Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems

Abstract

The theory of entanglement provides a fundamentally new language for describing interactions and correlations in many body systems. Its vocabulary consists of qubits and entangled pairs, and the syntax is provided by tensor networks. We review how matrix product states and projected entangled pair states describe many-body wavefunctions in terms of local tensors. These tensors express how the entanglement is routed, act as a novel type of non-local order parameter, and we describe how their symmetries are reflections of the global entanglement patterns in the full system. We will discuss how tensor networks enable the construction of real-space renormalization group flows and fixed points, and examine the entanglement structure of states exhibiting topological quantum order. Finally, we provide a summary of the mathematical results of matrix product states and projected entangled pair states, highlighting the fundamental theorem of matrix product vectors and its applications.

Review article. 72 pages

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• Bridging Rokhsar-Kivelson Type and Generic Quantum Phase Transitions via Thermofield Double States
• Chain-mapping methods for relativistic light-matter interactions
• Efficient Simulation of Quantum Many-body Thermodynamics by Tailoring Zero-temperature Tensor Network
• Stabilizer Rényi Entropy for Translation-Invariant Matrix Product States
• Spectral signatures of nonstabilizerness and criticality in infinite matrix product states
• Effects of Topological Boundary Conditions on Bell Nonlocality
• Non-onsite symmetries and quantum teleportation in split-index matrix product states
• Entanglement Spectrum as a diagnostic of chirality of Topological Spin Liquids: Analysis of an $\mathrm{SU}(3)$ PEPS
• Quantum Utility in Simulating the Real-time Dynamics of the Fermi-Hubbard Model using Superconducting Quantum Computers
• The ITransverse.jl library for transverse tensor network contractions
• Classifying fermionic states via many-body correlation measures
• Temperature dependence of energy transport in the ${\mathbb{Z}}_3$ chiral clock model
• ScarFinder: a detector of optimal scar trajectories in quantum many-body dynamics
• Parent Lindbladians for Matrix Product Density Operators
• Magnetic phases of the periodic Anderson model in two dimensions
• Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations
• Variational optimization of projected entangled-pair states on the triangular lattice
• Bound on entanglement in neural quantum states
• Entanglement dynamics from universal low-lying modes
• Variational Transformer Ansatz for the Density Operator of Steady States in Dissipative Quantum Many-Body Systems
• Reconfigurable dissipative entanglement between many spin ensembles: from robust quantum sensing to many-body state engineering
• Universal quasi-Fermi liquid physics of one-dimensional interacting fermions
• Computational Characterization of Symmetry-Protected Topological Phases in Open Quantum Systems
• Symmetries and field tensor network states
• Frustration-free free fermions
• The Tensor as an Informational Resource
• Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders
• Parent Hamiltonian for Fully-augmented Matrix Product States
• Gaussian matrix product states cannot efficiently describe critical systems
• Frustration-free free fermions and beyond
• Partial Degeneration of Tensors
• Pseudoentanglement from tensor networks
• Efficient inference of quantum system parameters by Approximate Bayesian Computation
• Threefold Way for Typical Entanglement
• Boundary-induced singularity in strongly-correlated quantum systems at finite temperature
• Non-Universality from Conserved Superoperators in Unitary Circuits
• A hybrid method integrating Green's function Monte Carlo and projected entangled pair states
• Post-measurement Quantum Monte Carlo
• Towards Symmetry-Aware Efficient Simulation of Quantum Systems and Beyond
• Quantum Magnetic Skyrmions on Kondo-type Lattices
• Efficient Eigenstate Preparation in an Integrable Model with Hilbert Space Fragmentation
• Critical Fermions are Universal Embezzlers
• A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems
• Robustness of near-thermal dynamics on digital quantum computers
• Efficiency of neural-network state representations of one-dimensional quantum spin systems
• Entanglement spectra of non-chiral topological (2+1)-dimensional phases with strong time-reversal breaking, Li-Haldane state counting, and PEPS
• Competing pair density wave orders in the square lattice $t$-$J$ model
• CTMRG study of the critical behavior of an interacting-dimer model
• Quantum Many-Body Linear Algebra, Hamiltonian Moments, and a Coupled Cluster Inspired Framework
• Revealing quantum phase string effect in doped Mott-insulator: a tensor network state approach
• Space-time first-order correlations of an open Bose-Hubbard model with incoherent pump and loss
• Entanglement scaling behaviors of free fermions on hyperbolic lattices
• Projected Entangled Pair States for Lattice Gauge Theories with Dynamical Fermions
• Les Houches Lecture Notes on Tensor Networks
• Gauging Quantum Phases: A Matrix Product State Approach
• Equivariant Parameter Families of Spin Chains: A Discrete MPS Formulation
• A Hermitian bypass to the non-Hermitian quantum theory
• Skeleton of isometric Tensor Network States for Abelian String-Net Models
• Tensorization of neural networks for improved privacy and interpretability
• Capturing reduced-order quantum many-body dynamics out of equilibrium via neural ordinary differential equations
• Quantitative Description of Strongly Correlated Materials by Combining Downfolding Techniques and Tensor Networks
• The Canonical Forms of Matrix Product States in Infinite-Dimensional Hilbert Spaces
• Dynamical cluster-based strategy for improving tensor network algorithms in quantum circuit simulations
• Emulation of large-scale qubit registers with a phase-space approach
• Emergence of topological defects and spin liquid in a two-orbital spin-fermion model on the honeycomb lattice
• Emergent Kitaev materials in synthetic Fermi-Hubbard bilayers
• Holographic Tensor Networks as Tessellations of Geometry
• Finding periodic orbits in projected quantum many-body dynamics
• Trainable Quantum Neural Network for Multiclass Image Classification with the Power of Pre-trained Tree Tensor Networks
• Non-local order parameters for fermion chains via the partial transpose
• Partons from stabilizer codes
• The abelian state hidden subgroup problem: Learning stabilizer groups and beyond
• TTNOpt: Tree tensor network package for high-rank tensor compression
• Algorithmic Aspects of Gauged Gaussian Fermionic PEPS: Gauge Fixing and Translation Invariance
• Universal entanglement correction induced by relevant deformations at the quantum critical point
• Topological crystals and soliton lattices in a Gross-Neveu model with Hilbert-space fragmentation
• General many-body entanglement swapping protocol: opportunities for distributed quantum computing
• Scalable Simulation of Quantum Many-Body Dynamics with Or-Represented Quantum Algebra
• Vanishing correlations in stochastic and bistochastic controlled circuits
• Energy Spectra of Compressed Quantum States
• Quantum Machine Learning for State Tomography Using Classical Data
• Who can compete with quantum computers? Lecture notes on quantum inspired tensor networks computational techniques
• Learning mixed quantum states in large-scale experiments
• Quantum Circuits for Matrix-Product Unitaries
• Classical simulation of noisy quantum circuits via locally entanglement-optimal unravelings
• Dispersion Relations in Two- and Three-Dimensional Quantum Systems
• Process Tensor Approaches to Non-Markovian Quantum Dynamics
• Exact large deviations and emergent long-range correlations in sequential quantum East circuits
• Anderson localization: a density matrix approach
• Programmable order by disorder effect and underlying phases through dipolar quantum simulators
• Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model
• Determination of ground states of one-dimensional quantum systems using the cluster iTEBD method
• Adaptive time Compressed QITE (ACQ) and its geometrical interpretation
• Tensor Network Loop Cluster Expansions for Quantum Many-Body Problems
• Chiral Edge Excitations of $\nu =1/2$ Fractional Chern Insulators in the Bosonic Hofstadter Model
• Exact Fractionalized Ground States in an Extended Spin-1 Kitaev Chain
• Grassmann tensor networks
• Continuous matrix product operators for quantum fields
• Measurement-Based Quantum Computation Using the Spin-1 XXZ Model with Uniaxial Anisotropy
• Entanglement scaling and criticality of infinite-size quantum many-body systems in continuous space addressed by a tensor network approach
• Efficient tomography of microwave photonic cluster states
• The product structure of MPS-under-permutations
• Excited states from local effective Hamiltonians of matrix product states and their entanglement spectrum transition
• Equivalence of residual entropy of hexagonal and cubic ices from tensor network methods
• Probing universal imaginary-time relaxation critical dynamics with infinite projected entangled pair states
• Entanglement as a topological marker in harmonically confined attractive Fermi-Hubbard chains
• A minimal tensor network beyond free fermions
• Symmetry-enforced minimal entanglement and correlation in quantum spin chains
• Using matrix-product states for time-series machine learning