Tensor Networks in a Nutshell

Abstract

Tensor network methods are taking a central role in modern quantum physics and beyond. They can provide an efficient approximation to certain classes of quantum states, and the associated graphical language makes it easy to describe and pictorially reason about quantum circuits, channels, protocols, open systems and more. Our goal is to explain tensor networks and some associated methods as quickly and as painlessly as possible. Beginning with the key definitions, the graphical tensor network language is presented through examples. We then provide an introduction to matrix product states. We conclude the tutorial with tensor contractions evaluating combinatorial counting problems. The first one counts the number of solutions for Boolean formulae, whereas the second is Penrose's tensor contraction algorithm, returning the number of $3$-edge-colorings of $3$-regular planar graphs.

to appear in Contemporary Physics, 34 pages

References in corpus (32)

• Entanglement of Formation of an Arbitrary State of Two Qubits
• Elementary gates for quantum computation
• Area laws for the entanglement entropy - a review
• Distributed Entanglement
• Efficient classical simulation of slightly entangled quantum computations
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Entanglement renormalization
• Thermodynamic limit of the density matrix renormalization for the spin-1 Heisenberg chain
• An Area Law for One Dimensional Quantum Systems
• A class of quantum many-body states that can be efficiently simulated
• Criticality, the area law, and the computational power of PEPS
• Classical simulation of quantum many-body systems with a tree tensor network
• A class of ansatz wave functions for 1D spin systems and their relation to DMRG
• Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
• High fidelity spin entanglement using optimal control
• Multi-particle entanglement and its applications to cryptography
• Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement, and Holography
• Quantum MERA Channels
• Consistency Conditions for an AdS/MERA Correspondence
• Density Matrix Renormalization Group in the Heisenberg Picture
• Simulation of time evolution with the MERA
• Quantum Criticality and the Kondo Lattice
• Entanglement and tensor network states
• Tensor Network Contractions for #SAT
• A Prehistory of n-Categorical Physics
• Categorical Quantum Circuits
• Algebraically contractible topological tensor network states
• Algebraic Geometry of Matrix Product States
• Solving search problems by strongly simulating quantum circuits
• Capturing long range correlations in two-dimensional quantum lattice systems using correlator product states
• Charged String Tensor Networks

Cited by in corpus (60)

• The Variational Quantum Eigensolver: a review of methods and best practices
• Quantum collision models: open system dynamics from repeated interactions
• Simulation methods for open quantum many-body systems
• Virtual Distillation for Quantum Error Mitigation
• Full-State Quantum Circuit Simulation by Using Data Compression
• Closing the "Quantum Supremacy" Gap: Achieving Real-Time Simulation of a Random Quantum Circuit Using a New Sunway Supercomputer
• Quantum Computational Advantage via High-Dimensional Gaussian Boson Sampling
• Variational Quantum Reinforcement Learning via Evolutionary Optimization
• Solving frustrated quantum many-particle models with convolutional neural networks
• Quantum Machine Learning: from physics to software engineering
• Constructing a virtual two-qubit gate by sampling single-qubit operations
• Equivalence Checking of Quantum Circuits with the ZX-Calculus
• Validating Quantum-Classical Programming Models with Tensor Network Simulations
• Quantum Entropy and Central Limit Theorem
• Resource-Efficient Quantum Computing by Breaking Abstractions
• Known Operator Learning and Hybrid Machine Learning in Medical Imaging -- A Review of the Past, the Present, and the Future
• QGOpt: Riemannian optimization for quantum technologies
• Lectures on Quantum Tensor Networks
• An application benchmark for fermionic quantum simulations
• Efficient classical simulation of noisy quantum computation
• Wave Matrix Lindbladization I: Quantum Programs for Simulating Markovian Dynamics
• Approximate optimization, sampling and spin-glass droplets discovery with tensor networks
• Mutual Information Scaling for Tensor Network Machine Learning
• Anomaly Detection with Tensor Networks
• Improved Efficiency of Open Quantum System Simulations Using Matrix Products States in the Interaction Picture
• Simulation Paths for Quantum Circuit Simulation with Decision Diagrams
• Qibolab: an open-source hybrid quantum operating system
• Thermodynamics of 3D Kitaev quantum spin liquids via tensor networks
• Topological ${\mathbb{Z}}_2$ RVB quantum spin liquid on the ruby lattice
• Special Session: Noisy Intermediate-Scale Quantum (NISQ) Computers -- How They Work, How They Fail, How to Test Them?
• Quantum Computing and Tensor Networks for Laminate Design: A Novel Approach to Stacking Sequence Retrieval
• Benchmarking treewidth as a practical component of tensor-network--based quantum simulation
• Fine-Grained Tensor Network Methods
• Efficient Quantum Circuits for Accurate State Preparation of Smooth, Differentiable Functions
• Thermal bosons in 3d optical lattices via tensor networks
• Interplay of entanglement structures and stabilizer entropy in spin models
• An Accurate and Efficient Analytic Model of Fidelity Under Depolarizing Noise Oriented to Large Scale Quantum System Design
• Efficient Contraction of Large Tensor Networks for Weighted Model Counting through Graph Decompositions
• Tensor network method for reversible classical computation
• Quantify the Non-Markovian Process with Intervening Projections in a Superconducting Processor
• Deep convolutional tensor network
• Interaction Decompositions for Tensor Network Regression
• Language Modeling with Reduced Densities
• State Diagrams to determine Tree Tensor Network Operators
• Infinitely Wide Tensor Networks as Gaussian Process
• Automated Verification of Silq Quantum Programs using SMT Solvers
• Multi-agent blind quantum computation without universal cluster states
• Lower and Upper Bounds on the VC-Dimension of Tensor Network Models
• At the Interface of Algebra and Statistics
• Positivity-preserving extensions of sum-of-squares pseudomoments over the hypercube
• Quantum Higher Order Singular Value Decomposition
• Perspectives on Utilization of Measurements in Quantum Algorithms
• BGLS: A Python Package for the Gate-by-Gate Sampling Algorithm to Simulate Quantum Circuits
• Efficient Construction of Functional Representations for Quantum Algorithms
• State of practice: evaluating GPU performance of state vector and tensor network methods
• Pushing Tensor Networks to the Limit
• Low-rank representation of tensor network operators with long-range pairwise interactions
• Entanglement criteria for the bosonic and fermionic induced ensembles
• Neural Tangent Kernel of Matrix Product States: Convergence and Applications
• Logical Abstractions for Noisy Variational Quantum Algorithm Simulation