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arXiv:2010.15281 · Physical Review A

Probing coherence and noise tolerance in discrete-time quantum walks: Unveiling self-focusing and breathing dynamics

A. R. C. Buarque, Wandearley S. Dias

Published Apr 16, 2021• 10 citations
arXivHTMLPDF

Abstract

The sensitivity of quantum systems to external disturbances is a fundamental problem for the implementation of functional quantum devices, quantum information, and computation. Based on remarkable experimental progress in optics and ultracold gases, we study the consequences of a short-time (instantaneous) noise while an intensity-dependent phase acquisition is associated with a qubit propagating on an $N$ cycle. By employing quantum coherence measures, we report emerging unstable regimes in which quantum walks arise, such as self-focusing and breathing dynamics. Our numerical and analytical results unveil appropriate settings which favor the stable regime, with the asymptotic distribution surviving for weak nonlinearities and disappearing in the thermodynamic limit with $1/N$. The diagram showing the threshold between different regimes reveals the quantum gates close to Pauli-Z as more noise tolerant. As we move towards the Pauli-X quantum gate, such aptness dramatically decreases and the threshold to the self-focusing regime becomes almost unavoidable. Quantum gates close to Hadamard exhibit an unusual aspect, in which an increment of the nonlinear strength can remove the dynamics from the self-focusing regime.

DOI
10.1103/physreva.103.042213

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The sensitivity of quantum systems to external disturbances is a fundamental problem for the implementation of functional quantum devices, quantum information, and computation. Based on remarkable experimental progress in optics and ultracold gases, we study the consequences of a short-time (instantaneous) noise while an intensity-dependent phase acquisition is associated with a qubit propagating on an NNN cycle. By employing quantum coherence measures, we report emerging unstable regimes in which quantum walks arise, such as self-focusing and breathing dynamics. Our numerical and analytical results unveil appropriate settings which favor the stable regime, with the asymptotic distribution surviving for weak nonlinearities and disappearing in the thermodynamic limit with 1/N1/N1/N. The diagram showing the threshold between different regimes reveals the quantum gates close to Pauli-Z as more noise tolerant. As we move towards the Pauli-X quantum gate, such aptness dramatically decreases and the threshold to the self-focusing regime becomes almost unavoidable. Quantum gates close to Hadamard exhibit an unusual aspect, in which an increment of the nonlinear strength can remove the dynamics from the self-focusing regime.
We study the existence and characterization of self-trapping phenomena in discrete-time quantum walks. By considering a Kerr-like nonlinearity, we associate an acquisition of the intensity-dependent phase with the walker while it propagates on the lattice. Adjusting the nonlinear parameter $\ensuremath{\chi}$ and the quantum gates $\ensuremath{\theta}$, we will show the existence of different quantum walking regimes, including those with traveling solitonlike structures or localized by self-trapping. The latter scenario is absent for quantum gates close enough to the Pauli-XXX gate. It appears for intermediate configurations and becomes predominant as quantum gates get closer to the Pauli-ZZZ gate. By using $\ensuremath{\chi}$ versus $\ensuremath{\theta}$ diagrams, we will show that the threshold between quantum walks with delocalized or localized regimes exhibits an unusual aspect in which an increment of the nonlinear strength can induce the system to transition from a localized to a delocalized regime.
We investigate the interplay between nonlinearity and inhomogeneities in discrete-time quantum walks on one-dimensional lattices. Nonlinear effects are introduced through a Kerr-like, intensity-dependent local phase, while spatial and temporal inhomogeneities are implemented via random variations of the quantum gate operations. By analyzing typical quantities, such as the return probability and the participation function, we identify distinct quantum walking regimes as the nonlinear parameter χχχ and the quantum gate parameter θθθ are varied. Spatial inhomogeneities weaken nonlinear self-trapping and constrict the region of robust localization. In this process, partially localized regimes emerge, characterized by the coexistence of a confined core and dispersive wave-packet components. In contrast, temporal inhomogeneities act as time-dependent perturbations that continuously disrupt the phase coherence required for self-trapped excitation, thereby enhancing dispersive emission and promoting delocalization. By using χχχ versus θθθ diagrams, we display a comprehensive characterization of how inhomogeneities modify the stability and extent of prevailing dynamical regimes, elucidating the competition between nonlinearity and inhomogeneities in discrete-time quantum walks.
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−1/2t^{-1/2}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.
The dynamics of nonlinear flip-flop quantum walk with amplitude-dependent phase shifts with pertubing potential barrier is investigated. Through the adjustment between uniform local perturbations and a Kerr-like nonlinearity of the medium we find a rich set of dynamic profiles. We will show the existence of different Hadamard quantum walking regimes, including those with mobile soliton-like structures or self-trapped states. The latter is predominant for perturbations with amplitudes that tend to $\ensuremath{\varphi}\ensuremath{\rightarrow}\ensuremath{\pi}/2$. In this system, the qubit shows an unusual behavior as we increase the amplitudes of the potential barriers and displays a monotonic decrease in the self-trapping ${\ensuremath{\varphi}}_{c}$ with respect to the nonlinear parameter. A chaotic-like regime becomes predominant for intermediate nonlinearity values. Furthermore, we show that, by changing the quantum coins ($\ensuremath{\theta}$) a nontrivial dynamic arises, where coins close to Pauli-XXX drives the system to a regime with predominant soliton-like structures, while the chaotic behavior are restricted to a narrow region in the $\ensuremath{\chi}\ensuremath{-}\ensuremath{\varphi}$ plane. We believe that is possible to implement and observe the proprieties of this model in an integrated photonic system.
The goal of research in quantum information is to investigate how quantum systems can be used to store, transmit and elaborate information and how the non-classical nature of their correlations allows defining protocols that outperform their classical counterparts. Despite of the many progresses, both theoretical and experimental, made in this field in the latest decades, many challenges lie ahead for practical implementations of quantum technologies. One of the most important ones is caused by the unavoidable interaction of quantum systems with their surroundings: The coupling to the environment is generally detrimental to the quantum information contained in the system as the system undergoes decoherence. In the quest for quantum technologies, it is fundamental to overcome the problem of decoherence and loss of information. Different physical implementations of qubits, such as superconducting and solid-state devices, are affected by the interaction with the environment in a way that can be described in terms of classical stochastic noise. The classical noise model can also be used to give an approximate, sometimes equivalent, description of full quantum models of system-environment interaction. This thesis contains my personal contribution to the study of the dynamics of discrete-variable quantum systems affected by classical noise. It covers in particular single- and two-qubit systems affected by Gaussian and non-Gaussian noise. It also discusses the dynamics of a quantum walk affected by spatially correlated classical noise. Analytical solutions for particular forms of noise and interactions, and a general numerical method for simulation of the dynamics are presented. Moreover, the thesis presents the experimental implementation of a quantum optical simulator of noisy dynamics of single-qubit systems. Finally, the use of quantum systems as probes of the spectral properties of large classical environments is discussed, showing that entanglement is a resource for improvements in the precision of the estimation.
We study how discrete-time quantum walks behave under short-range correlated noise. By considering noise as a source of inhomogeneity of quantum gates, we introduce a primitive relaxation in the assumption of uncorrelated stochastic noise: binary pair correlations manifesting in the random distribution. Using different quantum gates, we examined the transport properties for both spatial and temporal noise regimes. For spatial inhomogeneities, we unveil noise correlations driving quantum walks from the well-known exponentially localized regime to superdiffusive spreading. This scenario displays an intriguing performance in which the superdiffusive exponent is almost invariant to the degree of inhomogeneity. The time-asymptotic regime and the finite-size scaling also unveil an emergent superdiffusive behavior for quantum walks undergoing temporal noise correlation, replacing the diffusive regime exhibited when noise is random and uncorrelated. However, some quantum gates are insensitive to correlations, contrasting with the spatial noise scenario. Numerical and analytical results provide valuable insights into the underlying mechanism of superdiffusive quantum walks, including those arising from deterministic aperiodic inhomogeneities.