Limitations of discrete-time quantum walk on a one-dimensional infinite chain

How well can we manipulate the state of a particle via a discrete-time quantum walk? We show that the discrete-time quantum walk on a one-dimensional infinite chain with coin operators that are independent of the position can only realize product operators of the form $e^{i\xi }A?{\mathbb{1}}_p$, which cannot change the position state of the walker. We present a scheme to construct all possible realizations of all the product operators of the form $e^{i\xi }A?{\mathbb{1}}_p$. When the coin operators are dependent on the position, we show that the translation operators on the position can not be realized via a DTQW with coin operators that are either the identity operator $\mathbb{1}$ or the Pauli operator ${\sigma }_x$.     

Classical vacuum versus quantum vacuum

We discuss the concept of quantum vacuum in comparison with the classical concept of vacuum in the context of light-wave propagation. The role played by quantum fluctuations is examined with respect to the behaviour of photons as the quanta of electromagnetic field and also in relation to the fact that photons can be regarded as non-zero mass particles. In this context, photon mass is defined from Klein–Gordon equation.     

A restricted random walk on a one-dimensional random chain

It is found that under some special conditions the inverse velocity for a restricted random walk diverges even though it is finite for the corresponding unrestricted walk. This leads to an anomalous t^z (0<z<1) behaviour for the mean distance travelled in time t.     

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.     

A myopic random walk on a finite chain

We solve analytically the problem of a biased random walk on a finite chain of ‘sites’ (1,2,…,N) in discrete time, with ‘myopic boundary conditions’—a walker at 1 (orN) at timen moves to 2 (orN − 1) with probability one at time (n + 1). The Markov chain has period two; there is no unique stationary distribution, and the moments of the displacement of the walker oscillate about certain mean values asn → ∞, with amplitudes proportional to 1/N. In the continuous-time limit, the oscillating behaviour of the probability distribution disappears, but the stationary distribution is depleted at the terminal sites owing to the boundary conditions. In the limit of continuous space as well, the problem becomes identical to that of diffusion on a line segment with the standard reflecting boundary conditions. The first passage time problem is also solved, and the differences between the walks with myopic and reflecting boundaries are brought out.     

一维相位缺陷量子行走的共振传输

从分立时间量子行走理论出发, 分别在包含两个格点相位缺陷和一段格点相位缺陷(方相位势)的一维格点线上研究量子行走的静态共振传输. 利用系统独特的色散关系和边界点上的能量守恒条件, 获得量子行走通过缺陷区域的透射率, 讨论了相位缺陷的强度和宽度不同时透射率随入射动量的变化行为. 在相位缺陷强度π/2两侧, 透射率表现出不同的共振特性, 并给出了强缺陷强度下共振峰和缺陷宽度的关系.     

Laplace operator and random walk on one-dimensional nonhomogeneous lattice

A classical result of probability theory states that under suitable space and time renormalization, a random walk converges to Brownian motion. We prove an analogous result in the case of nonhomogeneous random walk on onedimensional lattice. Under suitable conditions on the nonhomogeneous medium, we prove convergence to Brownian motion and explicitly compute the diffusion coefficient. The proofs are based on the study of the spectrum of random matrices of increasing dimension.     

星图上的散射量子行走搜索算法

量子行走是一种典型的量子计算模型, 近年来开始受到量子计算理论研究者们的广泛关注. 本文首先证明了在星图上硬币量子行走与散射量子行走的酉等价关系, 之后提出了一个在星图上的散射量子行走搜索算法. 该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时搜索成功概率大于Grover算法.     

Cumulative quantum work-deficit versus entanglement in the dynamics of an infinite spin chain

We find that the dynamical phase transition (DPT) in nearest-neighbor bipartite entanglement of time-evolved states of the anisotropic infinite quantum XY spin chain, in a transverse time-dependent magnetic field, can be quantitatively characterized by the dynamics of an information-theoretic quantum correlation measure, namely, quantum work-deficit (QWD). We show that only those nonequilibrium states exhibit entanglement resurrection after death, on changing the field parameter during the DPT, for which the cumulative bipartite QWD is above a threshold. The results point to an interesting inter-relation between two quantum correlation measures that are conceptualized from different perspectives.     

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.     

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.     

Anomalous quantum heat transport in a one-dimensional harmonic chain with random couplings

We investigate quantum heat transport in a one-dimensional harmonic system with random couplings. In the presence of randomness, phonon modes may normally be classified as ballistic, diffusive or localized. We show that these modes can roughly be characterized by the local nearest-neighbor level spacing distribution, similarly to their electronic counterparts. We also show that the thermal conductance G(th) through the system decays rapidly with the system size (G(th)?～?L(-α)). The exponent α strongly depends on the system size and can change from α??1 with increasing system size, indicating that the system undergoes a transition from a heat conductor to a heat insulator. This result could be useful in thermal control of low-dimensional systems.     

Transient solution of a random walk with chemical rule

Conolly et al. [Math. Scientist 22 (1997) 83-91] have obtained the transient distribution for a random walk moving on the integers -∞<k<∞ of the real line. Their analysis is based on a generating function technique. In this paper, an alternative technique is used to derive elegant explicit expressions for the transient state distribution of an infinite random walk having “chemical” rule and starting initially at any arbitrary integer position (say i). As a special case of our result, Conolly et al.'s (1997) solution is easily obtained. Moreover, the transient solution of the infinite symmetric continuous random walk is also presented. Finally, numerical values testing the quality of our analytical results are illustrated.     

Non-local quantum functions and the distributed Deutsch-Jozsa algorithm

Non-local implementations of quantum gates are a vital part of quantum networks. We find an optimal non-local implementation of quantum functions, the quantum gate equivalent of a switch statement. Then, we apply this result to the Deutsch-Jozsa problem, obtaining a distributed Deutsch-Jozsa algorithm and we show the relative efficiency improvement. As an application, we find a non-cooperative game based upon the original Deutsch-Jozsa problem where a classical agent has at most a 50% probability of winning, while a quantum agent can win every time.     

Random walk on a one-dimensional lattice with a random distribution of traps

The probability of return to the starting point of a particle executing a random walk on a one-dimensional lattice with a static distribution of traps, is derived, for asymptotically large n, the number of steps. The probability decreases as exp ($\text{?n}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$), in agreement with some recent estimates of its upper and lower bounds for diffusion in d dimensions.     

Energy band modulation of GaAs/Al0.26Ga0.74As quantum well in 3D self-assembled nanomembranes

In this study, we investigate the modulation of energy band in 3D self-assembled nanomembranes containing GaAs/Al_0.26Ga_0.74As quantum wells (QWs). Photoluminescence (PL) characterizations demonstrate that the self-assembled structures have different optical transition properties and the modulation of the energy band is thus realized. Detailed spectral analyses disclose that the small strain change in structures with different curvatures cannot cause remarkable change in energy bands in Al_0.26Ga_0.74As layer. On the other hand, the optical transitions of GaAs QW layer is influenced by the strain evolution in term of light emission intensity. We also find the first order Stark effect in rolled-up nanomembrane with diameter of 150 μm, which is closely connected with the coupling effect between the deformation potential and the piezoelectric potential. Our work may pave a way for the fabrication of high performance rolled-QW infrared photo-detectors.     

The classical and quantum mechanics of a particle on a knot

A free particle is constrained to move on a knot obtained by winding around a putative torus. The classical equations of motion for this system are solved in a closed form. The exact energy eigenspectrum, in the thin torus limit, is obtained by mapping the time-independent Schrödinger equation to the Mathieu equation. In the general case, the eigenvalue problem is described by the Hill equation. Finite-thickness corrections are incorporated perturbatively by truncating the Hill equation. Comparisons and contrasts between this problem and the well-studied problem of a particle on a circle (planar rigid rotor) are performed throughout.     

The asymptotic behavior of a random walk on a dual-medium lattice

This paper considers the asymptotic distribution for the horizontal displacement of a random walk in a medium represented by a two-dimensional lattice, whose transitions are to nearest-neighbor sites, are symmetric in the horizontal and vertical directions, and depend on the column currently occupied. On either side of a change-point in the medium, the transition probabilities are assumed to obey an asymptotic density condition. The displacement, when suitably normalized, converges to a diffusion process of oscillating Brownian motion type. Various special cases are discussed.     

Localization and recurrence of a quantum walk in a periodic potential on a line

We present a numerical study of a model of quantum walk in a periodic potential on a line. We take the simple view that different potentials have different affects on the way in which the coin state of the walker is changed. For simplicity and definiteness, we assume that the walker＇s coin state is unaffected at sites without the potential, and rotated in an unbiased way according to the Hadamard matrix at sites with the potential. This is the simplest and most natural model of a quantum walk in a periodic potential with two coins. Six generic cases of such quantum walks are studied numerically. It is found that, of the six cases, four cases display significant localization effect where the walker is confined in the neighborhood of the origin for a sufficiently long time. Associated with such a localization effect is the recurrence of the probability of the walker returning to the neighborhood of the origin.