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.
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.
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.
Disorder in coined quantum walks generally leads to localization. We investigate the influence of the localization on the entanglement properties of coined quantum walks. Specifically, we consider quantum walks on the line and explore the effects of quenched disorder in the coin operations. After confirming that our choice of disorder localizes the walker, we study how the localization affects the properties of the coined quantum walk. We find that the mixing properties of the walk are altered nontrivially with mixing being improved at short time scales. Special focus is given to the influence of coin disorder on the properties of the quantum state and the coin-walker entanglement. We find that disorder alters the quantum state significantly even when the walker probability distribution is still close to the nondisordered case. We observe that, generically, coin disorder decreases the coin-walker entanglement and that the localization leaves distinct traces in the entanglement entropy and the entanglement negativity of the coined quantum walk.
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.
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.
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.
Abstract In this study, we investigate the emergence of the quantum boomerang effect in discrete-time quantum walks (DTQWs) subjected to random phase disorder. Our analysis shows that this effect can arise solely from the bias induced by the coin degree of freedom of the DTQW, without requiring external bias or asymmetry. We explore the evolution of the mean position of the quantum walker, denoted as X(t) X ¯ ( t ) , under various initial conditions of the walker and quantum coin operators. The results indicate a significant dependence of the observed phenomena on the choice of initial state, enabling the selective induction of the quantum boomerang effect in both or only one portion of the wavepacket associated with specific internal states. By varying the quantum coin parameter θ θ , we find that the maximum mean position follows a power-law decay near the Pauli-Z coin, characterized by XMax∼θ−2 X Max ∼ θ - 2 . Additionally, we identify a scaling behavior XMax∼W−2 X Max ∼ W - 2 , which is consistent with the localization length observed in disordered quantum systems.