We study a transport phenomenon in certain coined quantum walks where a subspace of states localized at a vertex gets transferred to another vertex. We first develop characterizations for perfect and pretty good subspace state transfer using the spectral properties of a Hermitian weighted digraph obtained from the underlying graph. We then provide a polynomial-time algorithm that tests whether pointwise perfect subspace state transfer occurs at an integer step, given that the subspace and coins are rational. Finally, we construct several infinite families of examples that admit pointwise perfect $d$-dimensional subspace state transfer where $d\ge 2$.
We study a transport phenomenon in certain coined quantum walks where a subspace of states localized at a vertex gets transferred to another vertex. We first develop characterizations for perfect and pretty good subspace state transfer using the spectral properties of a Hermitian weighted digraph obtained from the underlying graph. We then provide a polynomial-time algorithm that tests whether pointwise perfect subspace state transfer occurs at an integer step, given that the subspace and coins are rational. Finally, we construct several infinite families of examples that admit pointwise perfect d-dimensional subspace state transfer where d≥2.
Given a graph with Hermitian adjacency matrix H, perfect state transfer occurs from vertex a to vertex b if the (b,a)-entry of the unitary matrix exp(−iHt) has unit magnitude for some time t. This phenomenon is relevant for information transmission in quantum spin networks and is known to be monogamous under real symmetric matrices. We prove the following results: 1. For oriented graphs (whose nonzero weights are ±i), the oriented 3-cycle and the oriented edge are the only graphs where perfect state transfer occurs between every pair of vertices. This settles a conjecture of Cameron et al. On the other hand, we construct an infinite family of oriented graphs with perfect state transfer between any pair of vertices on a subset of size four. 2. There are infinite families of Hermitian graphs with one-way perfect state transfer, where perfect state transfer occurs without periodicity. In contrast, perfect state transfer implies periodicity whenever the adjacency matrix has algebraic entries (as shown by Godsil). 3. There are infinite families with non-monogamous pretty good state transfer in rooted graph products. In particular, we generalize known results on double stars (due to Fan and Godsil) and on paths with loops (due to Kempton, Lippner and Yau). The latter extends the experimental observation of quantum transport (made by Zimborás et al.) and shows non-monogamous pretty good state transfer can occur amongst distant vertices.
Abstract We establish the theory for pretty good state transfer in discrete-time quantum walks. For a class of walks, we show that pretty good state transfer is characterized by the spectrum of certain Hermitian adjacency matrix of the graph; more specifically, the vertices involved in pretty good state transfer must be strongly cospectral relative to this matrix, and the arccosines of its eigenvalues must satisfy some number theoretic conditions. Using normalized adjacency matrices, cyclic covers, and the theory on linear relations between geodetic angles, we construct several infinite families of walks that exhibits this phenomenon.
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.
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.
We introduce and study peak state transfer, a notion of high state transfer in qubit networks modeled by continuous-time quantum walks. Unlike perfect or pretty good state transfer, peak state transfer does not require fidelity arbitrarily close to 1, but crucially allows for an explicit determination of the time at which transfer occurs. We provide a spectral characterization of peak state transfer, which allows us to find many examples of peak state transfer, and we also establish tight lower bounds on fidelity and success probability. As a central example, we construct a family of weighted path graphs that admit peak state transfer over arbitrarily long distances with transfer probability approaching π/4≈0.78. These graphs offer exponentially improved sensitivity over known perfect state transfer examples such as the weighted paths related to hypercubes, making them practical candidates for efficient quantum wires.
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.