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Interdisciplinary Information Sciences

How to Realize One-dimensional Discrete-time Quantum Walk by Dirac Particle

Yutaka Shikano

Published Jan 1, 2017• 0 citations
Publisher

Abstract

One-dimensional discrete-time quantum walks (DTQWs) can simulate various quantum and classical dynamics and have already been implemented in several physical systems. This implementation needs a well-controlled quantum dynamical system, which is the same requirement for implementing quantum information processing tasks. Here, we consider how to realize DTQWs by Dirac particles toward a solid-state implementation of DTQWs.

DOI
10.4036/iis.2017.a.05

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One-dimensional discrete-time quantum walks (DTQWs) can simulate various quantum and classical dynamics and have already been implemented in several physical systems. This implementation needs a well-controlled quantum dynamical system, which is the same requirement for implementing quantum information processing tasks. Here, we consider how to realize DTQWs by Dirac particles toward a solid-state implementation of DTQWs.
Quantum walks have wide applications in quantum information, such as universal quantum computation, so it is important to explore properties of quantum walks thoroughly. We propose a novel method to implement discrete-time quantum walks (DTQWs) using only a single qubit, in which both coin and walker are encoded in the two-dimensional state space of a single qubit, operations are realized using single-qubit gates only, and high-dimensional final states of DTQWs can be obtained naturally. With this "one-qubit" approach, DTQW experiments can be realized much more easily, compared with previous methods, in most quantum systems and all properties based on quantum states of DTQWs (such as quantum correlation and coherence) can be investigated. By this approach, we experimentally implement one-particle and two-particle DTQWs with seven steps using single photons. Furthermore, we systematically investigate quantum correlations and coherence (based on the full state of the coin and walker) of the DTQW systems with different initial states of the coin, which have not been obtained and studied in DTQW experiments. As an application, we also study the assisted distillation of quantum coherence using the full state of the two-particle DTQW from the experiment. The maximal increase in distillable coherence for high-dimensional mixed states is investigated for the first time by obtaining its upper and lower bounds. Our work opens a new door to implement DTQW experiments and to better explore properties of quantum walks.
Quantum walks are a promising framework for developing quantum algorithms and quantum simulations. They represent an important test case for the application of quantum computers. Here we present different forms of discrete-time quantum walks (DTQWs) and show their equivalence for physical realizations. Using an appropriate digital mapping of the position space on which a walker evolves to the multiqubit states of a quantum processor, we present different configurations of quantum circuits for the implementation of DTQWs in one-dimensional position space. We provide example circuits for a five-qubit processor and address scalability to higher dimensions as well as larger quantum processors.
Discrete-time quantum walks (DTQWs) have broad applications in quantum computation and simulation, particularly in universal quantum computing. Prior research in circuit quantum electrodynamics (QED) has focused on implementing DTQWs in one-dimensional chains. However, to achieve practical quantum tasks, it is crucial to extend DTQWs to multidimensional networks with diverse structures. Here, we propose a protocol for realizing DTQWs in multidimensional networks, which accommodate nodes with different numbers of neighbors. Using this protocol, we numerically simulate a Grover walk search algorithm in a cubic network with circuit QED. Our simulations, which consider decoherence, align well with theoretical predictions. Our universal protocol can be adapted for various quantum systems and paves the way for DTQWs in multidimensional networks, which has potential applications in quantum computing and simulation.
The dynamics of a discrete-time quantum walk (DTQW) can be realized within a purely classical interacting particle system composed of some boxes and a large but finite number of balls, and can, in principle, be implemented in a tabletop experimental setting. The distribution of the balls evolves under stochastic, occupation-dependent update rules at each lattice site, producing quantum-walk dynamics without invoking a wavefunction. The update parameters are fixed by the parameters of coin and shift operations of the DTQW. This framework naturally yields a generalized active spin model and provides a minimal lattice-based microscopic understanding of the emergence of quantum-like dynamics in active matter systems. This interdisciplinary approach connects the classical models to the broad range of applications where DTQWs are successfully employed.
Dirac particle represents a fundamental constituent of our nature. Simulation of Dirac particle dynamics by a controllable quantum system using quantum walks will allow us to investigate the non-classical nature of dynamics in its discrete form. In this work, starting from a modified version of one-spatial dimensional general inhomogeneous split-step discrete quantum walk we derive an effective Hamiltonian which mimics a single massive Dirac particle dynamics in curved (1 + 1) space-time dimension coupled to U (1) gauge potential—which is a forward step towards the simulation of the unification of electromagnetic and gravitational forces in lower dimension and at the single particle level. Implementation of this simulation scheme in simple qubit-system has been demonstrated. We show that the same Hamiltonian can represent (2 + 1) space-time dimensional Dirac particle dynamics when one of the spatial momenta remains fixed. We also discuss how we can include U ( N ) gauge potential in our scheme, in order to capture other fundamental force effects on the Dirac particle. The emergence of curvature in the two-particle split-step quantum walk has also been investigated while the particles are interacting through their entangled coin operation.