The thesis presents research related to the dynamics of quantum systems, both isolated and in the presence of interactions with their environment. Generally, I employ matrix product state (MPS) techniques to explore quantum dynamics in open and closed systems. In the first part I present a study of quantum chaos and how mapping to a MPS variational manifold allow the use of techniques developed in the study of classical many-body systems. Using code developed for this project the Lyapunov spectrum is extracted to give an alternative perspective on eigenstate thermalization, pre-thermalization and integrability. In the second part, I present a novel combination of MPS methods with a Langevin description of the open system. I use this to show how coupling to the environment restricts the growth of entanglement. The consequences of this are relevant for simulations of open quantum systems and their use in in quantum technologies. Finally I discuss applications of these ideas to quantum search. I consider adiabatic and quantum walk algorithms for optimal scaling quantum search algorithms, and hybridizations between the two. The robustness of the different underlying physical mechanisms is investigated in a simple infinite-temperature model, and in a low-temperature limit using the MPS Langevin equation.
The thesis presents research related to the dynamics of quantum systems, both isolated and in the presence of interactions with their environment. Generally, I employ matrix product state (MPS) techniques to explore quantum dynamics in open and closed systems. In the first part I present a study of quantum chaos and how mapping to a MPS variational manifold allow the use of techniques developed in the study of classical many-body systems. Using code developed for this project the Lyapunov spectrum is extracted to give an alternative perspective on eigenstate thermalization, pre-thermalization and integrability. In the second part, I present a novel combination of MPS methods with a Langevin description of the open system. I use this to show how coupling to the environment restricts the growth of entanglement. The consequences of this are relevant for simulations of open quantum systems and their use in in quantum technologies. Finally I discuss applications of these ideas to quantum search. I consider adiabatic and quantum walk algorithms for optimal scaling quantum search algorithms, and hybridizations between the two. The robustness of the different underlying physical mechanisms is investigated in a simple infinite-temperature model, and in a low-temperature limit using the MPS Langevin equation.
This thesis investigates questions related with the quantification, creation and application of entanglement. We first show how the notion of entanglement is modified by the presence of superselection rules. Next, we develop protocols for the optimal generation of squeezing and entanglement from noisy operations. The second part of the thesis applies quantum information methods to quantum many-body problems. We start by investigating the relation of Hamiltonian and correlations for lattices of harmonic oscillators. Next, we turn towards the description of many-body systems in terms of Matrix Product States (MPS) and Projected Entangled Pair States (PEPS). We clarify the relation between entropy scaling laws and the approximability by MPS, and determine the computational complexity of creating and simulating PEPS. Finally, we define Gaussian MPS as a generalization of MPS to bosonic systems.
In this thesis we study problems relating the the structure and simulation of entangled many body quantum systems, their utility in adiabatic quantum computation, and the influence of the environment in thermalizing the system and degrading the usefulness of quantum dynamics in this setting. We then study a particular strongly coupled many body quantum system in order to better understand when quantum systems do not thermalize in this manner. In chapter 2 of this thesis we study the properties of quantum dynamics restricted to an efficiently representable sub-manifold of quantum states both the finite and infinite chain of spin- 1=2 subsystems. We investigate the trade-off between gains in efficiency due to this restriction against losses in fidelity. We find the integration to be very stable and shows significant gains in efficiency compared to the naively related matrix product states. However, much of this advantage is offset by a significant reduction in fidelity. We investigate the effect of explicit symmetry breaking in the ansatz and formulate the principles for determining when correlator product states may be a useful tool. We find that scaling with overlap/bond order may be more stable with correlator product states allowing a more efficient extraction of critical exponents and present an example in which the use of correlator product states is orders of magnitude quicker than matrix product states. In chapters 3, 4 and 5 we extend this picture to allow for the study of the dissipative and decohering dynamics of a quantum system interacting with a bath, and pay particular reference to its effect on adiabatic quantum computation. In chapter 3 we consider a system of mutually interacting superconducting flux qubits coupled to a thermal bath that generalises the dissipative model of Landau-Lifschitz-Gilbert to the case of anisotropic bath couplings. We show that the dissipation acts to bias the quantum trajectories towards a reduced phase space. We study the model in the context of the D-Wave computing device and recover dynamics closely related to several models proposed on phenomenological grounds. In chapter 4 we extend this analysis to study explicitly the influence of dissipative dynamics on the lifetime of entanglement. In chapter 5 we apply this understanding to develop a methodology for benchmarking the quantum correlations harnessed by an adiabatic computation and apply this process to the D-Wave Vesuvius machine. Further developing this interest in the effect of thermalisation of quantum dynamics in chapter 6 we consider systems which fail to thermalise even in the presence of strong coupling to their surroundings. This many body localised behaviour has been recently established to be a robust phase of matter in the presence of strong disorder in one dimension. Here we show the the low lying energy states of a many body system contain immobile excitations, this immobility results in an transition in the character of low lying eigenstates at arbitrarily weak disorder. This represents a novel appearance of localising behaviour in many body systems. Finally we consider possible avenues for future work stemming from this thesis.
The nonequilibrium effects of dissipation and drive play a key role in an immense variety of nanoscale and mesoscale quantum systems. To understand the behaviour of open quantum systems, we need accurate methods that capture the influence of the environment on the system, while managing the exponentially large Hilbert space required to describe the system. Tensor network algorithms offer an efficient way to approach this challenge. In this thesis, we develop and apply tensor network techniques to study the dynamics and steady states of various open quantum systems. The first part of the thesis focuses on the driven dissipative many body physics in coupled cavity arrays described by Born-Markov master equations. We extend transfer matrix product operator methods to Liouvillian dynamics, and utilize them to compute dynamical correlation functions and fluorescence spectrum of an infinite coupled cavity array in 1D. We also investigate thermalization, and observe the emergence of a quasi-thermal steady state with a negative effective temperature. In another study, we use infinite projected entangled pair state (iPEPS) methods to compute steady states of coupled cavity lattices in 2D. We find that a straightforward adaptation of iPEPS to Liouvillian dynamics is unstable, contradicting a recent publication in the field. The second part investigates more general systems involving strong couplings and structured environments that induce non-Markovian dynamics. We develop a powerful time-evolving matrix product operator (TEMPO) algorithm that builds on Feynman-Vernon influence functional formalism, and uses matrix product states (MPS) to represent the temporal non-Markovian correlations efficiently. We apply TEMPO to study the localization phase transition of the spin-boson model and the dynamics of two spatially separated two-level systems coupled to a common environment. Finally, we propose the Toblerone TEMPO algorithm, which extends TEMPO to many-body systems interacting with general bosonic environments.
The Thesis explores the application of Matrix Product States (MPS) in the domain of quantum many-body systems and quantum computing, show casing the versatility and effectiveness of MPS for addressing complex com putational challenges. The opening chapter introduces essential tools for Tensor Networks, addressing the curse of dimensionality and elucidating the main characteristics of Matrix Product States and Matrix Product Op erators. The discussion encompasses also important numerical techniques, including Density Matrix Renormalization Group and Time Evolving Block Decimation. Furthermore, the relation between MPS and quantum circuits is discussed. The subsequent Chapters delve into specific applications of MPS. In the second Chapter, we introduce a Tensor Network ansatz inspired by the backflow transformation for correlated systems. This extension of the MPS representation ensures an area law for entanglement in dimensions one or greater. We employ an optimization scheme combining DMRG and variational Monte Carlo algorithms for efficient ground-state search. Bench marking against spin models demonstrates high accuracy. The third Chapter explores quantum annealing for optimizing complex classical spin Hamil tonians, as the Hopfield model and the binary perceptron. We introduce an efficient Tensor Network representation for the adiabatic time evolution of quantum annealing, enabling scalable classical simulations. The use of MPS in mitigating Trotter errors and mapping to quantum circuits is also explored. In the fourth Chapter we present a novel method for evaluating the amount of nonstabilizerness (also known as quantum magic) contained in an MPS. We overcome the exponentially hard evaluation of this quan tity by employing a simple perfect sampling technique of Pauli string. Our innovative MPS approach enables efficient computation. Benchmarked on magic states and the quantum Ising chain ground state, this method offers easy access to the non-equilibrium dynamics of nonstabilizerness following a quantum quench. The Appendices provide additional insights, including: an introduction to variational Monte Carlo, technical details on MPS simula tions for the quantum annealing and a comprehensive overview on stabilizer formalism and nonstabilizerness.
Open quantum systems have drawn attention over decades due to its applicability in the foundation of theoretical physics, e.g. statistical mechanics, quantum mechanics and condensed matter physics. The dynamics of open quantum systems has been described as separate entities from their surrounding environment that consists of a very large number of modes, somehow coupled to the mode of the system. Even though the exact solution of the dynamical behavior of the system is impossible to calculate, we obtain a tentative solution using the crucial Markov approximation. The input-output formalism of the quantum Langevin equation (QLE) has been considered as a useful tool which provides a semi-classical description of the dynamics of the system, whereas the master equation provides a complete picture of the dynamics of the system expressed in terms of density matrix. While studying the dynamics of nonlinear system/environment coupling using QLE, we see, for a small value of external field, that the steady state system field does not change much from the steady state field obtained in the absence of nonlinear dissipation. However, in a case where a stronger external field is applied, we see that the deviation becomes substantial from the solution of linear system. We also see that the nonlinear coupling introduces significant difference in the cavity fluctuation spectrum. The description, therefore, provides a potential explanation of parametric effects in terms of nonlinear dissipation phenomena associated with the nonlinear coupling. Even though the theories developed in the context of open quantum systems have proven to be powerful tools, they do not provide a satisfactory platform to be implemented on non-linear Hamiltonians. We often approximate it by linearizing over nonlinear steady state field amplitude, and therefore, the interesting effects are often overlooked. The limitation of the analytics provokes us to simulate open quantum dynamics numerically. The numerical method consists of transformation of the environmental degrees of freedom to a one-dimensional many-body chain, and the computational technique includes numerical diagonalization and renormalization process. The time-adaptive density matrix renormalisation group (t-DMRG) is known as one of the most powerful techniques for the simulation of strongly-correlated many-body quantum systems. In this thesis, along with the theoretical modeling, we implement DMRG numerical scheme for the simulation of canonical S/B model by mapping it to one-dimensional harmonic chain with nearest neighbor interactions, and use the method to investigate the dynamics of the free dissipative system. The thermalization of open quantum systems is also studied by generating minimally entangled typical thermal states (METTS) through imaginary time evolution, and real-time evolving an empty system in the presence of the thermal bath. Further, we simulate coherently driven free dissipative Kerr nonlinear system numerically using Euler's method by solving Heisenberg equation of motion and t-DMRG algorithm, and demonstrate how the numerical results are analogous to classical bistability. By comparing with analytics, we see that the DMRG numerics is analogous to the quantum-mechanical exact solution obtained by mapping the equation of motion of the density matrix of the system to a Fokker-Plank equation. The comparison between two different numerical techniques shows that the semi-classical Euler's method determines the dynamics of the system field of one among two coherent branches, whereas DMRG numerics gives the superposition of both of them. Hence, DMRG-determined time dynamics undergoes generating non-classical states. Our approach of dealing with nonlinearity represents an important contribution in the developments of technique to study the dynamical and steady-state behavior of open quantum systems, which is a fundamental aspect of quantum physics.
Experimental advances have made it possible to realize and control quantum many-body systems, allowing the experimental study of non-equilibrium phenomena. Due to the Hilbert space growing exponentially with the number of particles, advanced numerical techniques are usually required to understand many-body effects at the theoretical level. In this thesis, we address the question of how to efficiently simulate the non-equilibrium dynamics of many-body systems using matrix product states (MPS). In particular, we consider quantum systems in interaction with macroscopic environments, as they appear in Anderson impurity problems, quantum thermodynamics, or open quantum systems. We develop a low-entanglement representation of the environment with only short-range interactions, perfectly suitable for simulations with MPS. We further show that an interleaved ordering of tight-binding chains can significantly reduce the creation of entanglement, as opposed to an intuitive implementation of the Hamiltonian's geometry. Our approach allows long-time simulations and an analysis of the environmental dynamics, in which, after a sudden quench, we find clear signatures of many-body effects. We employ our techniques to compute spectral functions of impurity models, with possible applications to dynamical mean-field theory impurity solvers, and to calculate dissipation in the non-equilibrium Anderson model with explicit time-dependence, as relevant for ongoing experiments with oscillating tip atomic force microscopes.