A discrete-time quantum walk is the quantum analogue of a Markov chain on a graph. We show that the evolution of a general discrete-time quantum walk that consists of two reflections satisfies a Chebyshev recurrence, under a projection. We apply this to study perfect state transfer in a model of discrete-time quantum walk whose transition matrix is given by two reflections, defined by the face and vertex incidence relations of a graph embedded in an orientable surface, proposed by Zhan [ J. Algebraic Combin. 53(4):1187–1213, 2020]. For this model, called the vertex-face walk, we prove results about perfect state transfer and periodicity and give infinite families of examples where these occur. In doing so, we bring together tools from algebraic and topological graph theory to analyze the evolution of this walk.
A discrete-time quantum walk is the quantum analogue of a Markov chain on a graph. We show that the evolution of a general discrete-time quantum walk that consists of two reflections satisfies a Chebyshev recurrence, under a projection. We apply this to study perfect state transfer in a model of discrete-time quantum walk whose transition matrix is given by two reflections, defined by the face and vertex incidence relations of a graph embedded in an orientable surface, proposed by Zhan [ J. Algebraic Combin. 53(4):1187–1213, 2020]. For this model, called the vertex-face walk, we prove results about perfect state transfer and periodicity and give infinite families of examples where these occur. In doing so, we bring together tools from algebraic and topological graph theory to analyze the evolution of this walk.
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.
In this paper, we analyze state transfer in quantum walks by using combinatorial methods. We generalize perfect state transfer in two-reflection discrete-time quantum walks to a notion that we call peak state transfer; we define peak state transfer as the highest state transfer that can be achieved between an initial and a target state under unitary evolution, even when perfect state transfer is unattainable. We give a spectral characterization of peak state transfer that allows us to fully characterize peak state transfer in the arc-reversal (Grover) walk on various families of graphs, including strongly regular graphs and incidence graphs of block designs (assuming that the walk starts at a point of the design). In addition, we provide many examples of peak state transfer, including an infinite family where the amount of peak state transfer tends to 1 as the number of vertices grows. We further demonstrate that peak state transfer properties extend to infinite families of graphs generated by vertex blow-ups, and we characterize periodicity in the vertex-face walk on toroidal grids. In our analysis, we make extensive use of the spectral decomposition of a matrix that is obtained by projecting the transition matrix down onto a subspace. Though we are motivated by a problem in quantum computing, we identify several open problems that are purely combinatorial, arising from the spectral conditions required for peak state transfer in discrete-time quantum walks.
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.
This thesis extends results about periodicity and perfect state transfer \nto oriented graphs. We prove that if a vertex a is periodic, then elements of \nthe eigenvalue support lie in Z √∆ for some squarefree negative integer \n∆. We find an infinite family of orientations of the complete graph that are \nperiodic. We find an example of a graph with both perfect state transfer \nand periodicity that is not periodic at an integer multiple of the period, and \nwe prove and use Gelfond-Schneider Theorem to show that every oriented \ngraph with perfect state transfer between two vertices will have both vertices \nperiodic. We find a complete characterization of when perfect state transfer \ncan occur in oriented graphs, and find a new example of a graph where one \nvertex has perfect state transfer to multiple other vertices.
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.
In this paper, we introduce a discrete quantum walk model called bipartite walks. Bipartite walks include many known discrete quantum walk models, like Grover’s walks, vertex-face walks. For the transition matrix of a quantum walk, there is a Hamiltonian associated with it. We will study the Hamiltonians of the bipartite walks. Let S be a skew-symmetric matrix. We are mainly interested in the Hamiltonians of the form iS. We show that the Hamiltonian can be written as iS if and only if the adjacency matrix of the bipartite graph is invertible. We show that Grover’s walks and vertex-face walks are special cases of bipartite walks. Via the Hamiltonians, phenomena of bipartite walks lead to phenomena of continuous walks. We show in detail how we use bipartite walks on paths to construct universal perfect state transfer in continuous walks.