We prove explicit upper bounds on the propagation velocity of one-dimensional quantum walks with periodic coins of arbitrary period. We treat two complementary settings. First, in a perturbative regime where one transmission parameter is small, we show that the corresponding almost reflecting coin acts as a bottleneck for transport: the velocity is bounded linearly in this parameter, with an explicit leading order estimate. Second, for arbitrary nonzero transmission parameters, we prove a general a priori bound in terms of their harmonic mean, together with a refined version that detects the spatial variation of neighboring coins. Moreover, we prove a general lower bound on the velocity. These bounds apply directly to the corresponding CMV setting.
We prove general sufficient conditions for zero velocity in position dependent one-dimensional quantum walks, and hence for the absence of ballistic transport. Our starting point is a general a priori upper bound on the velocity, formulated in terms of sparse bi-infinite sequences of sites, their gap structure, and the corresponding local coin parameters. This estimate yields several deterministic criteria for zero velocity that depend only on the behavior of suitable coin entries along selected subsequences and are independent of the values of the coins elsewhere. We also discuss the random case as an application of the general approach. All of our results remain valid in the CMV setting.
We establish exact scalings for the maximal group velocity of translation-invariant quantum walks in periodic electric fields. Our main result shows that the maximal group velocity decays exponentially with the period of the field in the whole parameter range, thus affirming a conjecture of arXiv:2302.01869 and at the same time augmenting it to an exact equality. We further demonstrate explicit revival relations and characterize the absolutely continuous spectrum in these models. Our results apply directly also to generalized CMV matrices.
We study the dynamics of discrete-time quantum walk using quantum coin operations, $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{C}({\ensuremath{\theta}}_{1})$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{C}({\ensuremath{\theta}}_{2})$, in time-dependent periodic sequence. For the two-period quantum walk with the parameters ${\ensuremath{\theta}}_{1}$ and ${\ensuremath{\theta}}_{2}$ in the coin operations we show that the standard deviation $[{\ensuremath{\sigma}}_{{\ensuremath{\theta}}_{1},{\ensuremath{\theta}}_{2}}(t)]$ is the same as the minimum of standard deviation obtained from one of the one-period quantum walks with coin operations ${\ensuremath{\theta}}_{1}$ or ${\ensuremath{\theta}}_{2}$, ${\ensuremath{\sigma}}_{{\ensuremath{\theta}}_{1},{\ensuremath{\theta}}_{2}}(t)=\mathrm{min}{{\ensuremath{\sigma}}_{{\ensuremath{\theta}}_{1}}(t),{\ensuremath{\sigma}}_{{\ensuremath{\theta}}_{2}}(t)}$. Our numerical result is analytically corroborated using the dispersion relation obtained from the continuum limit of the dynamics. Using the dispersion relation for one- and two-period quantum walks, we present the bounds on the dynamics of three- and higher-period quantum walks. We also show that the bounds for the two-period quantum walk will hold good for the split-step quantum walk which is also defined using two coin operators using ${\ensuremath{\theta}}_{1}$ and ${\ensuremath{\theta}}_{2}$. Unlike the previous known connection of discrete-time quantum walks with the massless Dirac equation where coin parameter $\ensuremath{\theta}=0$, here we show the recovery of the massless Dirac equation with nonzero $\ensuremath{\theta}$ parameters contributing to the intriguing interference in the dynamics in a totally nonrelativistic situation. We also present the effect of periodic sequence on the entanglement between coin and position space.
In this work, we extend the scattering quantum walk (SQW) framework to a lattice of energy-dependent point interactions. This yields, within the coined quantum walk (CQW) formalism, a coin operator that is directly related to the scattering matrix of zero-range potentials. The model thus provides a discrete-time quantum-walk (DTQW) analog of a periodic array of point interactions of the Kronig-Penney type, where the walker's wavenumber serves as a continuous, physically transparent control parameter for the coin operation. We analyze the spectra and the dynamics of position probability and entanglement, yielding distinct results for specific energies and point interactions. We relate the spectral structure to the spatial probability distribution and explicitly characterize the long-time entanglement behavior for each point interaction. The transmission modulus determines the quasienergy gap, bandwidth, and maximum group velocity, while also controlling the long-time coin-position entanglement for the initial state considered. The four families of one-dimensional point interactions (
δ,
δ′, crossed and asymmetric) realize qualitatively distinct transmission profiles and span the full range of behavior, including enhanced or strongly suppressed spreading and oscillatory entanglement.
We study the dynamics of discrete-time quantum walk using quantum coin operations,
C^(θ1) and
C^(θ2) in time dependent periodic sequence. For two-period quantum walk with the parameters
θ1 and
θ2 in the coin operations we show that the standard deviation (
σθ1,θ2(t)) is same as the minimum of standard deviation obtained from one of the one-period quantum walk with coin operations
θ1 or
θ2,
σθ1,θ2(t)=min{σθ1(t),σθ2(t)}. Our numerical result is analytically corroborated using the dispersion relation obtained from the continuum limit of the dynamics. Using the dispersion relation for one- and two-period quantum walk, we present the bounds on the dynamics of three- and higher period quantum walks. We also show that the bounds for the two-period quantum walk will hold good for the split-step quantum walk which is also defined using two coin operators using
θ1 and
θ2. Unlike the previous known connection of discrete-time quantum walks with the massless Dirac equation where coin parameter
θ=0, here we show the recovery of massless Dirac equation with non-zero
θ parameters contributing to the intriguing interference in the dynamics in a totally non-relativistic situation. We also present the effect of periodic sequence on the entanglement between coin and position space.
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.