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Research in Optical Sciences

Open Quantum Walks as a Tool for Dissipative Quantum Computing

Francesco Petruccione, Ilya Sinayskiy

Published Jan 1, 2012• 0 citations
Publisher

Abstract

Recently, open quantum walks (OQW) have been 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.
DOI
10.1364/qim.2012.qw4a.1

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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 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 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.
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.