We introduce an analytically treatable discrete time quantum walk in a one-dimensional lattice which combines non-Markovianity and hyperballistic diffusion associated with a Gaussian whose variance ${\ensuremath{\sigma}}_{t}^{2}$ grows cubicly with time $\ensuremath{\sigma}\ensuremath{\propto}{t}^{3}$. These properties have have been numerically found in several systems, namely, tight-binding lattice models. For its rules, our model can be understood as the quantum version of the classical non-Markovian ``elephant random walk'' process for which the quantum coin operator only changes the value of the diffusion constant although, contrarily, to the classical coin.
We introduce an analytically treatable discrete time quantum walk in a one-dimensional lattice which combines non-Markovianity and hyperballistic diffusion associated with a Gaussian whose variance \ensuremathσt2 grows cubicly with time \ensuremathσ\ensuremath∝t3. These properties have have been numerically found in several systems, namely, tight-binding lattice models. For its rules, our model can be understood as the quantum version of the classical non-Markovian ``elephant random walk'' process for which the quantum coin operator only changes the value of the diffusion constant although, contrarily, to the classical coin.
Quantum walks have been a platform for the study and modeling of many processes and physical systems, going from its application on the development of quantum algorithms to the study of energy transport in biological molecules.In the same way that in general aspects of quantum theory, it also aroused the necessity of studying the introduction of noise in the evolution of quantum walks, given that it is a major factor in experimental applications.In this context, in 2018 G. D. Molfetta et al. presented a quantum model of the classical elephant random walk, the elephant quantum walk, that analyzes a noisy unitary evolution.Posteriorly, M. A. Pires et al. generalized this walk into the generalized elephant quantum walk, showing to have an interesting property of controlling the diffusive behavior of the walker while maintaining the maximum generation of entanglement between its degrees of freedom, something unknown until then.In this work we proposed ourselves to study, through numerical experiments, the generation of entanglement in the generalized elephant quantum walk in general initial settings and evolutions.For such, first we introduce classical random walks, reviewing the necessary probability concepts.After, we approach quantum walks and restrict the description to coined discrete time quantum walks, the type of quantum walk to be studied.Our results indicate that this type of quantum walk potentially generates maximally entangled states for almost all initial states and evolutions, even with a low degree of noise.These results indicate that a dynamically random evolution, being it either in the coin operator, as previously studied for other authors, or in the shift operator as in this walk, generates maximally correlated states.
Abstract We extend to the gamut of functional forms of the probability distribution of the time-dependent step-length a previous model dubbed Elephant Quantum Walk, which considers a uniform distribution and yields hyperballistic dynamics where the variance grows cubicly with time, σ2 ∝ t3, and a Gaussian for the position of the walker. We investigate this proposal both locally and globally with the results showing that the time-dependent interplay between interference, memory and long-range hopping leads to multiple transitions between dynamical regimes, namely ballistic → diffusive → superdiffusive → ballistic → hyperballistic for non-hermitian coin whereas the first diffusive regime is quelled for implementations using the Hadamard coin. In addition, we observe a robust asymptotic approach to maximal coin-space entanglement.
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.
Quantum walk (QW) is the quantum analog of the random walk. QW is an integral part of the development of numerous quantum algorithms. Hence, an in-depth understanding of QW helps us to grasp the quantum algorithms. We revisit the one-dimensional discrete-time QW and discuss basic steps in detail by incorporating the most general coin operator, constant in both space and time, and a localized initial state using numerical modeling. We investigate the impact of each parameter of the general coin operator on the probability distribution of the quantum walker. We show that by tuning the parameters of the general coin, one can regulate the probability distribution of the walker. We provide an algorithm for the one-dimensional quantum walk driven by the general coin operator. The study on general coin operators also includes the popular coins – Hadamard, Grover, and Fourier.
It is believed that in the long time limit, the limiting behavior of the discrete-time quantum random walk will cross from quantum to classical if we take into account of the decoherence. The computer simulation has already shown that for the discrete-time one-dimensional Hadamard quantum random walk with coin decoherence such that the measurement operators on the coin space are defined by A0 = Ic √1 − p, A1 = |R > < R| √p and A2 = |L > < L > < L| √p is diffusive when 0 < p ≤ 1 and it is ballistic when P = 0. In this thesis, we are going to let p to be dynamical depending on the step t, that is, we consider p = 1/tß, ß ≥ 0 and we found that it has sub-ballistic behavior for 0 < ß < 1. Furthermore, we study not only the coin decoherence, but the total decoherence, that means the measurement operators apply on the Hilbert space H = Hp ⊗ Hc instead of the coin space only. We find that the results are both sub-ballistic for the coin and total decoherence when 0 < ß < 1. Moreover, according to the model given in [T. A. Brun, H. A. Carteret, and A. Ambainis, Phys. Rev. A 67, 032304 (2003)], we know that if the walker has chance to hop to the second nearest neighbor lattice in one step, the long-time behavior is also sub-ballistic and it is similar to that the walker can hop to the third nearest neighbor lattice in one step. By the way, we also find that if we combine the classical part of the model given in [Jing Zhao and Peiqing Tong. One-dimensional quantum walks subject to next nearest neighbor hopping decoherence, Nanjing Normal University, preprint (2014)] with different step length, then this decoherence will also cross from quantum to classical. Finally, we define the quantum γ-stable walk and obtain the quantum γ-stable law with decoherence. By this decoherence, we can see that the limiting behavior of the quantum stable walk will also cross from quantum to classical and we shows that it spreads out faster than the classical stable walk.
We introduce a discrete-time quantum random walk (QRW) framework for spatial epidemic modelling on a two-dimensional square lattice and compare its dynamics to classical random-walk SIR models. In our model, each infected site spawns a quantum walker whose coherent evolution (controlled by an amplitude-splitting coin and conditional shifts) can infect visited susceptible sites with probability p and persists for a lifetime of τ steps. We perform extensive quantum simulations on finite lattices and compute the basic reproduction number R0 across a broad grid of (p,τ) values. Results show that QRW dynamics interpolate between diffusive and super-diffusive regimes: at low p the QRW reproduces classical-like R0, while at higher p and τ ballistic propagation and interference produce markedly larger R0 and non-Gaussian spatial profiles. We compare the QRW R0 range to empirical estimates from historical outbreaks and discuss parameter regimes where QRW offers a closer qualitative match than classical diffusion. We conclude that QRWs provide a flexible, conceptually novel toy model for exploring rapid or heavy-tailed epidemic spread.