The dynamics of a discrete-time quantum walk subject to time-correlated noise is studied in this paper. The noise, which goes beyond the usual telegraph noise with random variables 1 and $\ensuremath{-}1$, is modeled as a unitary coin-type operator and it is generated by a sample path of the Ornstein-Uhlenbeck process. Up to the first order in the amplitude of noise, a master equation to describe the walk is derived. The dynamics governed by the master equation is in good agreement with those given by numerical simulations within a certain number of steps, depending on the parameters of noise. Two remarkable features of long-time dynamics are observed in the numerical simulations. (1) In the slow noise regime, with the increase of the noise amplitude, the quantum coherence is suppressed, and the dynamics of noisy discrete-time quantum walk gradually transits to that of classical random walk. (2) In the fast noise regime, the walker is confined into a few lattice sites, and the width of the wave packet is much narrower compared with that in the slow noise regime.
The dynamics of a discrete-time quantum walk subject to time-correlated noise is studied in this paper. The noise, which goes beyond the usual telegraph noise with random variables 1 and \ensuremath−1, is modeled as a unitary coin-type operator and it is generated by a sample path of the Ornstein-Uhlenbeck process. Up to the first order in the amplitude of noise, a master equation to describe the walk is derived. The dynamics governed by the master equation is in good agreement with those given by numerical simulations within a certain number of steps, depending on the parameters of noise. Two remarkable features of long-time dynamics are observed in the numerical simulations. (1) In the slow noise regime, with the increase of the noise amplitude, the quantum coherence is suppressed, and the dynamics of noisy discrete-time quantum walk gradually transits to that of classical random walk. (2) In the fast noise regime, the walker is confined into a few lattice sites, and the width of the wave packet is much narrower compared with that in the slow noise regime.
In the case of the discrete time coined quantum walk the reduced dynamics of the coin shows non-Markovian recurrence features due to information back-flow from the position degree of freedom. Here we study how this non-Markovian behavior is modified in the presence of open system dynamics. In the process, we obtain useful insights into the nature of non-Markovian physics. In particular, we show that in the case of (non-Markovian) random telegraph noise (RTN), a further discernbile recurrence feature is present in the dynamics. Moreover, this feature is correlated with the localization of the walker. On the other hand, no additional recurruence feature appears for other non-Markovian types of noise (Ornstein-Uhlenbeck and Power Law noise). We propose a power spectral method for comparing the relative strengths of the non-Markovian component due to the external noise and that due to the internal position degree of freedom.
We address the dynamics of continuous-time quantum walks on one-dimensional disordered lattices inducing dynamical noise in the system. Noise is described as time-dependent fluctuations of the tunneling amplitudes between adjacent sites, and attention is focused on non-Gaussian telegraph noise, going beyond the usual assumption of fast Gaussian noise. We observe the emergence of two different dynamical behaviors for the walker, corresponding to two opposite noise regimes: slow noise (i.e., strong coupling with the environment) confines the walker into few lattice nodes, while fast noise (weak coupling) induces a transition between quantum and classical diffusion over the lattice. A phase transition between the two dynamical regimes may be observed by tuning the ratio between the autocorrelation time of the noise and the coupling between the walker and the external environment generating the noise. We also address the non-Markovianity of the quantum map by assessing its memory effects, as well as evaluating the information backflow to the system. Our results suggest that the non-Markovian character of the evolution is linked to the dynamical behavior in the slow noise regime, and that fast noise induces a Markovian dynamics for the walker.
Discrete-time quantum walks exhibit ballistic spreading that is highly sensitive to decoherence. We study the long-time dynamics of a one-dimensional discrete-time quantum walk under Pauli noise acting on the coin. Choosing the unitary coin such that its invariant Bloch-sphere axis is orthogonal to the axis fixed by the conditional shift, we identify the distinct effects of noise aligned with either dynamical axis or orthogonal to both. Using a Fourier-space Pauli-superoperator formalism, we derive expressions for the first two position moments and long-time asymptotics for X-, Y-, Z-, and depolarizing noise. All channels induce a crossover from ballistic to diffusive spreading, with a second moment growing linearly at long times. The approach to this regime is channel dependent: it is exponentially fast for X-, Z-, and depolarizing noise, but algebraic, with a leading t−1/2 correction, for Y-noise. The position distributions retain distinct finite-noise signatures, including a Gaussian-like profile with a central depression for Y-noise. In the maximal-noise limit, depolarizing, Y-, and Z-noise yield the classical binomial distribution for arbitrary initial coin states, whereas X-noise does so only when the initial state has no component along the noise axis. Thus, the quantum-to-classical transition is governed by the relative alignment of the noise, the coherent walk dynamics, and the initial state.
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 Two models are first presented, of a one-dimensional discrete-time quantum walk (DTQW) with temporal noise on the internal degree of freedom (i.e., the coin): (i) a model with both a coin-flip and a phase-flip channel, and (ii) a model with random coin unitaries. It is then shown that both these models admit a common limit in the spacetime continuum, namely, a Lindblad equation with Dirac-fermion Hamiltonian part and, as Lindblad jumps, a chirality flip and a chirality-dependent phase flip, which are two of the three standard error channels for a two-level quantum system. This, as one may call it, Dirac Lindblad equation, provides a model of quantum relativistic spatial diffusion, which is evidenced both analytically and numerically. This model of spatial diffusion has the intriguing specificity of making sense only with original unitary models which are relativistic in the sense that they have chirality, on which the noise is introduced: the diffusion arises via the by-construction (quantum) coupling of chirality to the position. For a particle with vanishing mass, the model of quantum relativistic diffusion introduced in the present work, reduces to the well-known telegraph equation, which yields propagation at short times, diffusion at long times, and exhibits no quantumness. Finally, the results are extended to temporal noises which depend smoothly on position.
This study investigates the crossover dynamics of a one-dimensional discrete-time quantum walk (DTQW) in noisy environments using the density matrix formalism and Kraus operator representation of quantum operations. Decoherence is modeled by bit-flip, dephasing, and bit-phase noise channels, while dissipation is represented by generalized amplitude damping channels. The analysis of crossover dynamics involves examining the probability distribution, scaling exponent of variance, coherence measures, and entropy metrics such as Shannon and von Neumann entropy. The DTQW produces a symmetrical probability distribution of displacement under decoherence, with distinct features in scaling exponents of mean squared displacement or variance. Under dissipation, DTQW produces a non-symmetric probability distribution and spreads faster than classical walk under extreme decoherence. These suggest a broad class of crossover transport behavior, which can be classified as pure quantum walk, quantum-like walk, semi-classical-like walk, and classical-like walk. Remarkably, maximum Shannon entropy is achieved at lower degrees of decoherence. This study also analyzes the effect of temporal disorder in the coin operator modeled using fractional Gaussian noise (fGn). The crossover dynamic occurs in the form of weak localization, followed by enhanced entanglement and information backflow. Notably, the correlation of fGn is transferred to the coin's degree of entanglement and transpires in the two-point correlation of the von Neumann entropy fluctuation. These findings can be useful for implementing DTQW-based protocols in quantum error correction, quantum cryptography, and quantum memory on the noisy intermediate-scale quantum (NISQ) platform.