We construct families of graphs from linear groups SL ( 2 , q ) , GL ( 2 , q ) and GU ( 2 , q ) , where q is an odd prime power, with the property that the continuous-time quantum walks on the associated networks of qubits admit perfect state transfer.
A continuous-time quantum walk on a graph is represented by the complex matrix , where is the adjacency matrix of and is a non-negative time. If the graph models a network of interacting qubits, transfer of state among such qubits throughout time can be formalized as the action of the continuous-time quantum walk operator in the characteristic vectors of the vertices. Here, we are concerned with the problem of determining which graphs admit a perfect transfer of state. More specifically, we will study graphs whose adjacency matrix is a sum of tensor products of -matrices, focusing on the case where a graph is the tensor product of two other graphs. As a result, we will construct many new examples of perfect state transfer.
We study perfect state transfer on Cayley graphs from the point of view that state transfer is a property of a graph and not of a group. This paper is a bridge between the classical question about isomorphic Cayley graphs of non-isomorphic groups and quantum walks on graphs.
We show that a Cayley graph of a group with an abelian subgroup of index two is a Cayley graph of an abelian group under any one of three hypotheses, two drawn from the theory of isomorphic Cayley graphs. A statement of the same kind holds for extraspecial groups: every Cayley graph of an extraspecial p-group of order p2n+1 with a conjugacy-closed connection set is a Cayley graph of Zp2n+1. From these results we deduce that every explicit construction of perfect state transfer in the six papers we survey, on dihedral, dicyclic, generalized dihedral, V8n and extraspecial 2-groups, is a non-abelian presentation of an abelian Cayley graph. Moreover, we show that a non-abelian group with an abelian subgroup of index two admits a connected Cayley graph with perfect state transfer if and only if its order is divisible by four.
Genuinely non-abelian examples do exist. We prove that, for every odd prime power q≥5, the SL(2,q) graph of Pantangi and Sin, which they showed to admit perfect state transfer, is a Cayley graph of no abelian group; to our knowledge, this is the first infinite family of Cayley graphs with perfect state transfer provably admitting no abelian Cayley presentation. We also construct an infinite family of Cayley graphs with peak state transfer and determine all regular subgroups of the automorphism group of every member.
An appendix records a census of the connected vertex-transitive graphs with perfect state transfer on at most 30 vertices.
A continuous-time quantum walk on a graph is a matrix-valued function exp(−iAt) over the reals, where A is the adjacency matrix of the graph. Such a quantum walk has universal perfect state transfer if for all vertices u,v, there is a time where the (v,u) entry of the matrix exponential has unit magnitude. We prove new characterizations of graphs with universal perfect state transfer. This extends results of Cameron et al. (Linear Algebra and Its Applications, 455:115-142, 2014). Also, we construct non-circulant families of graphs with universal perfect state transfer. All prior known constructions were circulants. Moreover, we prove that if a circulant, whose order is prime, prime squared, or a power of two, has universal perfect state transfer then its underlying graph must be complete. This is nearly tight since there are universal perfect state transfer circulants with non-prime-power order where some edges are missing.
Let X be a graph, A its adjacency matrix, and t a non-negative real number. The matrix exp(i t A) determines the evolution in time of a certain quantum system defined on the graph. It represents a continuous-time quantum walk in X. We say that X admits perfect state transfer from a vertex u to a vertex v if there is a time t such that \n| exp(i t A)_(u,v) | = 1. \n \nThe main problem we study in this thesis is that of determining which simple graphs admit perfect state transfer. For some classes of graphs the problem is solved. For example, a path on n vertices admits perfect state transfer if and only if n=2 or n=3. However, the general problem of determining all graphs that admit perfect state transfer is substantially hard. \n \nIn this thesis, we focus on some special cases. We provide necessary and sufficient conditions for a distance-regular graph to admit perfect state transfer. In particular, we provide a detailed account of which distance-regular graphs of diameter three do so. \n \nA graph is said to be spectrally extremal if the number of distinct eigenvalues is equal to the diameter plus one. Distance-regular graphs are examples of such graphs. We study perfect state transfer in spectrally extremal graphs and explore rich connections to the topic of orthogonal polynomials. We characterize perfect state transfer in such graphs. \n \nWe also provide a general framework in which perfect state transfer in graph products can be studied. We use this to determine when direct products and double covers of graphs admit perfect state transfer. As a consequence, we provide many new examples of simple graphs admitting perfect state transfer. We also provide some advances in the understanding of perfect state transfer in Cayley graphs for the groups (Z_2)^d and Z_n. \n \nFinally, we consider the problem of determining which trees admit perfect state transfer. We show more generally that, except for the path on two vertices, if a connected bipartite graph contains a unique perfect matching, then it cannot admit perfect state transfer. We also consider this problem in the context of another model of quantum walks determined by the matrix exp(i t L), where L is the Laplacian matrix of the graph. In particular, we show that no tree on an odd number of vertices admits perfect state transfer according to this model.
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.
Quantum spin systems carry information in the form of quantum states, and the propagation of these quantum states is described by a quantum walk. In order to construct an operational quantum computer, the task of reliably transmitting quantum states from one part of a quantum computer to another must be accomplished. Quantum walks are known to be powerful tools in achieving this task. Pure states are quantum states represented by unit complex vectors. In this thesis, we use algebraic and combinatorial techniques to develop the theory of perfect state transfer on pure states in quantum walks, with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a spin network. We prove basic results about eigenvalue supports, periodicity, and strong cospectrality on pure states. We investigate a special type of strong cospectrality called m-strong cospectrality, and show that m-strongly cospectral pure states admit desirable quantum state transfer properties. Several characterisations of perfect state transfer are also given, one for general pure states, one for m-strongly pure states, and another one for real pure states. These characterisations give rise to algebraic and analytic properties of pure states admitting perfect state transfer. Moreover, we determine all complete graphs, complete bipartite graphs, cycles, and paths that admit perfect state transfer between m-strongly pure states. Constructions of infinite families of graphs admitting perfect state transfer are also given. We also characterise perfect state transfer between vertex states in joins and blowup graphs. We determine when the join operation preserves or induces perfect state transfer in the resulting graph. We also use blow-up graphs to construct new families of regular graphs with perfect state transfer. We use these two graph operations to demonstrate that graphs with perfect state transfer can be constructed from graphs that do not exhibit such a property. Finally, we investigate perfect state transfer on s-pair states – pure states that represent entanglement between two vertices. We establish combinatorial and spectral properties of s-pair states with perfect state transfer, and characterise its existence in complete graphs, complete bipartite graphs, paths and distance-regular graphs. Further, we construct infinite families of graphs with s-pair state transfer.