Let G be a graph with adjacency matrix A, let H(t) = exp(itA). G is called a periodic graph if there exists a time τ such that H(τ) is diagonal. If u and v are distinct vertices in G, we say that perfect state transfer occurs from u to v if there exists a time τ such that |H(τ)u,v| = 1. A necessary and sufficient condition for G is periodic is given. We give the existence for the perfect state ...

A duality between the properties of many spinor bosons on a regular lattice and those of a single particle on a weighted graph reveals that a quantum particle can traverse an infinite hierarchy of networks with perfect probability in polynomial time, even as the number of nodes increases exponentially. The one-dimensional "quantum wire" and the hypercube are special cases in this construction, ...

We present a communication protocol for chains of permanently coupled qubits which achieves perfect quantum state transfer and which is efficient with respect to the number chains employed in the scheme. The system consists of M uncoupled identical quantum chains. Local control (gates, measurements) is only allowed at the sending/receiving end of the chains. Under a quite general hypothesis on ...

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 w...

Let X be a graph on n vertices with with adjacency matrix A and let H(t) denote the matrix-valued function exp(iAt). If u and v are distinct vertices in X, we say perfect state transfer from u to v occurs if there is a time τ such that |H(τ)u,v| = 1. If u ∈ V (X) and there is a time σ such that |H(σ)u,u| = 1, we say X is periodic at u with period σ. We show that if perfect state transfer from u...

We suggest a scheme that allows arbitrarily perfect state transfer even in the presence of random fluctuations in the couplings of a quantum chain. The scheme performs well for both spatially correlated and uncorrelated fluctuations if they are relatively weak (say 5%). Furthermore, we show that given a quite arbitrary pair of quantum chains, one can check whether it is capable of perfect trans...

By considering distance-regular graphs as spin networks, first we introduce some particular spin Hamiltonians which are extended version of those of Refs.[1, 2]. Then, by using spectral analysis techniques and algebraic combinatoric structure of distanceregular graphs such as stratification introduced in [3, 4] and Bose-Mesner algebra, we give a method for finding a set of coupling constants in...

The issue of quantum states’ transfer – in particular, for socalled Perfect State Transfer (PST) – in the networks represented by the spin chains seems to be one of the major concerns in quantum computing. Especially, in the context of future communication methods that can be used in broadly defined computer science. The chapter presents a definition of Hamiltonian describing the dynamics of qu...