Scattering theory and discrete-time quantum walks

We study quantum walks on general graphs from the point of view of scattering theory. For a general finite graph we choose two vertices and attach one half line to each, and consider walks that proceed from one half line, through the graph, to the other. The probability of starting on one line and reaching the other after n steps can be expressed in terms of the transmission amplitude for the graph.     

Some mathematical aspects of Anderson localization: boundary effect,multimodality, and bifurcation

Anderson localization is a famous wave phenomenon that describes the absence of diffusion of waves in a disordered medium. Here we generalize the landscape theory of Anderson localization to general elliptic operators and complex boundary conditions using a probabilistic approach, and further investigate some mathematical aspects of Anderson localization that are rarely discussed before. First, we observe that under the Neumann boundary condition, the low energy quantum states are localized on the boundary of the domain with high probability. We provide a detailed explanation of this phenomenon using the concept of extended subregions and obtain an analytical expression of this probability in the one-dimensional case. Second, we find that the quantum states may be localized in multiple different subregions with high probability in the one-dimensional case and we derive an explicit expression of this probability for various boundary conditions. Finally, we examine a bifurcation phenomenon of the localization subregion as the strength of disorder varies. The critical threshold of bifurcation is analytically computed based on a toy model and the dependence of the critical threshold on model parameters is analyzed.     

Bringing order through disorder: localization of errors in topological quantum memories

Anderson localization emerges in quantum systems when randomized parameters cause the exponential suppression of motion. Here we consider this phenomenon in topological models and establish its usefulness for protecting topologically encoded quantum information. For concreteness we employ the toric code. It is known that in the absence of a magnetic field this can tolerate a finite initial density of anyonic errors, but in the presence of a field anyonic quantum walks are induced and the tolerable density becomes zero. However, if the disorder inherent in the code is taken into account, we demonstrate that the induced localization allows the topological quantum memory to regain a finite critical anyon density and the memory to remain stable for arbitrarily long times. We anticipate that disorder inherent in any physical realization of topological systems will help to strengthen the fault tolerance of quantum memories.     

Recurrence and Pólya number of quantum walks

We analyze the recurrence probability (Pólya number) for d-dimensional unbiased quantum walks. A sufficient condition for a quantum walk to be recurrent is derived. As a by-product we find a simple criterion for localization of quantum walks. In contrast with classical walks, where the Pólya number is characteristic for the given dimension, the recurrence probability of a quantum walk depends in general on the topology of the walk, choice of the coin and the initial state. This allows us to change the character of the quantum walk from recurrent to transient by altering the initial state.     

Trapping of continuous-time quantum walks on Erdös–Rényi graphs

We consider the coherent exciton transport, modeled by continuous-time quantum walks, on Erdös–Rény graphs in the presence of a random distribution of traps. The role of trap concentration and of the substrate dilution is deepened showing that, at long times and for intermediate degree of dilution, the survival probability typically decays exponentially with a (average) decay rate which depends non-monotonically on the graph connectivity; when the degree of dilution is either very low or very high, stationary states, not affected by traps, get more likely giving rise to a survival probability decaying to a finite value. Both these features constitute a qualitative difference with respect to the behavior found for classical walks.     

Quantum walk with jumps

We analyze a special class of 1-D quantum walks (QWs) realized using optical multi-ports. We assume non-perfect multi-ports showing errors in the connectivity, i.e. with a small probability the multi-ports can connect not to their nearest neighbor but to another multi-port at a fixed distance – we call this a jump. We study two cases of QW with jumps where multiple displacements can emerge at one timestep. The first case assumes time-correlated jumps (static disorder). In the second case, we choose the positions of jumps randomly in time (dynamic disorder). The probability distributions of position of the QW walker in both instances differ significantly: dynamic disorder leads to a Gaussian-like distribution, while for static disorder we find two distinct behaviors depending on the parity of jump size. In the case of even-sized jumps, the distribution exhibits a three-peak profile around the position of the initial excitation, whereas the probability distribution in the odd case follows a Laplace-like discrete distribution modulated by additional (exponential) peaks for long times. Finally, our numerical results indicate that by an appropriate mapping a universal functional behavior of the variance of the long-time probability distribution can be revealed with respect to the scaled average of jump size.     

Exact solutions and symmetry analysis for the limiting probability distribution of quantum walks

In the literature, there are numerous studies of one-dimensional discrete-time quantum walks (DTQWs) using a moving shift operator. However, there is no exact solution for the limiting probability distributions of DTQWs on cycles using a general coin or swapping shift operator. In this paper, we derive exact solutions for the limiting probability distribution of quantum walks using a general coin and swapping shift operator on cycles for the first time. Based on the exact solutions, we show how to generate symmetric quantum walks and determine the condition under which a symmetric quantum walk appears. Our results suggest that choosing various coin and initial state parameters can achieve a symmetric quantum walk. By defining a quantity to measure the variation of symmetry, deviation and mixing time of symmetric quantum walks are also investigated.     

Survival of Classical and Quantum Particles in the Presence of Traps

We present a detailed comparison of the motion of a classical and of a quantum particle in the presence of trapping sites, within the framework of continuous-time classical and quantum random walk. The main emphasis is on the qualitative differences in the temporal behavior of the survival probabilities of both kinds of particles. As a general rule, static traps are far less efficient to absorb quantum particles than classical ones. Several lattice geometries are successively considered: an infinite chain with a single trap, a finite ring with a single trap, a finite ring with several traps, and an infinite chain and a higher-dimensional lattice with a random distribution of traps with a given density. For the latter disordered systems, the classical and the quantum survival probabilities obey a stretched exponential asymptotic decay, albeit with different exponents. These results confirm earlier predictions, and the corresponding amplitudes are evaluated. In the one-dimensional geometry of the infinite chain, we obtain a full analytical prediction for the amplitude of the quantum problem, including its dependence on the trap density and strength.     

Decoherence and disorder in quantum walks: from ballistic spread to localization

We investigate the impact of decoherence and static disorder on the dynamics of quantum particles moving in a periodic lattice. Our experiment relies on the photonic implementation of a one-dimensional quantum walk. The pure quantum evolution is characterized by a ballistic spread of a photon's wave packet along 28 steps. By applying controlled time-dependent operations we simulate three different environmental influences on the system, resulting in a fast ballistic spread, a diffusive classical walk, and the first Anderson localization in a discrete quantum walk architecture.     

Localization under the effect of randomly distributed decoherence

Electron transport through disordered quasi one-dimensional quantum systems is studied. Decoherence is taken into account by a spatial distribution of virtual reservoirs, which represent local interactions of the conduction electrons with their environment. We show that the decoherence distribution has observable effects on the transport. If the decoherence reservoirs are distributed randomly without spatial correlations, a minimal degree of decoherence is necessary to obtain Ohmic conduction. Below this threshold the system is localized and thus, a decoherence driven metal-insulator transition is found. In contrast, for homogenously distributed decoherence, any finite degree of decoherence is sufficient to destroy localization. Thus, the presence or absence of localization in a disordered one-dimensional system may give important insight about how the electron phase is randomized.     

Classical random walk with memory versus quantum walk on a one-dimensional infinite chain

Quantum walk is a very useful tool for building quantum algorithms due to the faster spreading of probability distributions as compared to a classical random walk. Comparing the spreading of the probability distributions of a quantum walk with that of a mnemonic classical random walk on a one-dimensional infinite chain, we find that the classical random walk could have a faster spreading than that of the quantum walk conditioned on a finite number of walking steps. Quantum walk surpasses classical random walk with memory in spreading speed when the number of steps is large enough. However, in such a situation, quantum walk would seriously suffer from decoherence. Therefore, classical walk with memory may have some advantages in practical applications.     

逾渗分立时间量子行走的传输及纠缠特性

在一维分立时间量子行走中,通过静态和动态两种方式随机地断开连接边引入无序效应,研究了静态逾渗和动态逾渗对量子行走传输特性以及位置自由度和硬币自由之间纠缠的影响.随着演化时间的增加,静态逾渗会使得量子行走从弹道传输转变为安德森局域化,而动态逾渗则会使之转变为经典扩散.理想情况下,量子纠缠在较短的时间内就达到一个常数值E_0.静态逾渗量子行走的纠缠减小,并随着时间做无规振荡,而动态逾渗量子行走的纠缠则会随着时间光滑地增加,并在某一时间超过理想情况下的常数值,表现出动态逾渗增强量子纠缠的特性.     

Stopping time of a one-dimensional bounded quantum walk

The stopping time of a one-dimensional bounded classical random walk(RW) is defined as the number of steps taken by a random walker to arrive at a fixed boundary for the first time.A quantum walk(QW) is a non-trivial generalization of RW,and has attracted a great deal of interest from researchers working in quantum physics and quantum information.In this paper,we develop a method to calculate the stopping time for a one-dimensional QW.Using our method,we further compare the properties of stopping time for QW and RW.We find that the mean value of the stopping time is the same for both of these problems.However,for short times,the probability for a walker performing a QW to arrive at the boundary is larger than that for a RW.This means that,although the mean stopping time of a quantum and classical walker are the same,the quantum walker has a greater probability of arriving at the boundary earlier than the classical walker.     

Quantum Harmonic Oscillator Systems with Disorder

We study many-body properties of quantum harmonic oscillator lattices with disorder. A sufficient condition for dynamical localization, expressed as a zero-velocity Lieb-Robinson bound, is formulated in terms of the decay of the eigenfunction correlators for an effective one-particle Hamiltonian. We show how state-of-the-art techniques for proving Anderson localization can be used to prove that these properties hold in a number of standard models. We also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of one-particle eigenfunction correlators. In particular, we show that static correlations decay exponentially fast if the corresponding effective one-particle Hamiltonian exhibits localization at low energies, regardless of whether there is a gap in the spectrum above the ground state or not. Our results apply to finite as well as to infinite oscillator systems. The eigenfunction correlators that appear are more general than those previously studied in the literature. In particular, we must allow for functions of the Hamiltonian that have a singularity at the bottom of the spectrum. We prove exponential bounds for such correlators for some of the standard models.     

Uniqueness of Gibbs states of a quantum system on graphs

Gibbs states of an infinite system of interacting quantum particles are considered. Each particle moves on a compact Riemannian manifold and is attached to a vertex of a graph (one particle per vertex). Two kinds of graphs are studied: (a) a general graph with locally finite degree, (b) a graph with globally bounded degree. In case (a), the uniqueness of Gibbs states has been shown under the condition that the interaction potentials are uniformly bounded by a sufficiently small constant. In case (b), the interaction potentials are random. Here, under a certain condition imposed on the probability distribution of these potentials, the almost sure uniqueness of Gibbs states has been shown.     

Asymptotic distributions of continuous-time random walks: A probabilistic approach

We provide a systematic analysis of the possible asymptotic distributions o one-dimensional continuous-time random walks (CTRWs) by applying the limit theorems of probability theory. Biased and unbiased walks of coupled and decoupled memory are considered. In contrast to previous work concerning decoupled memory and Lévy walks, we deal also with arbitrary coupled memory and with jump densities asymmetric about its mean, obtaining asymmetric Lévy-stable limits. Suprisingly, it is found that in most cases coupled memory has no essential influence on the form of the limiting distribution. We discuss interesting properties of walks with an infinite mean waiting time between successive jumps.     

Discrete-time quantum walks on one-dimensional lattices

In this paper, we study discrete-time quantum walks on one-dimensional lattices. We find that the coherent dynamics depends on the initial states and coin parameters. For infinite size of lattices, we derive an explicit expression for the return probability, which shows scaling behavior P(0, t) ~ t ^-1 and does not depends on the initial states of the walk. In the long-time limit, the probability distribution shows various patterns, depending on the initial states, coin parameters and the lattice size. The time-averaged probability mixes to the limiting probability distribution in linear time, i.e., the mixing time M _ε is a linear function of N (size of the lattices) for large values of thresholds ϵ. Finally, we introduce another kind of quantum walk on infinite or even-numbered size of lattices, and show that by the method of mathematical induction, the walk is equivalent to the traditional quantum walk with symmetrical initial state and coin parameter.     

First passage time distributions for finite one-dimensional random walks

We present closed expressions for the characteristic function of the first passage time distribution for biased and unbiased random walks on finite chains and continuous segments with reflecting boundary conditions. Earlier results on mean first passage times for one-dimensional random walks emerge as special cases. The divergences that result as the boundary is moved out to infinity are exhibited explicitly. For a symmetric random walk on a line, the distribution is an elliptic theta function that goes over into the known Lévy distribution with exponent 1/2 as the boundary tends to ∞.     

A measure of the symmetry of random walks

We derive the probability density for a simple measure of the asymmetry of a one-dimensional random walk, namely the ratio of the minimum of the two maximum displacements in the positive and negative directions, to the maximum. We show that in the diffusion limit the asymmetry is independent of time. These results are shown to apply to random walks in which individual steps have a stable law distribution as well as to multidimensional random walks.     

Controllable simulation of topological phases and edge states with quantum walk

We simulate various topological phenomena in condense matter, such as formation of different topological phases, boundary and edge states, through two types of quantum walk with step-dependent coins. Particularly, we show that one-dimensional quantum walk with step-dependent coin simulates all types of topological phases in BDI family, as well as all types of boundary and edge states. In addition, we show that step-dependent coins provide the number of steps as a controlling factor over the simulations. In fact, with tuning number of steps, we can determine the occurrences of boundary, edge states and topological phases, their types and where they should be located. These two features make quantum walks versatile and highly controllable simulators of topological phases, boundary, edge states, and topological phase transitions. We also report on emergences of cell-like structures for simulated topological phenomena. Each cell contains all types of boundary (edge) states and topological phases of BDI family.