Most existing studies of quantum walk focus on unitary closed systems, while the nonunitary counterpart, relevant to open quantum systems, remains relatively unexplored. This paper analyzes the position distribution and variance of quantum walk under the specific nonunitary evolution operator. Our results reveal that the variance exhibits distinct dynamical behavior depending on angular parameters, spatial dimensions, and initial states. In particular, for the one-dimensional non-Hermitian two-band lattice model, we demonstrate that the variance grows quadratically with time steps in the short-time regime, indicating faster diffusion than classical random walk. Leveraging this acceleration, we further propose a teleportation protocol based on nonunitary quantum walk, which eliminates the need for entangled state preparation and thereby reduces resource overhead while enhancing information transfer efficiency.
Most existing studies of quantum walk focus on unitary closed systems, while the nonunitary counterpart, relevant to open quantum systems, remains relatively unexplored. This paper analyzes the position distribution and variance of quantum walk under the specific nonunitary evolution operator. Our results reveal that the variance exhibits distinct dynamical behavior depending on angular parameters, spatial dimensions, and initial states. In particular, for the one-dimensional non-Hermitian two-band lattice model, we demonstrate that the variance grows quadratically with time steps in the short-time regime, indicating faster diffusion than classical random walk. Leveraging this acceleration, we further propose a teleportation protocol based on nonunitary quantum walk, which eliminates the need for entangled state preparation and thereby reduces resource overhead while enhancing information transfer efficiency.
We study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary (i.e., nonrandom) case, we allow any unitary operator which commutes with translations and couples only sites at a finite distance from each other. For example, a single step of the walk could be composed of any finite succession of different shift and coin operations in the usual sense, with any lattice dimension and coin dimension. We find ballistic scaling and establish a direct method for computing the asymptotic distribution of position divided by time, namely as the distribution of the discrete time analog of the group velocity. In the random case, we let a Markov chain (control process) pick in each step one of finitely many unitary walks, in the sense described above. In ballistic order, we find a nonrandom drift which depends only on the mean of the control process and not on the initial state. In diffusive scaling, the limiting distribution is asymptotically Gaussian, with a covariance matrix (diffusion matrix) depending on momentum. The diffusion matrix depends not only on the mean but also on the transition rates of the control process. In the nonrandom limit, i.e., when the coins chosen are all very close or the transition rates of the control process are small, leading to long intervals of ballistic evolution, the diffusion matrix diverges. Our method is based on spatial Fourier transforms, and the first and second order perturbation theory of the eigenvalue 1 of the transition operator for each value of the momentum.
The quantum walk is a quantum counterpart of the classical random walk that exhibits nonclassical behaviors and outperforms the classical random walk in various aspects. It has been known that a single particle can be propagated by a discrete-time quantum walk with a quadratic time scaling in the variance of position distribution, beating the linear time scaling in a classical random walk. In this paper, we consider the discrete-time quantum walk for multiple particles in a one-dimensional lattice, and investigate the optimization of the joint coin state to enhance the spatial propagation of the particles in the lattice. We study the asymptotic evolution of position distribution for multiple particles in the long-time limit, and analytically optimize the joint coin state to derive the maximum variance of the position distribution between the particles after the evolution of the quantum walk. An interesting result is that an optimized coin state always possesses specific exchange symmetry which can be characterized by a graph consisting of two disconnected complete subgraphs and the exchange symmetry can significantly influence the position correlations between the particles, showing the critical role of coin symmetry in the propagation of multiple particles by the quantum walk. We further study the entanglement of the optimized coin states to show the relation of the coin correlations to the particle position distribution.
The quantum random walk has been much studied recently, largely due to its highly nonclassical behavior. In this paper, we study one possible route to classical behavior for the discrete quantum walk on the line: the presence of decoherence in the quantum ``coin'' which drives the walk. We find exact analytical expressions for the time dependence of the first two moments of position, and show that in the long-time limit the variance grows linearly with time, unlike the unitary walk. We compare this to the results of direct numerical simulation, and see how the form of the position distribution changes from the unitary to the usual classical result as we increase the strength of the decoherence.
In this thesis we study the propagation behavior of quantum particles which perform discrete-time quantum walks on spatial lattices.By definition, quantum walks are discrete-time evolutions such that a single time step couples only neighboring lattice sites.Additionally, we assume in this thesis that the dynamical law commutes with all translations of the spatial lattice, that is we consider translationally invariant quantum walks.The results presented in the following are separated into two parts: The first part is devoted to the study of single-particle quantum walks, and in the second part we investigate quantum walks of several particles.Single-particle quantum walks: Our aim is to study the asymptotic behavior of particles undergoing general translationally invariant quantum walks.Employing perturbation theory, we establish a direct method for computing the asymptotic distribution of the particle's position rescaled by an appropriate time-dependent factor.If the time evolution is implemented by a unitary operator commuting with all lattice translations and with empty pure point spectrum, we find linear scaling of the asymptotic position distribution.In this case, the asymptotic distribution of position divided by time is determined by the analog of the group velocity.If, on the other hand, the quantum walk is subject to decoherence, we expect that it behaves like a classical random walk.We provide explicit criteria under which this intuitive statement is true, that is, the asymptotic scaling of the particle's position distribution is that of a diffusion process.We find that the asymptotic distribution of position divided by the square root of time is a mixture of Gaussians in this case.However, we also provide examples of decoherent quantum walks which violate these criteria, and exhibit linear scaling of the asymptotic position distribution. Many-particle quantum walks:We investigate unitary quantum walks of two particles with collisional interactions.That is, the particles perform independent single-particle quantum walks if they are at distinct lattice sites, and an interaction takes place if both particles occupy the same lattice site.We prove that such quantum walks generically exhibit a molecular binding effect between the particles.More precisely, for a measurement of the particles' positions we have a bound on the probability to obtain a certain result which decays exponentially with the particles' distance.As a preparatory step for this analysis, we consider single-particle quantum walks with point defects, and provide a method to compute their eigenvalues in the spectral gap of the quantum walk without defect.We further generalize our model to arbitrary numbers of particles and study the behavior of local excitations in many-particle 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 (QW), the quantum mechanical counterpart of classical random walk, has recently been studied in various fields. The evolution of the discrete time quantum walk can be described as follows: the walker changes its spin state by the coin operator C, then takes one step left or right according to its spin state. For homogeneous quantum walk, the coin operator is independent of time and the standard deviation of the position grows linearly in time. It is quadratically faster than that in the classical random walk. In this work, we numerically study the dynamical behaviors of spreading in a one-dimensional discrete time quasiperiodic quantum walk (DTQQW). The DTQQW is that the coin operator is dependent on time and takes two different coins C() and C() arranged in generalized Fibonacci (GF) sequences. The GF sequences are constructed from A by the recursion relation: AAmBn, BA, for m, n are positive integers. They can be classified into two classes according to the wandering exponent . For 0, they belong to the first class, and for 0, they belong to the second class. For one dimensional system, the behaviors of two classes of GF systems are different either for the electronic spectrum of an electron in quasiperiodic potentials or for the quantum phase transitions of the quasiperiodic spin chains. In this paper, we discuss the cases of two different C operators (C();C()) arranged in GF sequences and find that the spreading behaviors are superdiffusion (the standard deviation of the position ~t; 0:5 1) for the two classes of GF DTQQW. For the second class of GF DTQQW, the exponent values are larger than those of the first class of GF DTQQW in the case of two identical C operators. By exploring the probability distribution in the real space, we find that for the first class of GF DTQQW, the probability distributions are almost the same for different initial states and are similar to the classical Gaussian distribution. For the probability distributions of the second class of GF DTQQW, there are two peaks at the two edges and the height of the two peaks can be different for different initial states. They are similar to the ballistic distribution of the homogeneous quantum walk. Therefore, we conclude that for the first class of GF DTQQW, the spreading behaviors are close to those of the classical random walk ( = 0:5) while for the second class of GF DTQQW, they are close to those of the homogeneous quantum walk ( = 1). This result is quite different from the characteristics of the quantum phase transitions in two classes of GF quasiperiodic quantum spin chains.