ABSTRACT The efficient implementation of quantum information processing relies on the reliable transfer of quantum states between specified vertices. This paper investigates the time required to achieve perfect state transfer (PST) on corona product graphs . To leverage the high symmetry of such graph structures, we employ a compression method to decompose their state space into several disjoint strata. By designing the coupling strengths between vertices, we analyze the time required for PST from a fixed vertex to vertices within each stratum. Furthermore, while keeping the number of vertices in the base graph fixed, we examine how the time required for PST on the corona product graph varies as the number of vertices in the other graph changes. Finally, we extend the obtained results to the general case where the base graph has an arbitrary number of vertices, thereby providing a complete characterization of the conditions for achieving PST.
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.
Perfect state transfer between two marked vertices of a graph by means of a discrete-time quantum walk is analyzed. We consider the quantum walk search algorithm with two marked vertices, sender and receiver. It is shown by explicit calculation that, for the coined quantum walks on a star graph and a complete graph with self-loops, perfect state transfer between the sender and receiver vertex is achieved for an arbitrary number of vertices N in O(N) steps of the walk. Finally, we show that Szegedy's walk with queries on a complete graph allows for state transfer with unit fidelity in the limit of large N.
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.
Perfect state transfer can be achieved between two marked vertices of graphs like a star graph, a complete graph with self-loops and a complete bipartite graph, and two-dimensional Lattice by means of discrete-time quantum walk. In this paper, we investigate the quality of quantum state transfer between two marked vertices of an unsymmetrical graph like the butterfly network. Our numerical results support the conjecture that the fidelity of state transfer depends on the quantum state to be transferred dynamically. The butterfly network is a typical example studied in networking coding. Therefore, these results can provide a clue to the construction of quantum network coding schemes.