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Quantum walk via open systems

J. P. Rodriguez

Published Jan 11, 2016• 0 citations
OpenAlex

Abstract

In this thesis, we present a study of quantum walks with an open quantum system formalism. In this way we allow the walkers to interact with a surrounding environment, and by doing so we are able to add directionality to the paths that the walkers take on a given graph. A theoretical and mathematical framework for performing a directed quantum walk is established, which works for general continuous-time quantum walks on any weighted and/or directed graphs. Since quantum correlations, such as entanglement and quantum discord, play an important role in quantum computation and quantum information processing, a detailed study is also carried out on their dynamical change while two quantum particles walk through specified graphs. Finally the electron transport process in molecules is simulated using quantum walks, in particular in the Titanium dioxide (TiO2) electrode as well as the Fenna-Matthews-Olson protein complex. Quantum correlations are measured in these two systems, showing some evidence of entanglement assisted transport of the excitons in the Fenna-Matthews-Olson protein complex, but not in the TiO2 electrode.

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Over the last two decades quantum walks have proven to be a very useful concept in quantum information processing, quantum communication and quantum simulations. The presented thesis reviews our contribution to this field of research. We provide a thorough introduction to methods used for the study of quantum walks and their potential applications. To keep the presentation intelligible we focus on specific examples. In particular, we analyze in detail the properties of homogeneous quantum walks on a one-dimensional lattice with two and three internal states. The application of the obtained results is illustrated on the examples persistence of unvisited sites and quantum transport to an absorbing sink. We also discuss quantum walks on finite graphs and their applications to quantum search and perfect state transfer. Extension of the presented results, which can be found in enclosed published papers, is discussed in detail.

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We investigate multipartite states in the Fenna-Matthews-Olson (FMO) pigment-protein complex of the green sulfur bacteria using a Lorentzian spectral density of the phonon reservoir fitted with typical parameter estimates of the species, Prosthecochloris aestuarii. The evolution of the entanglement measure of the excitonic W qubit states is evaluated in the picosecond time range, showing increased revivals in the non-Markovian regime. Similar trends are observed in the evolution dynamics of the Meyer-Wallach measure of the N-exciton multipartite state, with results showing that multipartite entanglement can last from 0.5 to 1 ps, between the bacteriochlorophylls of the FMO complex. The teleportation and quantum information splitting fidelities associated with the Greenberger-Horne-Zeilinger and W-like resource states formed by the excitonic qubit channels of the FMO complex show that revivals in fidelities increase with the degree of non-Markovian strength of the decoherent environment. Quantum information processing tasks involving teleportation followed by the decodification process involving W-like states of the FMO complex may play a critical role during coherent oscillations at physiological temperatures.

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In this thesis, we present a study of quantum walks with an open quantum system formalism. In this way we allow the walkers to interact with a surrounding environment, and by doing so we are able to add directionality to the paths that the walkers take on a given graph. A theoretical and mathematical framework for performing a directed quantum walk is established, which works for general continuous-time quantum walks on any weighted and/or directed graphs. Since quantum correlations, such as entanglement and quantum discord, play an important role in quantum computation and quantum information processing, a detailed study is also carried out on their dynamical change while two quantum particles walk through specified graphs. Finally the electron transport process in molecules is simulated using quantum walks, in particular in the Titanium dioxide (TiO2) electrode as well as the Fenna-Matthews-Olson protein complex. Quantum correlations are measured in these two systems, showing some evidence of entanglement assisted transport of the excitons in the Fenna-Matthews-Olson protein complex, but not in the TiO2 electrode.
Quantum walks are a fundamental tool in the study of quantum information, bridging the gap between quantum computation and modelling complex systems. In this thesis, we present a software package for simulation of multiparticle continuous-time quantum walks (CTQWs) using high performance computation, and detail a method of systematic dimensionalty reduction for fermionic CTQWs. We then consider CTQW applications; proving that the single particle CTQW is unsuited for determining graph isomorphism, and proposing a CTQW-based network centrality algorithm. This algorithm is utilized in the first successful physical quantum centrality implementation. Finally, we extend our centrality algorithm to PT-symmetric directed graphs.
Random walks are a powerful tool for the efficient implementation of algorithms in classical computation. Their quantum-mechanical analogues, called quantum walks, hold similar promise. Quantum walks provide a model of quantum computation that has recently been shown to be equivalent in power to the standard circuit model. As in the classical case, quantum walks take place on graphs and can undergo discrete or continuous evolution, though quantum evolution is unitary and therefore deterministic until a measurement is made. This thesis considers the usefulness of continuous-time quantum walks to quantum computation from the perspectives of both their fundamental power under various formulations, and their applicability in practical experiments. In one extant scheme, logical gates are effected by scattering processes. The results of an exhaustive search for single-qubit operations in this model are presented. It is shown that the number of distinct operations increases exponentially with the number of vertices in the scattering graph. A catalogue of all graphs on up to nine vertices that implement single-qubit unitaries at a specific set of momenta is included in an appendix. I develop a novel scheme for universal quantum computation called the discontinuous quantum walk, in which a continuous-time quantum walker takes discrete steps of evolution via perfect quantum state transfer through small `widget' graphs. The discontinuous quantum-walk scheme requires an exponentially sized graph, as do prior discrete and continuous schemes. To eliminate the inefficient vertex resource requirement, a computation scheme based on multiple discontinuous walkers is presented. In this model, n interacting walkers inhabiting a graph with 2n vertices can implement an arbitrary quantum computation on an input of length n, an exponential savings over previous universal quantum walk schemes. This is the first quantum walk scheme that allows for the application of quantum error correction. The many-particle quantum walk can be viewed as a single quantum walk undergoing perfect state transfer on a larger weighted graph, obtained via equitable partitioning. I extend this formalism to non-simple graphs. Examples of the application of equitable partitioning to the analysis of quantum walks and many-particle quantum systems are discussed.
Abstract The study of quantum walks has been shown to have a wide range of applications in areas such as artificial intelligence, the study of biological processes, and quantum transport. The quantum stochastic walk (QSW), which allows for incoherent movement of the walker, and therefore, directionality, is a generalization on the fully coherent quantum walk. While a QSW can always be described in Lindblad formalism, this does not mean that it can be microscopically derived in the standard weak-coupling limit under the Born–Markov approximation. This restricts the class of QSWs that can be experimentally realized in a simple manner. To circumvent this restriction, we introduce a technique to simulate open system evolution on a fully coherent quantum computer, using a quantum trajectories style approach. We apply this technique to a broad class of QSWs, and show that they can be simulated with minimal experimental resources. Our work opens the path towards the experimental realization of QSWs on large graphs with existing quantum technologies.
Quantum walks on a line with a single particle possess a classical analogue. Involving more walkers opens up the possibility of studying collective quantum effects, such as many-particle correlations. In this context, entangled initial states and the indistinguishability of the particles play a role. We consider the directional correlations between two particles performing a quantum walk on a line. For non-interacting particles, we find analytic asymptotic expressions and give the limits of directional correlations. We show that by introducing δ-interaction between the particles, one can exceed the limits for non-interacting particles.
Over the last two decades quantum walks have proven to be a very useful concept in quantum information processing, quantum communication and quantum simulations. The presented thesis reviews our contribution to this field of research. We provide a thorough introduction to methods used for the study of quantum walks and their potential applications. To keep the presentation intelligible we focus on specific examples. In particular, we analyze in detail the properties of homogeneous quantum walks on a one-dimensional lattice with two and three internal states. The application of the obtained results is illustrated on the examples persistence of unvisited sites and quantum transport to an absorbing sink. We also discuss quantum walks on finite graphs and their applications to quantum search and perfect state transfer. Extension of the presented results, which can be found in enclosed published papers, is discussed in detail.
We investigate multipartite states in the Fenna-Matthews-Olson (FMO) pigment-protein complex of the green sulfur bacteria using a Lorentzian spectral density of the phonon reservoir fitted with typical parameter estimates of the species, Prosthecochloris aestuarii. The evolution of the entanglement measure of the excitonic W qubit states is evaluated in the picosecond time range, showing increased revivals in the non-Markovian regime. Similar trends are observed in the evolution dynamics of the Meyer-Wallach measure of the N-exciton multipartite state, with results showing that multipartite entanglement can last from 0.5 to 1 ps, between the bacteriochlorophylls of the FMO complex. The teleportation and quantum information splitting fidelities associated with the Greenberger-Horne-Zeilinger and W-like resource states formed by the excitonic qubit channels of the FMO complex show that revivals in fidelities increase with the degree of non-Markovian strength of the decoherent environment. Quantum information processing tasks involving teleportation followed by the decodification process involving W-like states of the FMO complex may play a critical role during coherent oscillations at physiological temperatures.