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New Journal of Physics

Controlling probability transfer in the discrete-time quantum walk by modulating the symmetries

Tian Chen, Bo Wang, Xiangdong Zhang

Published Sep 29, 2017• 14 citations
Publisher

Abstract

Quantum walks are a useful platform for studying high-efficiency energy transport. In this paper, we study the probability transfer in the discrete-time quantum walk (DTQW) in detail. With intricate design of the initial state, the probability transfer in the DTQW system can exhibit similar behaviors to those in continuous-time quantum walks (CTQWs). Furthermore, without the assumption of a biased initial state and coupling to the environment in the DTQW system, we explore how to achieve directed probability transfer in the DTQW and find that it is closely connected to the time-reversal symmetry of the effective Hamiltonian of the walk. We find that regardless of whether the number of vertices in the DTQW is even or odd, we can control the probability transfer in the DTQW by modulating the time-reversal symmetry of the system. Such control of directed probability transfer can only work well in a CTQW with odd vertices. Even when temporal or spatial disorder emerges in the DTQW, the connection between the probability transfer and the symmetry of the walk still exists. Experimental proposals for observing the probability distributions in the DTQW are discussed.

DOI
10.1088/1367-2630/aa8fe4

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Quantum walks are a useful platform for studying high-efficiency energy transport. In this paper, we study the probability transfer in the discrete-time quantum walk (DTQW) in detail. With intricate design of the initial state, the probability transfer in the DTQW system can exhibit similar behaviors to those in continuous-time quantum walks (CTQWs). Furthermore, without the assumption of a biased initial state and coupling to the environment in the DTQW system, we explore how to achieve directed probability transfer in the DTQW and find that it is closely connected to the time-reversal symmetry of the effective Hamiltonian of the walk. We find that regardless of whether the number of vertices in the DTQW is even or odd, we can control the probability transfer in the DTQW by modulating the time-reversal symmetry of the system. Such control of directed probability transfer can only work well in a CTQW with odd vertices. Even when temporal or spatial disorder emerges in the DTQW, the connection between the probability transfer and the symmetry of the walk still exists. Experimental proposals for observing the probability distributions in the DTQW are discussed.
Discrete-time quantum walks are among the branches of quantum information and computation. They are platforms for developing quantum algorithms for quantum computers. In addition, due to their universal primitive nature, discrete-time quantum walks have been used to simulate other quantum systems and phenomena that are observed in physics and chemistry. To fully utilize the potentials that the discrete-time quantum walks hold in their applications, control over the discrete-time quantum walks and their properties becomes essential. In this dissertation, we propose two models for attaining a high level of control over the discrete-time quantum walks. In the first one, we incorporate a dynamical nature for the unitary operator performing the quantum walks. This enables us to readily control the properties of the walker and produce diverse behaviors for it. We show that with our proposal, the important properties of the discrete-time quantum walks such as variance would indeed improve. To explore the potential of this proposal, we apply it in the simulations of topological phases in condensed matter physics. With our proposal, we can control the simulations and determine the type of topological phenomena that should be simulated. In addition, we confirm simulations of topological phases and boundary states that can be observed in one-, two- and three-dimensional systems. Finally, we report the emergence of exotic phase structures in form of cell-like structures that contain all types of topological phases and boundary states of certain classes. In our second proposal, we take advantage of resources available in quantum mechanics, namely quantum entanglement and entangled qubits. In this proposal, we use entangled qubits in the structure of a quantum walk and show that by tuning the initial entanglement between these qubits and how these qubits are modified through the walk, one is able to produce diverse behaviors for the quantum walk and control its behavior.
We show that a quantum-state transfer, previously studied as a continuous-time process in networks of interacting spins, can be achieved within the model of discrete-time quantum walks with a position-dependent coin. We argue that, due to additional degrees of freedom, discrete-time quantum walks allow one to observe effects which cannot be observed in the corresponding continuous-time case. First, we study a discrete-time version of the engineered coupling protocol due to Christandl et al. [Phys. Rev. Lett. 92, 187902 (2004)] and then we discuss the general idea of conversion between continuous-time quantum walks and discrete-time quantum walks.
Abstract Continuous-time quantum walks (CTQWs) provide a versatile framework for exploring quantum transport on graphs. In this work, we investigate how the introduction of edge-weight modulation at a single vertex can suppress its occupation probability. We show that when the edges connected to the root vertex are enhanced by a factor J , the probability of detecting the walker at this vertex decays as 1/J21/J^21/J2 1 / J 2 , provided the initial state has no components on the vertex itself or its nearest neighbors. We derive the full eigenvalue and eigenvector structure of this system, revealing that the suppression arises from the decoupling of two symmetric line subgraphs and the destructive interference of higher-order contributions. The analysis is extended to tree graphs, where we demonstrate the same scaling behavior and identify the role of local graph geometry in controlling vertex probabilities. These results suggest edge-weight modulation as a mechanism for manipulating transport pathways in CTQWs, with potential applications in quantum information transfer and state engineering, and may serve as a probe of decoherence effects in open quantum systems.
Quantum walks can be defined in two quite distinct ways: discrete-time and continuous-time quantum walks (DTQWs and CTQWs). For classical random walks, there is a natural sense in which continuous-time walks are a limit of discrete-time walks. Quantum mechanically, in the discrete-time case, an additional ``coin space'' must be appended for the walk to have nontrivial time evolution. Continuous-time quantum walks, however, have no such constraints. This means that there is no completely straightforward way to treat a CTQW as a limit of a DTQW, as can be done in the classical case. Various approaches to this problem have been taken in the past. We give a construction for walks on ddd-regular, ddd-colorable graphs when the coin flip operator is Hermitian: from a standard DTQW we construct a family of discrete-time walks with a well-defined continuous-time limit on a related graph. One can think of this limit as a ``coined'' continuous-time walk. We show that these CTQWs share some properties with coined DTQWs. In particular, we look at a spatial search by a DTQW over the two-dimensional (2D) torus (a grid with periodic boundary conditions) of size $\sqrt{N}\ifmmode\times\else\texttimes\fi{}\sqrt{N}$, where it was shown that a coined DTQW can search in time O(NlogN)O(\sqrt{N}logN)O(N​logN), but a standard CTQW takes $\mathrm{\ensuremath{\Omega}}(N)$ time to search for a marked element. The continuous limit of the DTQW search over the 2D torus exhibits the O(NlogN)O(\sqrt{N}logN)O(N​logN) scaling, like the coined walk it is derived from. We also look at the effects of graph symmetry on the limiting walk, and show that the properties are similar to those of the DTQW as shown in Krovi and Brun, Phys. Rev. A 75, 062332 (2007).
Abstract. We study quantum transport on finite discrete structures and we model the process by means of continuous-time quantum walks. A direct and effective comparison between quantum and classical walks can be attained based on the average displacement of the walker as a function of time. Indeed, a fast growth of the average displacement can be advantageously exploited to build up efficient search algorithms. By means of analytical and numerical investigations, we show that the finiteness and the inhomogeneity of the substrate jointly weaken the quantum walk performance. We further highlight the interplay between the quantum-walk dynamics and the underlying topology by studying the temporal evolution of the transfer probability distribution and the lower bound of long time averages. PACS numbers: 05.60Gg, 71.35.-y, 05.60CdDynamics of continuous-time quantum walks in restricted geometries 2 1.
Discrete-time quantum walks (DTQWs) provide a convenient platform for a realization of many topological phases in noninteracting systems. They often offer more possibilities than systems with a static Hamiltonian. Nevertheless, researchers are still looking for DTQW symmetries protecting topological phases and for definitions of appropriate topological invariants. Although the majority of DTQW studies on this topic focus on the so-called split-step quantum walk, two distinct topological phases can be observed in more basic models. Here we infer topological properties of the basic DTQWs directly from the mapping of the Brillouin zone to the Bloch Hamiltonian. We show that for translation-symmetric systems they can be characterized by a homotopy relative to special points. We also propose a topological invariant corresponding to this concept. This invariant indicates the number of edge states at the interface between two distinct phases.