Understanding decoherence and dissipation in quantum systems is essential for exploiting quantum effects in emerging technological applications. A proper understanding requires accurate modelling of the system-environment interaction which can only be achieved through treating the system and environment on an equal footing. Simulating the entire system+environment wave-function is a challenging task due to the exponential scaling which is characteristic of quantum mechanics. By using many-body methods based on the powerful concept of a tensor network, one can, nonetheless, obtain numerically exact quantum dynamics for systems with hundreds of degrees of freedom. In addition, a recent innovation has shown that this many-body approach can be extended to finite temperatures without a significant increase in computational cost or algorithmic complexity. This has allowed us to directly compare quantum dynamical simulations with experimental results. Specifically, simulations of the linear vibronic coupling Hamiltonian, parameterised using abinitio electronic structure methods, were used to compute the absorption spectrum of the dye Methylene blue in water. This study has elucidated certain unexplained features of the absorption and shown the how non-adiabatic dynamics resulting from a conical intersection can lead to intensity borrowing between bright and dark excited states. Several methodological advances are also made, including an improvement to the underlying time evolution method TDVP which brings with it an order of magnitude speed-up compared to existing methods.
Understanding decoherence and dissipation in quantum systems is essential for exploiting quantum effects in emerging technological applications. A proper understanding requires accurate modelling of the system-environment interaction which can only be achieved through treating the system and environment on an equal footing. Simulating the entire system+environment wave-function is a challenging task due to the exponential scaling which is characteristic of quantum mechanics. By using many-body methods based on the powerful concept of a tensor network, one can, nonetheless, obtain numerically exact quantum dynamics for systems with hundreds of degrees of freedom. In addition, a recent innovation has shown that this many-body approach can be extended to finite temperatures without a significant increase in computational cost or algorithmic complexity. This has allowed us to directly compare quantum dynamical simulations with experimental results. Specifically, simulations of the linear vibronic coupling Hamiltonian, parameterised using abinitio electronic structure methods, were used to compute the absorption spectrum of the dye Methylene blue in water. This study has elucidated certain unexplained features of the absorption and shown the how non-adiabatic dynamics resulting from a conical intersection can lead to intensity borrowing between bright and dark excited states. Several methodological advances are also made, including an improvement to the underlying time evolution method TDVP which brings with it an order of magnitude speed-up compared to existing methods.
The simulation of open quantum dynamics is a critical tool for understanding how the non-classical properties of matter might be functionalised in future devices. However, unlocking the enormous potential of molecular quantum processes is highly challenging due to the very strong and non-Markovian coupling of 'environmental' molecular vibrations to the electronic 'system' degrees of freedom. Here, we present an advanced but general computational strategy that allows tensor network methods to effectively compute the non-perturbative, real-time dynamics of exponentially large vibronic wave functions of real molecules. We demonstrate how ab initio modelling, machine learning and entanglement analysis can enable simulations which provide real-time insight and direct visualisation of dissipative photophysics, and illustrate this with an example based on the ultrafast process known as singlet fission.
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.
Décrire les systèmes quantiques ouverts à température non-nulle à l’aide de Réseaux de Tenseurs avec application à la Spectroscopie Comprendre la décohérence et la dissipation dans les systèmes quantiques est essentiel pour exploiter les effets quantiques dans les applications technologiques émergentes. Une bonne compréhension nécessite une modélisation précise de l'interaction entre l'environnement et le système qui ne peut être obtenue qu'en traitant ceux-ci sur un pied d'égalité. Simuler la fonction d'onde entière du système + environnement est une tâche difficile en raison de la croissance exponentielle de l'espace de Hilbert qui est caractéristique de la mécanique quantique. En utilisant des méthodes à plusieurs corps basées sur le concept puissant de réseaux de tenseurs, on peut, néanmoins, obtenir une dynamique quantique numériquement exacte pour des systèmes avec des centaines de degrés de liberté. De plus, une innovation récente a montré que cette approche à plusieurs corps peut être étendue à des températures non-nulle sans augmentation du coût de calcul ou de la complexité algorithmique. Cela nous a permis de comparer directement les simulations dynamiques quantiques avec des résultats expérimentaux. Plus précisément, des simulations de l'hamiltonien de couplage vibronique linéaire, paramétrés à l'aide de méthodes de structure électronique ab initio, ont été utilisés pour calculer le spectre d'absorption du colorant Méthylène blue dans l'eau. Cette étude a élucidé certaines caractéristiques inexpliquées de l'absorption et a montré comment la dynamique non-adiabatique résultant d'une intersection conique peut conduire à des emprunts d'intensité entre et les états excités brillant et sombres. Plusieurs avancées méthodologiques sont également réalisées, dont une amélioration de la méthode d'évolution temporelle TDVP qui apporte un ordre de grandeur d'accélération par rapport aux méthodes existantes.
Simulating the non-perturbative and non-Markovian dynamics of open quantum systems is a very challenging many body problem, due to the need to evolve both the system and its environments on an equal footing. Tensor network and matrix product states (MPS) have emerged as powerful tools for open system models, but the numerical resources required to treat finite-temperature environments grow extremely rapidly and limit their applications. In this study we use time-dependent variational evolution of MPS to explore the striking theory of Tamascelli et al. (Phys. Rev. Lett. 2019, 123, 090402.) that shows how finite-temperature open dynamics can be obtained from zero temperature, i.e., pure wave function, simulations. Using this approach, we produce a benchmark dataset for the dynamics of the Ohmic spin-boson model across a wide range of coupling strengths and temperatures, and also present a detailed analysis of the numerical costs of simulating non-equilibrium steady states, such as those emerging from the non-perturbative coupling of a qubit to baths at different temperatures. Despite ever-growing resource requirements, we find that converged non-perturbative results can be obtained, and we discuss a number of recent ideas and numerical techniques that should allow wide application of MPS to complex open quantum systems.
The study of interacting quantum many-body systems is one of the central challenges in modern physics. Interestingly, by understanding the physical properties of such systems from a microscopic description, one can explain many complex quantum phenomena such as the high-temperature superconductivity, fractional quantum Hall effect, etc. However, due to high complexity of these systems, theoretical investigations such as analytical and perturbation theories are limited to studying only a small subset of problems. In the past few decades, numerical simulation has emerged as an indispensable method for studying many phenomena, and has led to immense success and progress in recent years. The main obstacle in addressing quantum many-body systems is that the number of parameters required to represent a state of the system scales exponentially with the system size. However, one hardly needs the full Hilbert space but only a small subspace to describe physical properties of the system. From the point of view in quantum information theory, quantum entanglement is one of the main factors that is responsible for the complexity of studying quantum many-body systems. In particular, the amount of quantum entanglement in such system quantifies the correlation between constituents and therefore it determines how big the subspace is required to represent the system. In this thesis, we are interested in using tensor network states, a new emerging numerical technique to study dynamical and ground state properties of the system. The key ingredient is that a tensor network state which consists of a fairly small number of parameters can represent very well the complete wave function of a system. In one spatial dimension, Matrix Product State (MPS) has become a standard tool, combined with related algorithms such as Time Evolving Block Decimation algorithm (TEBD) to study dynamical properties of a locally-perturbed quantum lattice system. In this thesis, we introduce a technique called ``infinite boundary conditions'' that allows us to use a finite-size MPS to represent the state of an infinite-size system. As a result, one can simulate dynamics of a system for a longer time with a significant computational cost reduction compared to traditional methods. In two spatial dimensions, the tensor network known as Projected Entangled Pair States (PEPS) is emerging as a useful approach. However algorithms for optimizing the matrix elements of these tensors are complex and not very efficient. In particular, the efficiency of the optimization depends critically on the quality of the so-called ``environment'', which is expensive to compute. We have introduced a new method to accelerate the convergence of calculating the PEPS ground state by recycling the environment tensors for many optimization steps, vastly reducing the number of times that the full environment needs to be calculated.
Predicting the properties of correlated materials and molecules often requires a many-body description that goes beyond single-particle and mean-field approximations. Tensor networks address this challenge by exploiting the entanglement structure of quantum states to construct compact, systematically improvable representations. Although developed largely for idealized model Hamiltonians, these methods are increasingly being integrated into material-specific calculations. This Perspective introduces their underlying principles and examines how network geometry, bond-dimension truncation, and approximate contraction determine the accuracy and computational reach of these approaches. We survey applications across regimes of increasing complexity, ranging from ground-state properties of quantum magnets and itinerant-fermion systems to excitation spectra, nonequilibrium dynamics, finite-temperature and open systems, and molecular electronic structure. We then examine four routes connecting these methods to real materials, including experimentally constrained models, downfolded Hamiltonians, dynamical mean-field theory with tensor network impurity solvers, and quantum embedding. Throughout, we evaluate the maturity of existing capabilities and their principal limitations, synthesizing this assessment into a practical method-selection guide. We argue that reliable materials predictions require quantifying uncertainties introduced at each stage, from constructing material-specific Hamiltonians to extracting experimentally accessible observables. Finally, we identify the algorithmic advances and reproducible computational practices needed to establish tensor networks as a routine tool for predictive materials science and quantum chemistry.