Disorder in parity-time symmetric quantum walks

We experimentally investigate the impact of static disorder and dynamic disorder on the non-unitary dynamics of parity-time(PT)-symmetric quantum walks.Via temporally alternating photon losses in an interferometric network,we realize the passive PT-symmetric quantum dynamics for single photons.Controllable coin operations allow us to simulate different environmental influences,which result in three different behaviors of quantum walkers:a standard ballistic spread,a diffusive behavior,and a localization,respectively,in a PT-symmetric quantum walk architecture.     

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.     

A quantum walk in phase space with resonator-assisted double quantum dots

We implement a quantum walk in phase space with a new mechanism based on the superconducting resonator-assisted double quantum dots.By analyzing the hybrid system,we obtain the necessary factors implementing a quantum walk in phase space:the walker,coin,coin flipping and conditional phase shift.The coin flipping is implemented by adding a driving field to the resonator.The interaction between the quantum dots and resonator is used to implement conditional phase shift.Furthermore,we show that with different driving fields the quantum walk in phase space exhibits a ballistic behavior over 25 steps and numerically analyze the factors influencing the spreading of the walker in phase space.     

Edge states in a two-dimensional quantum walk with disorder

We investigate the effect of spatial disorder on the edge states localized at the interface between two topologically different regions. Rotation disorder can localize the quantum walk if it is strong enough to change the topology, otherwise the edge state is protected. Nonlinear spatial disorder, dependent on the walker’s state, attracts the walk to the interface even for very large coupling, preserving the ballistic transport characteristic of the clean regime.     

基于相位匹配的量子行走搜索算法及电路实现

量子行走是经典随机行走在量子力学框架下的对应, 理论上可以用来解决一类无序数据库的搜索问题. 因为携带信息的量子态的扩散速度与经典相比有二次方式的增长, 所以量子行走优于经典随机行走, 量子行走的特性值得加以利用. 量子行走作为一种新发现的物理现象的数学描述, 引发了一种新的思维方式, 孕育了一种新的理论计算模型. 最新研究表明, 量子行走本身也是一种通用计算模型, 可被视为设计量子算法的高级工具, 因此受到部分计算机理论科学领域学者的关注和研究. 对于多数问题求解方案的量子算法的设计, 理论上可以只在量子行走模型下进行考虑. 基于Grover算法的相位匹配条件, 本文提出了一个新的基于量子行走的搜索算法. 理论演算表明: 一般情况下本算法的时间复杂度与Grover算法相同, 但是当搜索的目标数目多于总数的1/3时, 本算法搜索成功的概率要大于Grover算法. 本文不但利用Grover算法中相位匹配条件构造了一个新的量子行走搜索算法, 而且在本研究室原有的量子电路设计研究成果的基础上给出了该算法的量子电路表述.     

Self-Attractive Random Walks: The Case of Critical Drifts

Self-attractive random walks (polymers) undergo a phase transition in terms of the applied drift (force): If the drift is strong enough, then the walk is ballistic, whereas in the case of small drifts self-attraction wins and the walk is sub-ballistic. We show that, in any dimension d ≥ 2, this transition is of first order. In fact, we prove that the walk is already ballistic at critical drifts, and establish the corresponding LLN and CLT.     

Ballistic transport and quantum interference in InSb nanowire devices

An experimental realization of a ballistic superconductor proximitized semiconductor nanowire device is a necessary step towards engineering topological quantum electronics. Here, we report on ballistic transport in In Sb nanowires grown by molecular-beam epitaxy contacted by superconductor electrodes. At an elevated temperature, clear conductance plateaus are observed at zero magnetic field and in agreement with calculations based on the Landauer formula. At lower temperature, we have observed characteristic Fabry–Pérot patterns which confirm the ballistic nature of charge transport.Furthermore, the magnetoconductance measurements in the ballistic regime reveal a periodic variation related to the Fabry–Pérot oscillations. The result can be reasonably explained by taking into account the impact of magnetic field on the phase of ballistic electron's wave function, which is further verified by our simulation. Our results pave the way for better understanding of the quantum interference effects on the transport properties of In Sb nanowires in the ballistic regime as well as developing of novel device for topological quantum computations.     

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.     

On the Relationship Between Continuous- and Discrete-Time Quantum Walk

Quantum walk is one of the main tools for quantum algorithms. Defined by analogy to classical random walk, a quantum walk is a time-homogeneous quantum process on a graph. Both random and quantum walks can be defined either in continuous or discrete time. But whereas a continuous-time random walk can be obtained as the limit of a sequence of discrete-time random walks, the two types of quantum walk appear fundamentally different, owing to the need for extra degrees of freedom in the discrete-time case. In this article, I describe a precise correspondence between continuous- and discrete- time quantum walks on arbitrary graphs. Using this correspondence, I show that continuous-time quantum walk can be obtained as an appropriate limit of discrete-time quantum walks. The correspondence also leads to a new technique for simulating Hamiltonian dynamics, giving efficient simulations even in cases where the Hamiltonian is not sparse. The complexity of the simulation is linear in the total evolution time, an improvement over simulations based on high-order approximations of the Lie product formula. As applications, I describe a continuous-time quantum walk algorithm for element distinctness and show how to optimally simulate continuous-time query algorithms of a certain form in the conventional quantum query model. Finally, I discuss limitations of the method for simulating Hamiltonians with negative matrix elements, and present two problems that motivate attempting to circumvent these limitations.     

Random Walk Quantum Clustering Algorithm Based on Space

In the random quantum walk, which is a quantum simulation of the classical walk, data points interacted when selecting the appropriate walk strategy by taking advantage of quantum-entanglement features; thus, the results obtained when the quantum walk is used are different from those when the classical walk is adopted. A new quantum walk clustering algorithm based on space is proposed by applying the quantum walk to clustering analysis. In this algorithm, data points are viewed as walking participants, and similar data points are clustered using the walk function in the pay-off matrix according to a certain rule. The walk process is simplified by implementing a space-combining rule. The proposed algorithm is validated by a simulation test and is proved superior to existing clustering algorithms, namely, Kmeans, PCA + Kmeans, and LDA-Km. The effects of some of the parameters in the proposed algorithm on its performance are also analyzed and discussed. Specific suggestions are provided.     

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.     

Two models of quantum random walk

We present an overview of two models of quantum random walk. In the first model, the discrete quantum random walk, we present the explicit solution for the recurring amplitude of the quantum random walk on a one-dimensional lattice. We also introduce a new method of solving the problem of random walk in the most general case and use it to derive the hitting amplitude for quantum random walk on the hypercube. The second is a special model based on a local interaction between neighboring spin-1/2 particles on a one-dimensional lattice. We present explicit results for the relevant quantities and obtain an upper bound on the speed of convergence to limiting probability distribution.     

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.

     

Continuous-time quantum walks on star graphs

In this paper, we investigate continuous-time quantum walk on star graphs. It is shown that quantum central limit theorem for a continuous-time quantum walk on star graphs for N-fold star power graph, which are invariant under the quantum component of adjacency matrix, converges to continuous-time quantum walk on K₂ graphs (complete graph with two vertices) and the probability of observing walk tends to the uniform distribution.     

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.     

Ballistic Aharonov–Bohm quantum bits and quantum gates

We propose an architecture to perform quantum computation, using ballistic electrons as qubits and coupled quantum rings as quantum gates. In the proposed architecture two adjacent one-dimensional wires, creating a single qubit, are connected to two coupled quantum rings, where the required magnetic flux is provided by enclosed nano-sized magnets. The phase modulation of the wave function of the ballistic electrons under the Aharonov–Bohm effect is carefully designed to facilitate reprogrammable and dynamically controllable quantum gates. Arbitrary single-qubit quantum gates with high fidelity can be constructed on the basis of this architecture.     

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.     

Controllable Quantum Network Coding Scheme Based on Quantum Walk

Quantum entanglement is one of the key methods in quantum information processing, but it is difficult to prepare quantum entanglement. Quantum walk is widely used in quantum computation and quantum simulation, it can be applied to the preparation of quantum entangled states. In this paper, a controllable quantum network coding scheme based on quantum walk is proposed. With the help of quantum walk, the scheme preliminary realized the entanglement distribution of butterfly network, reduced entanglement resources and enhanced scalability. According to the existing technology, it is feasible to implement the quantum network coding scheme proposed in this paper.

     

Irreversibility of a quantum walk induced by controllable decoherence employing random unitary operations

Quantum walk is different from random walk in reversibility and interference. Observation of the reduced re- versibility in a realistic quantum walk is of scientific interest in understanding the unique quantum behavior. We propose an idea to experimentally investigate the decoherence-induced irreversibility of quantum walks with trapped ions in phase space via the average fidelity decay. By introducing two controllable decoherence sources, i.e., the phase damping channel (i.e., dephasing) and the high temperature amplitude reservoir (i.e., dissipation), in the intervals between the steps of quantum walk, we find that the high temperature amplitude reservoir shows more detrimental effects than the phase damping channel on quantum walks. Our study also shows that the average fidelity decay works better than the position variance for characterizing the transition from quantum walks to random walk. Experimental feasibility to monitor the irreversibility is justified using currently available techniques.