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arXiv:1812.05158 · Europhysics Letters (EPL)

Mimicking disorder on a clean graph: Interference-induced inhibition of spread in a cyclic quantum random walk

Jayanth Jayakumar, Sreetama Das, Aditi Sen, Ujjwal Sen

Published Jan 3, 2020• 3 citations
arXivHTMLPDF

Abstract

We quantitatively differentiate between the spreads of discrete-time quantum and classical random walks on a cyclic graph. Due to the closed nature of any cyclic graph, there is additional “collision”-like interference in the quantum random walk along with the usual interference in any such walk on any graph, closed or otherwise. We find that the quantum walker exhibits inhibition of spread in comparison to the classical one, even in the absence of disorder, a phenomenon that is potentially attributable to the additional interference in the quantum case. This is to be contrasted with the situation on open graphs, where the quantum walker, being effectively denied the collision-like interference, garners a much higher spread than its classical counterpart. Inhibition of spread also occurs on open graphs, but with insertion of disorder. We use the Shannon entropy of the position probability distribution to quantify the spread of the walker in both quantum and classical cases. We find that for a given number of vertices on a cyclic graph, the entropy with respect to number of steps for the quantum walker saturates, on average, to a value lower than that for the corresponding classical one. We also analyze variations of the entropies with respect to system size, and look at the corresponding asymptotic growth rates.

DOI
10.1209/0295-5075/128/20007

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We explore static noise in a discrete quantum random walk over a homogeneous cyclic graph, focusing on the spectral and dynamical properties of the system. Using a three-parameter unitary coin, we control the spectral structure of the noiseless step operator on the unit circle in the complex plane. One parameter governs the probability amplitudes and induces two spectral bandgaps proportional to its value. The half-sum of the two phase parameters rotates the spectrum and induces twofold degeneracy under specific conditions. Degenerate spectra yield eigenstates with sinusoidal probability distributions, while non-degenerate spectra produce flat distributions. We introduce static phase noise in the sites and analyze its impact across two distinct propagation regimes. In the walk-on-the-line regime, which precedes a full traversal of the graph, we extract the spreading exponent β from the step-resolved mean squared displacement. We find that low participation ratios correlate with sub-diffusive spread, while high ratios correspond to ballistic or super-diffusive evolution. After the walker completes a cycle, finite-size effects dominate. In this walk-on-the-cycle regime, the spreading exponent no longer characterizes the behavior. We quantify localization using a convergence criterion based on the coefficient of variation of the mean squared displacement. Across both regimes, we identify a sharp crossover near static site noise strength ϕs = π/3. This transition coincides with a drop in the participation ratio, a transition from diffusive to sub-diffusive spread in the walk-on-the-line regime, and a reduction in the saturation level of the mean squared displacement in the walk-on-the-cycle regime. Our results demonstrate that the eigenstate participation ratio provides a computationally efficient spectral diagnostic that anticipates localization across both regimes, offering an alternative to full dynamical simulations.
We study the absorption time and spreading rate of the discrete-time quantum walk propagating on a line in the presence or absence of an absorber. We analytically establish that in the presence of an absorber, the average absorption time of the quantum walker is finite, contrary to the behavior of a classical random walker, indicating an infinite resource reduction on moving over to a quantum version of a walker. Furthermore, numerical simulations indicate a reversal of this behavior due to the insertion of disorder in the walker's step lengths. Additionally, we demonstrate that in the presence of an absorber, there is a speed-up in the spreading rate, and that a disordered quantum walk that is sub-ballistic regains the ballistic spreading of a clean quantum walk.
The phenomenon of localization usually happens due to the existence of disorder in a medium. Nevertheless, certain quantum systems allow dynamical localization solely due to the nature of internal interactions. We study a discrete time quantum walker which exhibits disorder-free localization. The quantum walker moves on a one-dimensional lattice and interacts with on-site spins by coherently rotating them around a given axis at each step. Since the spins do not have dynamics of their own, the system poses the local spin components along the rotation axis as an extensive number of conserved moments. When the interaction is weak, the spread of the walker shows subdiffusive behavior having downscaled ballistic tails in the evolving probability distribution at intermediate timescales. However, as the interaction gets stronger the walker gets completely localized in total absence of disorder in both lattice and initial state. Using a matrix-product-state ansatz, we investigate the relaxation and entanglement dynamics of the on-site spins due to their coupling with the quantum walker. Surprisingly, we find that, even in the delocalized regime, entanglement growth and relaxation occur slowly, unlike majority of the other models displaying a localization transition.
We consider a quantum particle (walker) on a line that coherently chooses to jump to the left or right depending on the result of a toss of a quantum coin. The lengths of the jumps are considered to be independent and identically distributed quenched Poisson random variables. We find that the spread of the walker is significantly inhibited, whereby it resides in the near-origin region, with respect to the case when there is no disorder. The scaling exponent of the quenched-averaged dispersion of the walker is subballistic but superdiffusive. We also show that the features are universal to a class of sub- and super-Poissonian-distributed quenched randomized jumps.
Abstract Quantum walks are known to propagate quadratically faster than their classical counterparts and are used to model dynamics in various quantum systems. The spread of the quantum walk in position space shows anomalous diffusion behavior. By controlling the action of quantum coin operation on the corresponding coin degree of freedom of the walker, one can demonstrate control over the diffusion behavior. In this work, we report different forms of coin operations on quantum walks exhibiting anomalous diffusion behavior. Homogeneous and accelerated quantum walks display superdiffusive behavior, whereas uncorrelated static and dynamic disorders in the evolution induce strong and weak localization of the particle indicating subdiffusive and normal diffusive behavior. The role played by the interference effects in the spreading of the walker has remained elusive and our aim in this work is to present the interplay between quantum coherence and mean squared displacement of the walker. We employ two reliable measures of coherence for conclusively establishing the role of quantum interference as the driving force behind the anomalous diffusive behavior in the dynamics of quantum walks.
We study the response of spreading behavior, of two-dimensional discrete-time quantum walks, to glassy disorder in the jump length. We consider different discrete probability distributions to mimic the disorder, and three types of coin operators, viz., Grover, Fourier, and Hadamard, to analyze the scale exponent of the disorder-averaged spreading. We find that the ballistic spreading of the clean walk is inhibited in presence of disorder, and the walk becomes sub-ballistic but remains super-diffusive. The resilience to disorder-induced inhibition is stronger in two-dimensional walks, for all the considered coin operations, in comparison to the same in one dimension. The quantum advantage of quantum walks is therefore more secure in two dimensions than in one.