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.
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.
This is a set of lectures on tensor networks with a strong emphasis on the core algorithms involving Matrix Product States (MPS) and Matrix Product Operators (MPO). Compared to other presentations, particular care has been given to disentangle aspects of tensor networks from the quantum many-body problem: MPO/MPS algorithms are presented as a way to deal with linear algebra on extremely (exponentially) large matrices and vectors, regardless of any particular application. The lectures include well-known algorithms to find eigenvectors of MPOs (the celebrated DMRG), solve linear problems, and recent learning algorithms that allow one to map a known function into an MPS (the Tensor Cross Interpolation, or TCI, algorithm). The lectures end with a discussion of how to represent functions and perform calculus with tensor networks using the "quantics" representation. They include the detailed analytical construction of important MPOs such as those for differentiation, indefinite integration, convolution, and the quantum Fourier transform. Three concrete applications are discussed in detail: the simulation of a quantum computer (either exactly or with compression), the simulation of a quantum annealer, and techniques to solve partial differential equations (e.g. Poisson, diffusion, or Gross-Pitaevskii) within the "quantics" representation. The lectures have been designed to be accessible to a first-year PhD student and include detailed proofs of all statements.
This book serves as an introductory yet thorough guide to tensor networks and their applications in quantum computation and quantum information, designed for advanced undergraduate and graduate-level readers. In Part I, foundational topics are covered, including tensor structures and network representations like Matrix Product States (MPS) and Tree Tensor Networks (TTN). These preliminaries provide readers with the core mathematical tools and concepts necessary for quantum physics and quantum computing applications, bridging the gap between multi-linear algebra and complex quantum systems. Part II explores practical applications of tensor networks in simulating quantum dynamics, with a particular focus on the efficiency they offer for systems of high computational complexity. Key topics include Hamiltonian dynamics, quantum annealing, open system dynamics, and optimization strategies using TN frameworks. A final chapter addresses the emerging role of “quantum magic” in tensor networks. It delves into non-stabilizer states and their contribution to quantum computational power beyond classical simulability, featuring methods such as stabilizer-enhanced MPS and the Clifford-dressed TDVP. In presenting tensor networks as tools for understanding quantum complexity, this book aims to foster a deeper collaboration between the many-body physics and quantum computing communities, inviting a broader audience to engage with these recent developments.
These notes provide an introduction to tensor-network methods in quantum many-body physics, with an emphasis on matrix-product states (MPS). They develop the basic tensor-network language, including graphical notation, virtual indices, bond dimensions, gauge freedom, canonical forms, QR and singular-value decompositions, and the role of entanglement in controlling the efficiency of the representation. The main MPS algorithms are then introduced, including contractions, correlation functions, matrix-product operators, DMRG, and time-evolution methods. The notes also briefly discuss projected entangled-pair states (PEPS) as a higher-dimensional generalization of MPS, together with the basic ideas behind approximate PEPS contraction. Finally, tensor-network representations of mixed states, quantum channels, and Lindblad dynamics are presented, with applications to thermal states and open quantum systems. The presentation is accompanied by short Julia code examples based on ITensor, ITensorMPS, and TensorMixedStates. These notes were written for the 9th Les Houches Summer School on Computational Physics: Open Quantum Systems, held in June 2026.
This thesis focuses on the development and application of tensor networks methods for simulating one-dimensional quantum many-body systems. The general aim is to unveil numerical evidence for a new universality class of 1D interacting fermion systems referred to as quasi-Fermi liquids (qFLs). Realizing such systems beyond a purely analytical context and elucidating their peculiar characteristics is the principle value of this thesis.Chapter 1 begins by introducing the essential physics of one-dimensional systems, known as Luttinger liquids relative to their higher dimensional counterparts. Then it introduces non-linear Luttinger liquids juxtaposed against the defining quasi-Fermi liquid (qFL) field theory that motivates the following lattice studies. Chapter 2 provides an in-depth review of matrix product states (MPS), the main methodology used to explore quasi-Fermi liquids. The chapter serves as a pedagogical introduction to the field and details those novel algorithmic developments necessary for the study of fermionic systems. The central result of this chapter is the section on finite entanglement scaling techniques applied to dynamic correlation functions. Chapter 3 begins the study of lattice models that exhibit quasi-Fermi liquid characteristics. Here we explore the ground states exclusively by computing their static correlation functions. The central results include Luttinger parameter phase space diagrams which demonstrate the existence of the quasi-Fermi liquid across multiple lattice models. Chapter 4 studies the excitations of the qFL lattice models. We begin with results from non-MPS based methods which were essential for benchmarking our novel application of MPS to fermionic spectral functions. The central results of the chapter are the power-law exponents that demonstrate the low-energy universality of the qFL and our analysis of the dynamic structure factor. Chapter 5 concludes with a recapitulation of the main results and an outlook on future work. The Appendix contains those tensor diagrams necessary for a numerical implementation of those algorithms introduced in Chapter 2.--Author's abstract
This thesis focuses on the development, application, and high-performance implementation of numerical methods for simulating quantum many-body systems. In the first chapter, I provide an overview of several most successful conventional approaches for simulating quantum many-body systems, along with their current limitations. Then, the basic properties of the quantum spin model, which will be the subject of study in the following chapters, and its ground state are briefly summarized. Additionally, I discuss the background and the motivations of the research projects carried out during my PhD research. Chapter 2 introduces two numerical algorithms that are extensively used in this thesis, which are exact diagonalization (ED) and variational Monte Carlo. I outline the process of constructing a Hamiltonian matrix and employing iterative methods for exact diagonalization. Subsequently, I introduce the mechanism of the four important components in the ``standard" variational Monte Carlo that are widely used in the most recent research. In Chapter 3, we propose a symmetry-projected variational ansatz in the form of linear combinations of simple restricted Boltzmann machines. This construction allows one to explore states outside of the original variational manifold and increase the representation power with moderate computational effort. Besides allowing one to restore spatial symmetries, an expansion in terms of Krylov states using a Lanczos recursion offers a solution that can further improve the quantum state accuracy. We illustrate these ideas with an application to the Heisenberg J1−J2 model on the square lattice, a paradigmatic problem under debate in condensed matter physics,and achieve state-of-the-art accuracy in the representation of the ground state. In Chapter 4, we present a variational approach based on a Chebyshev expansion of the spectral function, and each Chebyshev polynomial is represented by a neural network quantum state. The Chebyshev moments are obtained by recursively applying the Hamiltonian and projecting on the space of variational states using a modified natural gradient descent method. We compare this approach with a modified approximation of the spectral function which uses a Krylov subspace constructed from the "Chebyshev wave functions." We present results for the one-dimensional and two-dimensional Heisenberg models on the square lattice and compare them with those obtained by other methods in the literature. In Chapter 5, we describe a method to carry out thermodynamic simulations of quantum many-body models using typicality and variational representations of quantum states. Minimally entangled typical thermal states (METTS) are a construction that allows one to solve for the imaginary time evolution of quantum many-body systems. By using wave functions that are weakly entangled, one can take advantage of efficient representations in the form of matrixproduct states. We generalize these ideas to arbitrary variational wave functions and we focus, as an illustration, on the particular case of restricted Boltzmann machines. We demonstrate the application of the method to quantum spin systems in one and two spatial dimensions. In Chapter 6, we study the properties of the impurity band in heavily doped nonmagnetic semiconductors using the Jacobi-Davidson algorithm and the supervised deep learning method. The disorder averaged inverse participation ratio and Thouless number calculation show us the rich structure inside the impurity band. A convolutional neural network model, which is trained to distinguish the extended/localized phase of the Anderson model with high accuracy, shows us the results in good agreement with the conventional approach. Together, we find that there are three mobility edges in the impurity band for a specific on-site impurity potential, which means the presence of the extended states while filling the impurity band. In the final chapter, Chapter 7, we discuss the implementation of architecture-aware optimizations for spin dynamics simulations on x86 CPUs and Nvidia GPUs. The time cost of evaluating the spin-spin correlation functions in real space increases quadratically as the number of spins N, leading to significant computational effort, making the simulation of large spin systems very challenging. We propose to use a highly optimized general matrix multiply (GEMM) subroutine to calculate the dynamical spin-spin correlation function that can achieve near-optimal hardware utilization. Furthermore, we fuse the element-wise operations in the calculation of S(q,t) into the in-house GEMM kernel to reduce memory latency. On Nvidia GPU, we perform the on-the-fly calculation in the epilogue of the GEMM subroutine to avoid saving intermediate results on global memory, which significantly reduces memory usage and achieves better performance than cuBLAS.--Author's abstract
Tensor product state (TPS) based methods are powerful tools to efficiently simulate quantum many-body systems in and out of equilibrium. In particular, the one-dimensional matrix-product (MPS) formalism is by now an established tool in condensed matter theory and quantum chemistry. In these lecture notes, we combine a compact review of basic TPS concepts with the introduction of a versatile tensor library for Python (TeNPy) [1]. As concrete examples, we consider the MPS based time-evolving block decimation and the density matrix renormalization group algorithm. Moreover, we provide a practical guide on how to implement abelian symmetries (e.g., a particle number conservation) to accelerate tensor operations.