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 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.
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.
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.
Tensor Networks are a relatively new machine learning approach. The architectures proposed initially are inspired by approaches from quantum many-body physics simulations. One common layout is the matrix product state (MPS) also known as a tensor train optimized with gradient descent techniques. We introduce a global normalization condition, so that the MPS represents a quantum state. We investigate two optimization methods that find the locally optimal tensors and compare them regarding their effectiveness. One is based on gradient descent and the other on an adaptation of DMRG.
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.
Quantum computing has long promised to revolutionize the way we solve complex problems. At the same time, tensor networks are widely used across various fields due to their computational efficiency and capacity to represent intricate systems. While both technologies can address similar problems, the primary aim of this thesis is not to compare them. Such comparison would be unfair, as quantum devices are still in an early stage, whereas tensor network algorithms represent the state-of-the-art in quantum simulation. Instead, we explore a potential synergy between these technologies, focusing on how two flagship algorithms from each paradigm, the Density Matrix Renormalization Group (DMRG) and quantum annealing, might collaborate in the future. Furthermore, a significant challenge in the DMRG algorithm is identifying an appropriate tensor network representation for the quantum system under study. The representation commonly used is called Matrix Product Operator (MPO), and it is notoriously difficult to obtain for certain systems. This thesis outlines an approach to this problem using finite automata, which we apply to construct the MPO for our case study. Finally, we present a practical application of this framework through the quadratic knapsack problem (QKP). Despite its apparent simplicity, the QKP is a fundamental problem in computer science with numerous practical applications. In addition to quantum annealing and the DMRG algorithm, we implement a dynamic programming approach to evaluate the quality of our results. Our results highlight the power of tensor networks and the potential of quantum annealing for solving optimization problems. Moreover, this thesis is designed to be self-explanatory, ensuring that readers with a solid mathematical background can fully understand the content without prior knowledge of quantum mechanics.
In optimization, one of the well-known classical algorithms is power iterations. Simply stated, the algorithm recovers the dominant eigenvector of some diagonalizable matrix. Since numerous optimization problems can be formulated as an eigenvalue/eigenvector search, this algorithm features wide applicability. Operationally, power iterations consist of performing repeated matrix-to-vector multiplications (or MatVec) followed by a renormilization step in order to converge to the dominant eigenvalue/eigenvector. However, classical realizations, including novel tensor network based approaches, necessitate an exponential scaling for the algorithm's run-time. In this paper, we propose a quantum realiziation to circumvent this pitfall. Our methodology involves casting low-rank representations; Matrix Product Operators (MPO) for matrices and Matrix Product States (MPS) for vectors, into quantum circuits. Specifically, we recover a unitary approximation by variationally minimizing the Frobenius distance between a target MPO and an MPO ansatz wherein the tensor cores are constrained to unitaries. Such an unitary MPO can easily be implemented as a quantum circuit with the addition of ancillary qubits. Thereafter, with appropriate initialization and post-selection on the ancillary space, we realize a single iteration of the classical algorithm. With our proposed methodology, power iterations can be realized entirely on a quantum computer via repeated, static circuit blocks; therefore, a run-time advantage can indeed be guaranteed. Moreover, by exploiting Riemannian optimization and cross-approximation techniques, our methodology becomes instance agnostic and thus allows one to address black-box optimization within the framework of quantum computing.