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arXiv:2609.03272

Discrete-time quantum walks with energy-dependent coins

Alana Spak dos Santos, Thiago T. Tsutsui, Fabiano M. Andrade

quant-phmath-phSubmitted Sep 3, 2026• Announced Sep 4, 2026• 0 citations
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Abstract

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.

Comments
11 pages, 9 figures, comments are welcome

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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 (δδδ, δ′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.
One of the unique features of discrete-time quantum walks is called trapping, meaning the inability of the quantum walker to completely escape from its initial position, although the system is translationally invariant. The effect is dependent on the dimension and the explicit form of the local coin. A four-state discrete-time quantum walk on a square lattice is defined by its unitary coin operator, acting on the four-dimensional coin Hilbert space. The well-known example of the Grover coin leads to a partial trapping, i.e., there exists some escaping initial state for which the probability of staying at the initial position vanishes. On the other hand, some other coins are known to exhibit strong trapping, where such an escaping state does not exist. We present a systematic study of coins leading to trapping, explicitly construct all such coins for discrete-time quantum walks on the two-dimensional square lattice, and classify them according to the structure of the operator and the manifestation of the trapping effect. We distinguish three types of trapping coins exhibiting distinct dynamical properties, as exemplified by the existence or nonexistence of the escaping state and the area covered by the spreading wave packet.
We study the localization properties, energy spectra, and coin-position entanglement of the aperiodic discrete-time quantum walks. The aperiodicity is described by spatially dependent quantum coins distributed on the lattice, whose distribution is neither periodic (Bloch-like) nor random (Anderson-like). Within transport properties we identified delocalized and localized quantum walks mediated by a proper adjusting of aperiodic parameter. Both scenarios are studied by exploring typical quantities (inverse participation ratio, survival probability, and wave packet width), as well as the energy spectra of an associated effective Hamiltonian. By using the energy spectra analysis, we show that the early stage the inhomogeneity leads to a vanishing gap between two main bands, which justifies the predominantly delocalized character observed for ν<0.5. With increase of ν arise gaps and flat bands on the energy spectra, which corroborates the suppression of transport detected for ν>0.5. For ν high enough, we observe an energy spectra, which resembles that described by the one-dimensional Anderson model. Within coin-position entanglement, we show many settings in which an enhancement in the ability to entangle is observed. This behavior brings new information about the role played by aperiodicity on the coin-position entanglement for static inhomogeneous systems, reported before as almost always reducing the entanglement when comparing with the homogeneous case. We extend the analysis in order to show that systems with static inhomogeneity are able to exhibit asymptotic limit of entanglement.
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Properties of one-dimensional discrete-time quantum walks (DTQWs) are sensitive to the presence of inhomogeneities in the substrate, which can be generated by defining position-dependent coin operators. Deterministic aperiodic sequences of two or more symbols provide ideal environments where these properties can be explored in a controlled way. Based on an exhaustive numerical study, this work discusses a two-coin model resulting from the construction rules that lead to the usual fractal Cantor set. Although the fraction of the less frequent coin [Formula: see text] as the size of the chain is increased, it leaves peculiar properties in the walker dynamics. They are characterized by the wave function, from which results for the probability distribution and its variance, as well as the entanglement entropy, were obtained. A number of results for different choices of the two coins are presented. The entanglement entropy has shown to be very sensitive to uncovering subtle quantum effects present in the model.
We investigate the role of different aperiodic sequences in the dynamics of single quantum particles in discrete space and time. For this we consider three aperiodic sequences, namely, the Fibonacci, Thue-Morse, and Rudin-Shapiro sequences, as examples of tilings the diffraction spectra of which have pure point, singular continuous, and absolutely continuous support, respectively. Our interest is to understand how the order, intrinsically introduced by the deterministic rule used to generate the aperiodic sequences, is reflected in the dynamical properties of the quantum system. For this system we consider a single particle undergoing a discrete-time quantum walk (DTQW), where the aperiodic sequences are used to distribute the coin operations at different lattice positions (inhomogeneous DTQW) or by applying the same coin operation at all lattice sites at a given time but choosing different coin operation at each time step according to the chosen aperiodic sequence (time dependent DTQW). We study the energy spectra and the spreading of an initially localized wave packet for different cases, finding that in the case of Fibonacci and Thue-Morse tilings the system is superdiffusive, whereas in the Rudin-Shapiro case it is strongly subdiffusive. Trying to understand this behavior in terms of the energy spectra, we look at the survival amplitude as a function of time. By means of the echo we present strong evidence that, although the three orderings are very different as evidenced by their diffraction spectra, the energy spectra are all singular continuous except for the inhomogeneous DTQW with the Rudin-Shapiro sequence where it is discrete. This is in agreement with the observed strong localization both in real space and in the Hilbert space. Our paper is particularly interesting because quantum walks can be engineered in laboratories by means of ultracold gases or in optical waveguides, and therefore would be a perfect playground to study singular continuous energy spectra in a completely controlled quantum setup.
We investigate a discrete-time quantum walk of a three-component quantum particle on a one-dimensional lattice. As coin operators, we employ parameterized rotations generated by the Gell-Mann matrices, which enable systematic tuning of the couplings between the internal components. We analyze how these couplings influence and control the dynamics by systematically exploring the position-space probability distribution across a broad region of the parameter space. To quantify the impact of different inter-component couplings, we further examine the ratio of the mean position to the variance in each half of the lattice. Our results indicates that the system supports a rich variety of transport regimes, ranging from nearly symmetric, rapidly spreading walks to strongly anisotropic dynamics with partial localization. This framework thus provides a new avenue for engineering targeted spreading and trapping behavior in multicomponent discrete-time quantum walks.