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.
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 discrete-time quantum walks with Fourier, Grover, and Hadamard coin operators on disordered square lattices, alongside classical random walks. Edges in the lattice are absent with a probability,
p. In this case, both classical and quantum walks exhibit an essentially identical degradation of transport. We then study a general version of this model, in which the existing edges are retained. However, the absent edges are no longer absent, but now become partially transmitting, with a transmission coefficient
f. We find that the response is now quite sensitive to the walk protocol and coin symmetry. The classical walk exhibits asymptotic stability of the anomalous diffusion exponent,
β(t). In comparison, quantum walks exhibit a monotonically decaying
β(t) beyond the initial transient regime, leading to the absence of quantum diffusion in two dimensions. We map the transport regimes in the
(f,p) plane and identify the contour for
β=1, which separates the subdiffusive and superdiffusive regimes. We also examine the effective diffusion coefficient, the inverse participation ratio, and the first two moments of edge transmission. We restrict ourselves to analyzing bulk propagation prior to finite-size boundary reflections. The results indicate that quantum transport depends significantly on coin-specific interference and cannot be fully described by transmission statistics alone.
Abstract We study a 2D disordered discrete-time quantum walk (QW) based on 1D generalized elephant QW where an entangling coin operator is assumed. We show that considering a given disorder in one direction, it is possible to control the degree of spreading and entanglement in the other direction. This observation helps assert that the random QWs of this ilk serve as a controllable decoherence channel with the degree of randomness being the tunable parameter and highlight the role of dimensionality in quantum systems regarding information and transport.
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.
Quantum Walk (QW) has very different transport properties to its classical counterpart due to interference effects. Here we study the discrete-time quantum walk (DTQW) with on-site static/dynamic phase disorder following either binary or uniform distribution in both one and two dimensions. For one dimension, we consider the Hadamard coin; for two dimensions, we consider either a 2-level Hadamard coin (Hadamard walk) or a 4-level Grover coin (Grover walk) for the rotation in coin-space. We study the transport properties e.g. inverse participation ratio (IPR) and the standard deviation of the density function (σ) as well as the coin-position entanglement entropy (EE), due to the two types of phase disorders and the two types of coins. Our numerical simulations show that the dimensionality, the type of coins, and whether the disorder is static or dynamic play a pivotal role and lead to interesting behaviors of the DTQW. The distribution of the phase disorder has very minor effects on the quantum walk.
The dynamical and static disordered quantum walks were extensively studied recently. It is found that, for the dynamical disorder case, the transport behavior of particles is diffusive, and for the static disorder case the transport behavior is localized. In this work, we study the effect of quantum coins on the transport behaviors of the binary disordered quantum walks. We find that once the coins satisfy certain conditions, the sub-ballistic spreading could be found in binary dynamical disorder quantum walks, and the sub-ballistic, diffusive and sub-diffusive spreadings could be found in binary static disorder quantum walks. We obtain the necessary conditions for those abnormal diffusive behaviors.