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arXiv:1402.2146 · Journal of Physics Conference Series

Open Quantum Walks: a short introduction

Ilya Sinayskiy, Francesco Petruccione

Published Jun 10, 2013• 29 citations
arXivHTMLPDF

Abstract

The concept of open quantum walks (OQW), quantum walks exclusively driven by the interaction with the external environment, is reviewed. OQWs are formulated as discrete completely positive maps on graphs. The basic properties of OQWs are summarised and new examples of OQWs on and their simulation by means of quantum trajectories are presented.

DOI
10.1088/1742-6596/442/1/012003

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The concept of open quantum walks (OQW), quantum walks exclusively driven by the interaction with the external environment, is reviewed. OQWs are formulated as discrete completely positive maps on graphs. The basic properties of OQWs are summarised and new examples of OQWs on and their simulation by means of quantum trajectories are presented.
A model of non-reversal quantum walk is introduced. The process is introduced in 1D and 2D using the formalism of open quantum walks (OQWs). In such a walk, a particle cannot go back to previously visited sites but it can jump on the same site or move to a new site. Examples of some non-reversal quantum trajectories and distributions are given and their difference with normal OQWs is discussed. Finally, a linear relationship is formulated between the radius of the spread and the number of steps of the walk that might have interesting applications.
Open quantum walks (OQWs) describe a quantum walker on an underlying graph whose dynamics is purely driven by dissipation and decoherence. Mathematically, they are formulated as completely positive trace preserving (CPTP) maps on the space of density matrices for the walker on the graph. Any microscopically derived OQW must include the possibility of remaining on the same site on the graph when the map is applied. We extend the CPTP map to describe a lazy OQW. We derive a central limit theorem for lazy OQWs on a ddd-dimensional lattice, where the distribution converges to a Gaussian. We show that the properties of this Gaussian computed using conventional methods agree with the general formulas derived from our central limit theorem.
Open Quantum Walks (OQW) are a type of quantum walk governed by the system's interaction with its environment. We explore the time evolution and the limit behavior of the OQW framework for Quantum Computation and show how we can represent random unitary quantum channels, such as the dephasing and depolarizing channels, in this model. We also develop a simulation protocol with circuit representation for this model, which is heavily inspired by the fact that graphs represent OQW and are, thereby, local (meaning that the state in a particular node interacts only with its neighborhood). We obtain asymptotic advantages in the system's dimension, circuit depth, and CNOT count compared to other simulation methods.
Open quantum walks (OQWs) are a new type of quantum walks which are entirely driven by the dissipative interaction with external environments and are formulated as completely positive trace-preserving maps on graphs. A generalized quantum optical scheme for implementing OQWs that includes non-zero temperature of the environment is suggested. In the proposed quantum optical scheme, a two-level atom plays the role of the "walker", and the Fock states of the cavity mode correspond to the lattice sites for the "walker". Using the small unitary rotations approach the effective dynamics of the system is shown to be an OQW. For the chosen set of parameters, an increase in the temperature of the environment causes the system to reach the asymptotic distribution much faster compared to the scheme proposed earlier where the temperature of the environment is zero. For this case the asymptotic distribution is given by a steady Gaussian distribution.
Open quantum walks (OQW) are formulated as quantum Markov chains on graphs. It is shown that OQWs are a very useful tool for the formulation of dissipative quantum computing algorithms and for dissipative quantum state preparation. In particular, single qubit gates and the CNOT-gate are implemented as OQWs on fully connected graphs. Also, dissipative quantum state preparation of arbitrary single qubit states and of all two-qubit Bell-states is demonstrated. Finally, the discrete time version of dissipative quantum computing is shown to be more efficient if formulated in the language of OQWs.
Open quantum systems interact with their environment, leading to nonunitary dynamics. We investigate the thermodynamics of linear Open Quantum Walks (OQWs), a class of quantum walks whose dynamics is entirely driven by the environment. We define an equilibrium temperature, identify a population inversion near a finite critical value of a control parameter, analyze the thermalization process, and develop the statistical mechanics needed to describe the thermodynamical properties of linear OQWs. We also study nonequilibrium thermodynamics by analyzing the time evolution of entropy, energy, and temperature, while providing analytical tools to understand the system's evolution as it converges to the thermalized state. We examine the validity of the second and third laws of thermodynamics in this setting. Finally, we employ these developments to shed light on dissipative quantum computation within the OQW framework.