We study the distributions of the continuous-time quantum walk on a one-dimensional lattice. In particular we will consider walks on unbounded lattices, walks with one and two boundaries and Dirichlet boundary conditions, and walks with periodic boundary conditions. We will prove that all continuous-time quantum walks can be written as a series of Bessel functions of the first kind and show how to approximate these series.
We study the distributions of the continuous-time quantum walk on a one-dimensional lattice. In particular we will consider walks on unbounded lattices, walks with one and two boundaries and Dirichlet boundary conditions, and walks with periodic boundary conditions. We will prove that all continuous-time quantum walks can be written as a series of Bessel functions of the first kind and show how to approximate these series.
The propagation by continuous-time quantum walks (CTQWs) on one-dimensional lattices shows structures in the transition probabilities between different sites reminiscent of quantum carpets. For a system with periodic boundary conditions, we calculate the transition probabilities for a CTQW by diagonalizing the transfer matrix and by a Bloch function ansatz. Remarkably, the results obtained for the Bloch function ansatz can be related to results from (discrete) generalized coined quantum walks. Furthermore, we show that here the first revival time turns out to be larger than for quantum carpets.
I obtain the dynamics of the continuous time quantum walk on a d-dimensional lattice, with periodic boundary conditions, as an appropriate limit of the dynamics of the discrete time quantum walk on the same lattice. This extends the main result of [8] which proved this limit for the case of the quantum walk on the infinite line and the quantum cellular automaton proposed in [1]. By highlighting the main features of the limiting procedure, I then extend it to general graphs. For a given discrete time quantum walk on a general graph, I single out the type of continuous dynamics (Hamiltonians) that can be obtained as a limit of the discrete time dynamics. 1 Generalities on quantum walks; continuous and discrete time Consider a graph G: = {V, E} with a set of vertices V of cardinality N and a set of edges E. We assume G to be undirected and without self-loops. In the interval of time ∆t a certain fraction γ∆t of a quantity pj leaves the location of the j-th vertex to move to a neighboring vertex k. The quantity pj, j = 1,..., N, at time t + ∆t is given by 1 pj(t + ∆t) = pj(t) − deg (j)γ∆tpj(t) + ∑ (k,j)∈E γ∆tpk(t). (1) Define ⃗p the N−vector whose components are the pj’s quantities, and L the Laplacian matrix defined by Ljj = − deg (j), Ljk = 1 if (j, k) ∈ E and Ljk = 0 otherwise. L is
The mathematical relationship between the discrete-time and continuous-time quantum walks and the one-dimensional Dirac equation is explored by studying a class of solutions for each, expressed in terms of the generalized, regular, and modified Bessel functions, respectively. Rigorous limits connecting these solutions are established. In addition, new analytical and numerical results are presented for quantum walks and the Dirac equation, including entanglement, relativistic localization and wave packet spreading, and normal and anomalous Zitterbewegung.
Quantum walks can be defined in two quite distinct ways: discrete-time and continuous-time quantum walks (DTQWs and CTQWs). For classical random walks, there is a natural sense in which continuous-time walks are a limit of discrete-time walks. Quantum mechanically, in the discrete-time case, an additional ``coin space'' must be appended for the walk to have nontrivial time evolution. Continuous-time quantum walks, however, have no such constraints. This means that there is no completely straightforward way to treat a CTQW as a limit of a DTQW, as can be done in the classical case. Various approaches to this problem have been taken in the past. We give a construction for walks on d-regular, d-colorable graphs when the coin flip operator is Hermitian: from a standard DTQW we construct a family of discrete-time walks with a well-defined continuous-time limit on a related graph. One can think of this limit as a ``coined'' continuous-time walk. We show that these CTQWs share some properties with coined DTQWs. In particular, we look at a spatial search by a DTQW over the two-dimensional (2D) torus (a grid with periodic boundary conditions) of size $\sqrt{N}\ifmmode\times\else\texttimes\fi{}\sqrt{N}$, where it was shown that a coined DTQW can search in time O(NlogN), but a standard CTQW takes $\mathrm{\ensuremath{\Omega}}(N)$ time to search for a marked element. The continuous limit of the DTQW search over the 2D torus exhibits the O(NlogN) scaling, like the coined walk it is derived from. We also look at the effects of graph symmetry on the limiting walk, and show that the properties are similar to those of the DTQW as shown in Krovi and Brun, Phys. Rev. A 75, 062332 (2007).
In this paper, we consider continuous-time quantum walks (CTQWs) on a one-dimensional ring lattice of N nodes in which every node is connected to its 2m nearest neighbors ( m on either side). In the framework of the Bloch function ansatz, we calculate the space-time transition probabilities between two nodes of the lattice. We find that the transport of CTQWs between two different nodes is faster than that of the classical continuous-time random walks (CTRWs). The transport speed, which is defined by the ratio of the shortest path length and propagating time, increases with the connectivity parameter m for both CTQWs and CTRWs. For fixed parameter m , the transport of CTRWs gets slower with the increase of the shortest distance while the transport (speed) of CTQWs turns out to be a constant value. In the long-time limit, depending on the network size N and connectivity parameter m , the limiting probability distributions of CTQWs show various patterns. When the network size N is an even number, the probability of being at the original node differs from that of being at the opposite node, which also depends on the precise value of parameter m .
In this paper, we present an extensive analytical study of continuous-time quantum walks (CTQWs) occurring on two-dimensional lattices under various boundary conditions, focusing on the M × N lattice wrapped on Mobius strips and Klein bottles, which are featured by the twisted boundary conditions. We find that the eigenvalue spectra and transport efficiency (both quantum mechanical and classical) for the two structures show comparable results with those for other well-known two-dimensional lattices, including rectangles, cylinders and tori. We also demonstrate that the behaviors of CTQWs depend on the initial node of the excitation and the network size, both on short and long timescales. In addition, we discover the asymmetric behaviors of limiting probabilities for Mobius strips and Klein bottles, which are quite different from each other and are also compared to those discovered in other two-dimensional networks. Our work provides a comprehensive understanding of recent results about CTQWs on two-dimensional lattices, and sheds light on quantum dynamics on lattices, especially those with different boundary conditions.