Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks

Abstract

The curse of dimensionality associated with the Hilbert space of spin systems provides a significant obstruction to the study of condensed matter systems. Tensor networks have proven an important tool in attempting to overcome this difficulty in both the numerical and analytic regimes. These notes form the basis for a seven lecture course, introducing the basics of a range of common tensor networks and algorithms. In particular, we cover: introductory tensor network notation, applications to quantum information, basic properties of matrix product states, a classification of quantum phases using tensor networks, algorithms for finding matrix product states, basic properties of projected entangled pair states, and multiscale entanglement renormalisation ansatz states. The lectures are intended to be generally accessible, although the relevance of many of the examples may be lost on students without a background in many-body physics/quantum information. For each lecture, several problems are given, with worked solutions in an ancillary file.

Published version. Introductory lecture notes, worked solutions to problems in ancillary file. Comments welcome

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Cited by in corpus (254)

• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Quantum Error Mitigation
• Non-Markovian quantum processes: complete framework and efficient characterisation
• Mixed-state entanglement from local randomized measurements
• The Expressive Power of Parameterized Quantum Circuits
• Variational optimization algorithms for uniform matrix product states
• Virtual Distillation for Quantum Error Mitigation
• Hyper-optimized tensor network contraction
• Quantum stochastic processes and quantum non-Markovian phenomena
• Quantum Machine Learning for Chemistry and Physics
• Efficient classical simulation of random shallow 2D quantum circuits
• Computational advantage of quantum random sampling
• The SLH framework for modeling quantum input-output networks
• Quantum principal component analysis only achieves an exponential speedup because of its state preparation assumptions
• Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial
• Tensor Network Algorithms: a Route Map
• A review of Quantum Cellular Automata
• Tensor Networks in a Nutshell
• Models of quantum complexity growth
• Symmetry Protected Topological Order in Open Quantum Systems
• Holographic tensor network models and quantum error correction: A topical review
• Confinement and lack of thermalization after quenches in the bosonic Schwinger model
• Finite correlation length scaling with infinite projected entangled-pair states
• Enhancing Generative Models via Quantum Correlations
• Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter
• Loop Quantum Gravity, Exact Holographic Mapping, and Holographic Entanglement Entropy
• Gauge fixing, canonical forms and optimal truncations in tensor networks with closed loops
• Entanglement barrier and its symmetry resolution: theory and experiment
• TensorNetwork: A Library for Physics and Machine Learning
• Rigorous RG algorithms and area laws for low energy eigenstates in 1D
• TensorNetwork for Machine Learning
• Benchmarking highly entangled states on a 60-atom analog quantum simulator
• Tensor Network Python (TeNPy) version 1
• Sewing spacetime with Lorentzian threads: complexity and the emergence of time in quantum gravity
• Contracting projected entangled pair states is average-case hard
• Quantum error mitigation via matrix product operators
• Calderbank-Steane-Shor Holographic Quantum Error Correcting Codes
• Jet: Fast quantum circuit simulations with parallel task-based tensor-network contraction
• Finite correlation length scaling with infinite projected entangled pair states at finite temperature
• Practical adaptive quantum tomography
• Generative Tensor Network Classification Model for Supervised Machine Learning
• Tensor-Network Approach for Quantum Metrology in Many-Body Quantum Systems
• Limitations of Linear Cross-Entropy as a Measure for Quantum Advantage
• O(3) nonlinear sigma model in 1+1 dimensions with matrix product states
• Parallel time-dependent variational principle algorithm for matrix product states
• Tensor networks for quantum machine learning
• Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices
• Continuous Tensor Network States for Quantum Fields
• Operator content of entanglement spectra after global quenches in the transverse field Ising chain
• A tensor network view of multilayer multiconfiguration time-dependent Hartree methods
• Efficient description of many-body systems with Matrix Product Density Operators
• Efficient Simulation of Dynamics in Two-Dimensional Quantum Spin Systems with Isometric Tensor Networks
• Quantum many-body systems in thermal equilibrium
• Scaling of neural-network quantum states for time evolution
• Classical versus Quantum: comparing Tensor Network-based Quantum Circuits on LHC data
• Gauging tensor networks with belief propagation
• Tensor Networks contraction and the Belief Propagation algorithm
• Sparse sampling and tensor network representation of two-particle Green's functions
• Anomalies and entanglement renormalization
• Deep learning black hole metrics from shear viscosity
• Central charges of aperiodic holographic tensor network models
• Group Field theory and Tensor Networks: towards a Ryu-Takayanagi formula in full quantum gravity
• Variational quantum algorithm with information sharing
• Probabilistic Modeling with Matrix Product States
• An adaptive algorithm for quantum circuit simulation
• Entanglement order parameters and critical behavior for topological phase transitions and beyond
• Quantum Teleportation-Inspired Algorithm for Sampling Large Random Quantum Circuits
• QGOpt: Riemannian optimization for quantum technologies
• Tensor-network codes
• Efficient Quantum State Sample Tomography with Basis-dependent Neural-networks
• Holographic rainbow networks for colorful Motzkin and Fredkin spin chains
• Entanglement-guided architectures of machine learning by quantum tensor network
• Lectures on Quantum Tensor Networks
• Matrix product state algorithms for Gaussian fermionic states
• General tensor network decoding of 2D Pauli codes
• Trimer states with ${\mathbb{Z}}_3$ topological order in Rydberg atom arrays
• Tensor Networks with a Twist: Anyon-permuting domain walls and defects in PEPS
• Projected Entangled Pair States: Fundamental analytical and numerical limitations
• Fisher Metric, Geometric Entanglement and Spin Networks
• Time-dependent variational principle with controlled bond expansion for matrix product states
• Quantum gravity states, entanglement graphs and second-quantized tensor networks
• Efficient variational contraction of two-dimensional tensor networks with a non-trivial unit cell
• Simulating quantum circuits using tree tensor networks
• Augmenting Density Matrix Renormalization Group with Disentanglers
• Quantum computational advantage with string order parameters of 1D symmetry-protected topological order
• Emergent behaviors in group ring flocks
• Density-matrix renormalization group: a pedagogical introduction
• MPSDynamics.jl: Tensor network simulations for finite-temperature (non-Markovian) open quantum system dynamics
• QSpace - An open-source tensor library for Abelian and non-Abelian symmetries
• Operational Metric for Quantum Chaos and the Corresponding Spatiotemporal Entanglement Structure
• Mutual Information Scaling for Tensor Network Machine Learning
• Approximate encoding of quantum states using shallow circuits
• Importance of the spectral gap in estimating ground-state energies
• On the stability of topological order in tensor network states
• Tensor networks for quantum computing
• Simple heuristics for efficient parallel tensor contraction and quantum circuit simulation
• Exploring the CP-violating Dashen phase in the Schwinger model with tensor networks
• Group Field Theory and Holographic Tensor Networks: Dynamical Corrections to the Ryu-Takayanagi formula
• An introduction to infinite projected entangled-pair state methods for variational ground state simulations using automatic differentiation
• Decoding Holographic Codes with an Integer Optimisation Decoder
• Energy cost of entanglement extraction in complex quantum systems
• Exact holographic tensor networks for the Motzkin spin chain
• Opening the Black Box Inside Grover's Algorithm
• Tangent-Space Gradient Optimization of Tensor Network for Machine Learning
• Certified algorithms for equilibrium states of local quantum Hamiltonians
• Tensor Network Decoding Beyond 2D
• First-Order Phase Transition of the Schwinger Model with a Quantum Computer
• Reduced density matrix and entanglement in interacting quantum field theory with Hamiltonian truncation
• Positive maps and trace polynomials from the symmetric group
• Weak first-order phase transitions in the frustrated square lattice J1-J2 classical Ising model
• Ryu-Takayanagi Formula for Symmetric Random Tensor Networks
• Holographic maps from quantum gravity states as tensor networks
• Gauss Law, Minimal Coupling and Fermionic PEPS for Lattice Gauge Theories
• Calibrating the Classical Hardness of the Quantum Approximate Optimization Algorithm
• Quantum-Inspired Fluid Simulation of 2D Turbulence with GPU Acceleration
• Classifying One-Dimensional Quantum States Prepared by a Single Round of Measurements
• Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States
• Quantum Computing and Tensor Networks for Laminate Design: A Novel Approach to Stacking Sequence Retrieval
• Conformal Properties of Hyperinvariant Tensor Networks
• Detecting Violations of Differential Privacy for Quantum Algorithms
• Efficient Quantum Simulation of QCD Jets on the Light Front
• Invertible bimodule categories and generalized Schur orthogonality
• Uhlmann fidelities from tensor networks
• Residual Tensor Train: A Quantum-inspired Approach for Learning Multiple Multilinear Correlations
• From Tree Tensor Network to Multiscale Entanglement Renormalization Ansatz
• Tensor network noise characterization for near-term quantum computers
• Quantum-inspired event reconstruction with Tensor Networks: Matrix Product States
• Symmetry Resolution in non-Lorentzian Field Theories
• Local tensor-network codes
• Gauging the bulk: generalized gauging maps and holographic codes
• Classification of symmetry-protected topological many-body localized phases in one dimension
• Decohering Tensor Network Quantum Machine Learning Models
• Classification and reconstruction of optical quantum states with deep neural networks
• Fermionic wave packet scattering: a quantum computing approach
• Entropy Constraints for Ground Energy Optimization
• A graphical calculus for integration over random diagonal unitary matrices
• On the descriptive power of Neural-Networks as constrained Tensor Networks with exponentially large bond dimension
• Bulk area law for boundary entanglement in spin network states: entropy corrections and horizon-like regions from volume correlations
• The Basis of Design Tools for Quantum Computing: Arrays, Decision Diagrams, Tensor Networks, and ZX-Calculus
• Tensor Networks for Medical Image Classification
• Faster variational quantum algorithms with quantum kernel-based surrogate models
• Quantum message-passing algorithm for optimal and efficient decoding
• A unified diagrammatic approach to topological fixed point models
• Symmetry-resolved entanglement of 2D symmetry-protected topological states
• Locally purified density operators for noisy quantum circuits
• K-theoretic classification of inductive limit actions of fusion categories on AF-algebras
• Dynamical heterogeneity and large deviations in the open quantum East glass model from tensor networks
• Tensor Renormalization Group Centered About a Core Tensor
• Quantum trimer models and topological SU(3) spin liquids on the kagome lattice
• Continuum approach to real time dynamics of 1+1D gauge field theory: out of horizon correlations of the Schwinger model
• Optimizing one-axis twists for variational Bayesian quantum metrology
• Magic entanglement renormalization for quantum fields
• AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum states
• Optimal Symbolic Construction of Matrix Product Operators and Tree Tensor Network Operators
• Contrasting pseudo-criticality in the classical two-dimensional Heisenberg and ${\mathrm{RP}}^2$ models: zero-temperature phase transition versus finite-temperature crossover
• Approaching Collateral Optimization for NISQ and Quantum-Inspired Computing
• Capturing long-range memory structures with tree-geometry process tensors
• Topological Spin Liquids: Robustness under perturbations
• Holographic networks for (1+1)-dimensional de Sitter spacetime
• Parton Distribution Functions in the Schwinger model from Tensor Network States
• Deep convolutional tensor network
• Tensorized orbitals for computational chemistry
• Low-rank Tensor Grid for Image Completion
• Variationally optimizing infinite projected entangled-pair states at large bond dimensions: A split corner transfer matrix renormalization group approach
• Towards topological fixed-point models beyond gappable boundaries
• Loop Series Expansions for Tensor Networks
• State Diagrams to determine Tree Tensor Network Operators
• Interaction Decompositions for Tensor Network Regression
• Topological dualities via tensor networks
• Méthodes de calcul avec réseaux de tenseurs en physique (Basic tensor network computations in physics)
• Locality optimization for parent Hamiltonians of Tensor Networks
• Phase transitions of a 2D deformed-AKLT model
• Tensor-network approach to thermalization in open quantum many-body systems
• Approximate symmetries of Hamiltonians
• Language Modeling with Reduced Densities
• Complete aggregation of the Lohe tensor model with the same free flow
• Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators
• Triggering Boundary Phase Transitions through Bulk Measurements in 2D Cluster States
• Equivariant Tensor Network Potentials
• Matrix product state ansatz for the variational quantum solution of the Heisenberg model on Kagome geometries
• Segmenting two-dimensional structures with strided tensor networks
• On the Schrödinger-Lohe hierarchy for aggregation and its emergent dynamics
• Quantum fields for unitary representations of Thompson's groups F and T
• A geometrical description of the universal $1\to 2$ asymmetric quantum cloning region
• The State Preparation of Multivariate Normal Distributions using Tree Tensor Network
• A Matrix Product State Model for Simultaneous Classification and Generation
• The Asymmetric Quantum Cloning Region
• Large-scale stochastic simulation of open quantum systems
• Gauging Defects in Quantum Spin Systems
• Controlling quantum many-body systems using reduced-order modelling
• Thermalization at Low Temperatures via Weakly-Damped Multi-Site Baths
• Survey on Computational Applications of Tensor Network Simulations
• At the Interface of Algebra and Statistics
• Two Dimensional Isometric Tensor Networks on an Infinite Strip
• Graphical Framework for Non-Gaussian Quantum States
• Efficient and systematic calculation of arbitrary observables for the matrix product state excitation ansatz
• Finite temperature tensor network algorithm for frustrated two-dimensional quantum materials
• Quantum Gauge Networks: A New Kind of Tensor Network
• Simulating quantum circuits using the multi-scale entanglement renormalization ansatz
• TensorKrowch: Smooth integration of tensor networks in machine learning
• Efficient tensor networks for control-enhanced quantum metrology
• A Practical Introduction to Tensor Network Renormalization with TNRKit.jl
• Equivalence checking of quantum circuits via intermediary matrix product operator
• Redundant string symmetry-based error correction: Demonstrations on quantum devices
• Estimating entanglement in 2D Heisenberg model in the strong rung-coupling limit
• A quantum inspired approach to learning dynamical laws from data -- block-sparsity and gauge-mediated weight sharing
• Scale-Invariant Continuous Entanglement Renormalization of a Chern Insulator
• Parallel Quantum Signal Processing Via Polynomial Factorization
• Efficient modeling of superconducting quantum circuits with tensor networks
• Dynamical Structure Factor from Weak Measurements
• Carving-width and contraction trees for tensor networks
• Quick design of feasible tensor networks for constrained combinatorial optimization
• Logarithmic growth of peripheral entanglement concentrated via noisy measurements in a star network of spins
• Exact average many-body interatomic interaction model for random alloys
• Experimental measurement and a physical interpretation of quantum shadow enumerators
• The Complexity of Contracting Planar Tensor Network
• Deformations of the Boundary Theory of the Square Lattice AKLT Model
• Entanglement renormalization for quantum fields with boundaries and defects
• Fast Tensor Disentangling Algorithm
• Symmetry Properties of Quantum Dynamical Entropy
• On Infinite Tensor Networks, Complementary Recovery and Type II Factors
• Non-onsite symmetries and quantum teleportation in split-index matrix product states
• QAS-QTNs: Curriculum Reinforcement Learning-Driven Quantum Architecture Search for Quantum Tensor Networks
• Boundary-induced singularity in strongly-correlated quantum systems at finite temperature
• Studies of braided non-Abelian anyons using anyonic tensor networks
• Quantum Fisher information from tensor network integration of Lyapunov equation
• Variational optimization of projected entangled-pair states on the triangular lattice
• Les Houches Lecture Notes on Tensor Networks
• Stochastic Representation of the Quantum Quartic Oscillator
• A Synthesis of Hidden Subgroup Quantum Algorithms and Quantum Chemical Dynamics
• Tensor Completion via Tensor Networks with a Tucker Wrapper
• Robustness of critical U(1) spin liquids and emergent symmetries in tensor networks
• paces: Parallelized Application of Co-Evolving Subspaces, a method for computing quantum dynamics on GPUs
• Finite graphs and configurations of points
• From barren plateaus through fertile valleys: Conic extensions of parameterised quantum circuits
• Imaginary Time Spectral Transforms for Excited State Preparation
• QMetro++ -- Python optimization package for large scale quantum metrology with customized strategy structures
• Convolutional Polar Codes on Channels with Memory using Tensor Networks
• Patch-based Medical Image Segmentation using Matrix Product State Tensor Networks
• Tensor networks for quantum causal histories
• New perspectives on quantum kernels through the lens of entangled tensor kernels
• Multi-Tensor Network Representation for High-Order Tensor Completion
• The Canonical Forms of Matrix Product States in Infinite-Dimensional Hilbert Spaces
• Variational Tensor Network Simulation of Gaussian Boson Sampling and Beyond
• Pauli weight requirement of the matrix elements in time-evolved local operators: dependence beyond the equilibration temperature
• Enhanced Krylov Methods for Molecular Hamiltonians: Reduced Memory Cost and Complexity Scaling via Tensor Hypercontraction
• A quantum eigenvalue solver based on tensor networks
• Tensorization of neural networks for improved privacy and interpretability
• Matrix product state classification of 1D multipole symmetry protected topological phases
• Simultaneous superadditivity of the direct and complementary channel capacities
• Who can compete with quantum computers? Lecture notes on quantum inspired tensor networks computational techniques
• Group-invariant tensor train networks for supervised learning
• A minimal tensor network beyond free fermions
• Process Tensor Approaches to Non-Markovian Quantum Dynamics