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Annalen der Physik

Quantum Walks and Perfect State Transfer on the Corona Graphs Pn∘Cm

Qi Han, Rong Zhang

Published May 1, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1002/andp.70223

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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.
The edge corona product of a graph GGG with size mmm and another graph HHH, usually expressed as G⋄HG\diamond HG⋄H, is formed by taking one copy of GGG and mmm copies of HHH, and then connecting each endpoint of the iii-th edge of GGG to each vertex of the iii-th copy of HHH. This paper investigates perfect state transfer (PST) and pretty good state transfer (PGST) in continuous-time quantum walks on edge corona product of graphs. We establish some sufficient conditions under which this product has no periodic vertices; consequently, PST does not occur. We also exhibit an infinite family of periodic edge corona graphs that do not admit PST, and further give sufficient conditions for the occurrence of PGST on edge corona product of graphs.
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.
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 NNN in O(N)O(\sqrt{N})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 NNN.
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 GGG contains 2N2N2N vertices, and PST occurs between NNN pairs of vertices at the same time TTT, then we say GGG 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.
We provide a brief survey of perfect state transfer in quantum walks on finite graphs. The ability to transfer a quantum state from one part of a quantum computer to another is a key ingredient of scalable architectures. Transfer through structures that do not require locally varying dynamic control simplifies the design and hence reduces the opportunities for errors to arise. Continuous time walks quantum walks on highly structured graphs exhibit perfect state transfer for the complete graph of size 2, the path of length 3, and the cycle of size 4. From these, larger graphs can be constructed, and the use of edge weights widens this set considerably. Discrete-time quantum walks have more flexibility through exploiting the coin degrees of freedom, but with the disadvantage that local control of the coin is required if the degree of the vertices varies. The closely related property of periodicity (exact return to the starting state) is also mentioned.
Graphs that exhibit perfect state transfer (PST), by which a quantum walk beginning at one vertex is guaranteed to end at another after a nonzero displacement in time, are of great interest to researchers in the field of quantum computing. The conditions imposed by PST are strong, and many of the graphs that are known to exhibit PST are highly symmetric, which provides motivation to relax the restrictions imposed by perfect state transfer. This project introduces the property of group state transfer (GST), by which quantum walks are guaranteed to transfer states between nontrivial vertex sets. We define GST, examine its implications for finite simple graphs, and provide examples of graphs that exhibit GST.