Quantum Walks on Hypergraphs

In this work we introduce the concept of a quantum walk on a hypergraph. We show that the staggered quantum walk model is a special case of a quantum walk on a hypergraph.

     

Random Time-Dependent Quantum Walks

We consider the discrete time unitary dynamics given by a quantum walk on the lattice \mathbb Z^d{\mathbb {Z}^d} performed by a quantum particle with internal degree of freedom, called coin state, according to the following iterated rule: a unitary update of the coin state takes place, followed by a shift on the lattice, conditioned on the coin state of the particle. We study the large time behavior of the quantum mechanical probability distribution of the position observable in \mathbb Z^d{\mathbb {Z}^d} when the sequence of unitary updates is given by an i.i.d. sequence of random matrices. When averaged over the randomness, this distribution is shown to display a drift proportional to the time and its centered counterpart is shown to display a diffusive behavior with a diffusion matrix we compute. A moderate deviation principle is also proven to hold for the averaged distribution and the limit of the suitably rescaled corresponding characteristic function is shown to satisfy a diffusion equation. A generalization to unitary updates distributed according to a Markov process is also provided.     

Open Quantum Random Walks

A new model of quantum random walks is introduced, on lattices as well as on finite graphs. These quantum random walks take into account the behavior of open quantum systems. They are the exact quantum analogues of classical Markov chains. We explore the “quantum trajectory” point of view on these quantum random walks, that is, we show that measuring the position of the particle after each time-step gives rise to a classical Markov chain, on the lattice times the state space of the particle. This quantum trajectory is a simulation of the master equation of the quantum random walk. The physical pertinence of such quantum random walks and the way they can be concretely realized is discussed. Differences and connections with the already well-known quantum random walks, such as the Hadamard random walk, are established.     

Quantum Random Walks and Thermalisation

It is shown how to construct quantum random walks with particles in an arbitrary faithful normal state. A convergence theorem is obtained for such walks, which demonstrates a thermalisation effect: the limit cocycle obeys a quantum stochastic differential equation without gauge terms. Examples are presented which generalise that of Attal and Joye (J Funct Anal 247:253–288, 2007).     

Transport Efficiency of Continuous-Time Quantum Walks on Graphs

Continuous-time quantum walk describes the propagation of a quantum particle (or an excitation) evolving continuously in time on a graph. As such, it provides a natural framework for modeling transport processes, e.g., in light-harvesting systems. In particular, the transport properties strongly depend on the initial state and specific features of the graph under investigation. In this paper, we address the role of graph topology, and investigate the transport properties of graphs with different regularity, symmetry, and connectivity. We neglect disorder and decoherence, and assume a single trap vertex that is accountable for the loss processes. In particular, for each graph, we analytically determine the subspace of states having maximum transport efficiency. Our results provide a set of benchmarks for environment-assisted quantum transport, and suggest that connectivity is a poor indicator for transport efficiency. Indeed, we observe some specific correlations between transport efficiency and connectivity for certain graphs, but, in general, they are uncorrelated.     

Spectral Properties of Quantum Walks on Rooted Binary Trees

We define coined quantum walks on the infinite rooted binary tree given by unitary operators $U(C)$ on an associated infinite dimensional Hilbert space, depending on a unitary coin matrix $C\in U(3)$ , and study their spectral properties. For circulant unitary coin matrices $C$ , we derive an equation for the Carathéodory function associated to the spectral measure of a cyclic vector for $U(C)$ . This allows us to show that for all circulant unitary coin matrices, the spectrum of the quantum walk has no singular continuous component. Furthermore, for coin matrices $C$ which are orthogonal circulant matrices, we show that the spectrum of the Quantum Walk is absolutely continuous, except for four coin matrices for which the spectrum of $U(C)$ is pure point.     

Quantum Random Walks with General Particle States

A convergence theorem is obtained for quantum random walks with particles in an arbitrary normal state. This unifies and extends previous work on repeated-interactions models, including that of Attal and Pautrat (Ann Henri Poincaré 7:59–104 2006) and Belton (J Lond Math Soc 81:412–434, 2010; Commun Math Phys 300:317–329, 2010). When the random-walk generator acts by ampliation and either multiplication or conjugation by a unitary operator, it is shown that the quantum stochastic cocycle which arises in the limit is driven by a unitary process.     

Limit Theorems for Open Quantum Random Walks

We consider the limit distributions of open quantum random walks on one-dimensional lattice space. We introduce a dual process to the original quantum walk process, which is quite similar to the relation of Schrödinger-Heisenberg representation in quantum mechanics. By this, we can compute the distribution of the open quantum random walks concretely for many examples and thereby we can also obtain the limit distributions of them. In particular, it is possible to get rid of the initial state when we consider the evolution of the walk, it appears only in the last step of the computation.     

基于量子核函数算法的图像压缩研究

为了提高图像压缩的质量,提出量子核函数算法。首先采用自适应灰度阈值使图像归一化处理,用量子比特表示图像白色和黑色状态的出现概率;然后每个像素点的值被编码为量子角度向量;接着构造适应度函数的极小值获得高斯径向基核函数的最优核宽,在量子核影响半径以外的像素量子编码不再起引导作用;最后给出了算法流程。实验仿真显示本文算法的压缩重构效果清晰,PSNR最大,压缩时间最少。     

Dynamical Localization of Quantum Walks in Random Environments

We study shock statistics in the scalar conservation law ∂ _t u+∂ _x f(u)=0, x∈ℝ, t>0, with a convex flux f and spatially random initial data. We show that the Markov property (in x) is preserved for a large class of random initial data (Markov processes with downward jumps and derivatives of Lévy processes with downward jumps). The kinetics of shock clustering is then described completely by an evolution equation for the generator of the Markov process u(x,t), x∈ℝ. We present four distinct derivations for this evolution equation, and show that it takes the form of a Lax pair. The Lax equation admits a spectral parameter as in Manakov (Funct. Anal. Appl. 10:328–329, 1976), and has remarkable exact solutions for Burgers equation (f(u)=u ²/2). This suggests the kinetic equations of shock clustering are completely integrable.     

Brun-Type Formalism for Decoherence in Two-Dimensional Quantum Walks

We study decoherence in the quantum walk on the xy-plane.We generalize the method of decoherent coin quantum walk,introduced by [T.A.Brun,et al.,Phys.Rev.A 67(2003) 032304],which could be applicable to all sorts of decoherence in two-dimensional quantum walks,irrespective of the unitary transformation governing the walk.As an application we study decoherence in the presence of broken line noise in which the quantum walk is governed by the two-dimensional Hadamard operator.     

Return Probability of Quantum and Correlated Random Walks

The analysis of the return probability is one of the most essential and fundamental topics in the study of classical random walks. In this paper, we study the return probability of quantum and correlated random walks in the one-dimensional integer lattice by the path counting method. We show that the return probability of both quantum and correlated random walks can be expressed in terms of the Legendre polynomial. Moreover, the generating function of the return probability can be written in terms of elliptic integrals of the first and second kinds for the quantum walk.     

Transport in Quantum Multi-barrier Systems as Random Walks on a Lattice

Journal of Statistical Physics - Coulomb and log-gases are exchangeable singular Boltzmann–Gibbs measures appearing in mathematical physics at many places, in particular in random matrix...     

Topological Boundary Invariants for Floquet Systems and Quantum Walks

A Floquet systems is a periodically driven quantum system. It can be described by a Floquet operator. If this unitary operator has a gap in the spectrum, then one can define associated topological bulk invariants which can either only depend on the bands of the Floquet operator or also on the time as a variable. It is shown how a K-theoretic result combined with the bulk-boundary correspondence leads to edge invariants for the half-space Floquet operators. These results also apply to topological quantum walks.     

Quantum Walks,Weyl Equation and the Lorentz Group

Quantum cellular automata and quantum walks provide a framework for the foundations of quantum field theory, since the equations of motion of free relativistic quantum fields can be derived as the small wave-vector limit of quantum automata and walks starting from very general principles. The intrinsic discreteness of this framework is reconciled with the continuous Lorentz symmetry by reformulating the notion of inertial reference frame in terms of the constants of motion of the quantum walk dynamics. In particular, among the symmetries of the quantum walk which recovers the Weyl equation—the so called Weyl walk—one finds a non linear realisation of the Poincaré group, which recovers the usual linear representation in the small wave-vector limit. In this paper we characterise the full symmetry group of the Weyl walk which is shown to be a non linear realization of a group which is the semidirect product of the Poincaré group and the group of dilations.