State transfer on two-fold Cayley trees via quantum walks

Perfect state transfer(PST)has great significance due to its applications in quantum information processing and quantum computation.The main problem we study in this paper is to determine whether the two-fold Cayley tree,an extension of the Cayley tree,admits perfect state transfer between two roots using quantum walks.We show that PST can be achieved by means of the so-called nonrepeating quantum walk[Phys.Rev.A 89042332(2014)]within time steps that are the distance between the two roots;while both the continuous-time quantum walk and the typical discrete-time quantum walk with Grover coin approaches fail.Our results suggest that in some cases the dynamics of a discrete-time quantum walk may be much richer than that of the continuous-time quantum walk.     

On Continuous-Time Self-Avoiding Random Walk in Dimension Four

We study the distribution of the end-to-end distance of continuous-time self-avoiding random walks (CTRW) in dimension four from two viewpoints. From a real-space renormalization-group map on probabilities, we conjecture the asymptotic behavior of the end-to-end distance of a weakly self-avoiding random walk (SARW) that penalizes two-body interactions of random walks in dimension four on a hierarchical lattice. Then we perform the Monte Carlo computer simulations of CTRW on the four-dimensional integer lattice, paying special attention to the difference in statistical behavior of the CTRW compared with the discrete-time random walks. In this framework, we verify the result already predicted by the renormalization-group method and provide new results related to enumeration of self-avoiding random walks and calculation of the mean square end-to-end distance and gyration radius of continous-time self-avoiding random walks.     

On the non-equivalence of two standard random walks

We focus on two models of nearest-neighbour random walks on d$d$-dimensional regular hyper-cubic lattices that are usually assumed to be identical—the discrete-time Polya walk, in which the walker steps at each integer moment of time, and the Montroll–Weiss continuous-time random walk in which the time intervals between successive steps are independent, exponentially and identically distributed random variables with mean 1$1$. We show that while for symmetric random walks both models indeed lead to identical behaviour in the long time limit, when there is an external bias they lead to markedly different behaviour.     

Finding tree symmetries using continuous-time quantum walk

Quantum walk, the quantum counterpart of random walk, is an important model and widely studied to develop new quantum algorithms. This paper studies the relationship between the continuous-time quantum walk and the symmetry of a graph, especially that of a tree. Firstly, we prove in mathematics that the symmetry of a graph is highly related to quantum walk. Secondly, we propose an algorithm based on the continuous-time quantum walk to compute the symmetry of a tree. Our algorithm has better time complexity O(N3) than the current best algorithm. Finally, through testing three types of 10024 trees, we find that the symmetry of a tree can be found with an extremely high efficiency with the help of the continuous-time quantum walk.     

Non-Markovian decoherent quantum walks

Quantum walks act in obviously different ways from their classical counterparts, but decoherence will lessen and close this gap between them. To understand this process, it is necessary to investigate the evolution of quantum walks under different decoherence situations. In this article, we study a non-Markovian decoherent quantum walk on a line. In a short time regime, the behavior of the walk deviates from both ideal quantum walks and classical random walks. The position variance as a measure of the quantum walk collapses and revives for a short time, and tends to have a linear relation with time. That is, the walker's behavior shows a diffusive spread over a long time limit, which is caused by non-Markovian dephasing affecting the quantum correlations between the quantum walker and his coin. We also study both quantum discord and measurement-induced disturbance as measures of the quantum correlations, and observe both collapse and revival in the short time regime, and the tendency to be zero in the long time limit. Therefore, quantum walks with non-Markovian decoherence tend to have diffusive spreading behavior over long time limits, while in the short time regime they oscillate between ballistic and diffusive spreading behavior, and the quantum correlation collapses and revives due to the memory effect.     

Quantum walk under coherence non-generating channels

We investigate the probability distribution of the quantum walk under coherence non-generating channels. We definea model called generalized classical walk with memory. Under certain conditions, generalized classical random walk withmemory can degrade into classical random walk and classical random walk with memory. Based on its various spreadingspeed, the model may be a useful tool for building algorithms. Furthermore, the model may be useful for measuring thequantumness of quantum walk. The probability distributions of quantum walks are generalized classical random walkswith memory under a class of coherence non-generating channels. Therefore, we can simulate classical random walkand classical random walk with memory by coherence non-generating channels. Also, we find that for another class ofcoherence non-generating channels, the probability distributions are influenced by the coherence in the initial state of thecoin. Nevertheless, the influence degrades as the number of steps increases. Our results could be helpful to explore therelationship between coherence and quantum walk.     

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.     

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 probabilities in coherent exciton transfer with trapping

In the quest for signatures of coherent transport we consider exciton trapping in the continuous-time quantum walk framework. The survival probability displays different decay domains, related to distinct regions of the spectrum of the Hamiltonian. For linear systems and at intermediate times the decay obeys a power law, in contrast with the corresponding exponential decay found in incoherent continuous-time random walk situations. To differentiate between the coherent and incoherent mechanisms, we present an experimental protocol based on a frozen Rydberg gas structured by optical dipole traps.     

The two-state random walk

We develop asymptotic results for the two-state random walk, which can be regarded as a generalization of the continuous-time random walk. The two-state random walk is one in which a particle can be in one of two states for random periods of time, each of the states having different spatial transition probabilities. When the sojourn times in each of the states and the second moments of transition probabilities are finite, the state probabilities have an asymptotic Gaussian form. Several known asymptotic results are reproduced, such as the Gaussian form for the probability density of position in continuous-time random walks, the time spent in one of these states, and the diffusion constant of a two-state diffusing particle.     

Survival,Extinction and Approximation of Discrete-time Branching Random Walks

We consider a general discrete-time branching random walk on a countable set X. We relate local, strong local and global survival with suitable inequalities involving the first-moment matrix M of the process. In particular we prove that, while the local behavior is characterized by M, the global behavior cannot be completely described in terms of properties involving M alone. Moreover we show that locally surviving branching random walks can be approximated by sequences of spatially confined and stochastically dominated branching random walks which eventually survive locally if the (possibly finite) state space is large enough. An analogous result can be achieved by approximating a branching random walk by a sequence of multitype contact processes and allowing a sufficiently large number of particles per site. We compare these results with the ones obtained in the continuous-time case and we give some examples and counterexamples.     

Walks on Apollonian networks

We carry out comparative studies of random walks on deterministic Apollonian networks (DANs) and random Apollonian networks (RANs). We perform computer simulations for the mean first-passage time, the average return time, the mean-square displacement, and the network coverage for the unrestricted random walk. The diffusions both on DANs and RANs are proved to be sublinear. The effects of the network structure on the dynamics and the search efficiencies of walks with various strategies are also discussed. Contrary to intuition, it is shown that the self-avoiding random walk, which has been verified as an optimal local search strategy in networks, is not the best strategy for the DANs in the large size limit.     

Search algorithm on strongly regular graphs based on scattering quantum walks

Janmark, Meyer, and Wong showed that continuous-time quantum walk search on known families of strongly regular graphs(SRGs) with parameters(N, k, λ, μ) achieves full quantum speedup. The problem is reconsidered in terms of scattering quantum walk, a type of discrete-time quantum walks. Here, the search space is confined to a low-dimensional subspace corresponding to the collapsed graph of SRGs. To quantify the algorithm's performance, we leverage the fundamental pairing theorem, a general theory developed by Cottrell for quantum search of structural anomalies in star graphs.The search algorithm on the SRGs with k scales as N satisfies the theorem, and results can be immediately obtained, while search on the SRGs with k scales as√N does not satisfy the theorem, and matrix perturbation theory is used to provide an analysis. Both these cases can be solved in O(√N) time steps with a success probability close to 1. The analytical conclusions are verified by simulation results on two SRGs. These examples show that the formalism on star graphs can be applied more generally.     

Shannon Entropy and Diffusion Coefficient in Parity-Time Symmetric Quantum Walks

Non-Hermitian topological edge states have many intriguing properties, however, to date, they have mainly been discussed in terms of bulk–boundary correspondence. Here, we propose using a bulk property of diffusion coefficients for probing the topological states and exploring their dynamics. The diffusion coefficient was found to show unique features with the topological phase transitions driven by parity–time (PT)-symmetric non-Hermitian discrete-time quantum walks as well as by Hermitian ones, despite the fact that artificial boundaries are not constructed by an inhomogeneous quantum walk. For a Hermitian system, a turning point and abrupt change appears in the diffusion coefficient when the system is approaching the topological phase transition, while it remains stable in the trivial topological state. For a non-Hermitian system, except for the feature associated with the topological transition, the diffusion coefficient in the PT-symmetric-broken phase demonstrates an abrupt change with a peak structure. In addition, the Shannon entropy of the quantum walk is found to exhibit a direct correlation with the diffusion coefficient. The numerical results presented herein may open up a new avenue for studying the topological state in non-Hermitian quantum walk systems.     

Approximate locality for quantum systems on graphs

In this Letter we make progress on a long-standing open problem of Aaronson and Ambainis [Theory Comput. 1, 47 (2005)]: we show that if U is a sparse unitary operator with a gap Delta in its spectrum, then there exists an approximate logarithm H of U which is also sparse. The sparsity pattern of H gets more dense as 1/Delta increases. This result can be interpreted as a way to convert between local continuous-time and local discrete-time quantum processes. As an example we show that the discrete-time coined quantum walk can be realized stroboscopically from an approximately local continuous-time quantum walk.     

Continuity and Anomalous Fluctuations in Random Walks in Dynamic Random Environments: Numerics,Phase Diagrams and Conjectures

We perform simulations for one dimensional continuous-time random walks in two dynamic random environments with fast (independent spin-flips) and slow (simple symmetric exclusion) decay of space-time correlations, respectively. We focus on the asymptotic speeds and the scaling limits of such random walks. We observe different behaviors depending on the dynamics of the underlying random environment and the ratio between the jump rate of the random walk and the one of the environment. We compare our data with well known results for static random environment. We observe that the non-diffusive regime known so far only for the static case can occur in the dynamical setup too. Such anomalous fluctuations give rise to a new phase diagram. Further we discuss possible consequences for more general static and dynamic random environments.     

Quantum walks with coins undergoing different quantum noisy channels

Quantum walks have significantly different properties compared to classical random walks, which have potential applications in quantum computation and quantum simulation. We study Hadamard quantum walks with coins undergoing different quantum noisy channels and deduce the analytical expressions of the first two moments of position in the longtime limit. Numerical simulations have been done, the results are compared with the analytical results, and they match extremely well. We show that the variance of the position distributions of the walks grows linearly with time when enough steps are taken and the linear coefficient is affected by the strength of the quantum noisy channels.     

Continuous-Time Quantum Walk on Integer Lattices and Homogeneous Trees

This paper is concerned with the continuous-time quantum walk on ℤ, ℤ ^d , and infinite homogeneous trees. By using the generating function method, we compute the limit of the average probability distribution for the general isotropic walk on ℤ, and for nearest-neighbor walks on ℤ ^d and infinite homogeneous trees. In addition, we compute the asymptotic approximation for the probability of the return to zero at time t in all these cases.     

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.