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Tensor networks for complex quantum systems

Roman Orus

arXiv:1812.04011v2cond-mat.str-elhep-latquant-ph

TL;DR

Tensor-network research develops tractable representations and algorithms for complex quantum many-body states. This review surveys foundational concepts, major tensor-network structures, numerical methods, and applications across quantum physics and other disciplines. It presents tensor networks as an active, interdisciplinary framework spanning problems from two-dimensional quantum models to quantum gravity and artificial intelligence.

  • Problem

    Complex quantum many-body states have exponentially large direct descriptions, motivating representations that capture their entanglement structure for numerical simulation.

  • Method

    The paper provides an overview of tensor-network concepts, structures, algorithms, and developments across quantum many-body physics and related disciplines.

  • Results

    The review covers tensor-network applications including two-dimensional Hubbard and antiferromagnetic models, entanglement Hamiltonians, quantum gravity, and artificial intelligence.

  • Takeaways & Limitations

    Tensor networks provide a broadly useful framework for representing correlations in quantum many-body systems and other scientific settings.

Abstract

from arXiv · show

Tensor network states and methods have erupted in recent years. Originally developed in the context of condensed matter physics and based on renormalization group ideas, tensor networks lived a revival thanks to quantum information theory and the understanding of entanglement in quantum many-body systems. Moreover, it has been not-so-long realized that tensor network states play a key role in other scientific disciplines, such as quantum gravity and artificial intelligence. In this context, here we provide an overview of basic concepts and key developments in the field. In particular, we briefly discuss the most important tensor network structures and algorithms, together with a sketch on advances related to global and gauge symmetries, fermions, topological order, classification of phases, entanglement Hamiltonians, AdS/CFT, artificial intelligence, the 2d Hubbard model, 2d quantum antiferromagnets, conformal field theory, quantum chemistry, disordered systems, and many-body localization.

I. INTRODUCTION

Tensor networks represent complex quantum states through interconnected tensor building blocks that encode entanglement, enabling numerical methods beyond earlier simulation limits. The field developed from renormalization ideas and condensed-matter physics into an interdisciplinary framework spanning quantum information, quantum gravity, and artificial intelligence.

  • Foundations: Tensor network states represent complex quantum states using tensor building blocks whose interconnections encode quantum entanglement.This representation organizes the wavefunction’s structure rather than treating all exponentially many coefficients independently.
  • History: Tensor networks evolved from approximations of classical statistical models through developments linking them to quantum lattice wavefunctions and DMRG.DMRG was later understood as variational optimization over Matrix Product States.
  • Quantum information: Entanglement theory clarified why MPS suit gapped one-dimensional systems with local interactions.Entanglement entropy and its structure helped motivate this connection.
  • Interdisciplinary scope: Tensor networks now connect condensed matter and quantum information with quantum gravity and artificial intelligence.Examples include MERA’s proposed relation to geometry through AdS/CFT and tensor-network structures in neural networks.
  • Core tools: Singular Value Decomposition is closely tied to Schmidt decomposition and supports tensor-network numerical algorithms.Canonical forms simplify contractions, although exact simultaneous canonical forms are unavailable for tensor networks with loops.

II. MAIN TENSOR NETWORK STRUCTURES

The review introduces tensor-network structures for states, operators, and mixed states, emphasizing how their geometry determines entanglement representation, correlations, and computational tractability. It also relates Schmidt decomposition and SVD to tensor-network approximation and describes continuum extensions.

  • MPS: MPS are one-dimensional tensor arrays suited to gapped one-dimensional systems, with efficient local-observable calculations and finite correlation length.Their one-dimensional area-law structure prevents formal representation of critical-system entanglement.
  • PEPS: PEPS are two-dimensional tensor arrays that capture two-dimensional area-law correlations and critical correlation functions, but cannot be contracted both efficiently and exactly.They can represent chiral and non-chiral topological order, while gapped chiral topological order remains incompletely understood.
  • Renormalization structures: TTNs encode tree-like coarse-graining and are naturally suited to gapped one-dimensional systems, while MERA adds disentanglers to handle critical entanglement efficiently.Branching MERA extends the entanglement scaling available across renormalization scales.
  • Operators and mixed states: MPOs and PEPOs represent operators, while MPDOs represent mixed states and are positive by construction.These structures extend tensor-network representations beyond pure quantum states.
  • Schmidt decomposition and SVD: The Schmidt decomposition is equivalent to SVD of wavefunction coefficients in orthonormal bases, making singular-value truncation a method for truncating Schmidt coefficients and entanglement.The Schmidt rank satisfies χ ≤ min(d_A, d_B).
  • Continuum tensor networks: Tensor-network structures admit continuum limits such as cMPS, cMERA, and cPEPS for quantum-field-theory functionals and operators.The review notes that cMPS and cMERA already have applications, whereas cPEPS remain comparatively unexplored.

III. MAIN ALGORITHMS

The review frames tensor-network algorithms as families of methods for optimizing states or contracting networks, while noting that they cannot be classified by a single criterion.

  • Algorithmic overview: Tensor-network numerical algorithms are organized by their basic strategies for obtaining states or contracting networks, rather than by one unique classification criterion.The review presents key method families while leaving implementation details to specialized references.
  • Algorithmic overview: The algorithmic discussion emphasizes explaining core ideas and method families while omitting technical implementation details.This scope positions the section as an entry point rather than a full implementation manual.
  • Algorithmic overview: Tensor-network methods address both variational state construction and the contraction of networks needed to evaluate physical quantities.The review treats these as distinct but related algorithmic tasks.

A. Methods to obtain 1d states

Methods for one-dimensional states include variational MPS optimization, time evolution, tree and MERA algorithms, and tangent-space approaches, while two-dimensional methods target PEPS environments and contractions.

  • Methods to obtain 1d states: DMRG optimizes an MPS variationally by sweeping through tensors and minimizing the energy expectation value.Canonical forms improve stability and performance, and extensions include infinite systems, periodic boundaries, and excitations.
  • Methods to obtain 1d states: TEBD evolves MPS in real or imaginary time, with computational cost O(χ3) for open-boundary MPS of bond dimension χ.Real-time evolution remains practical while entanglement stays limited; imaginary-time evolution can approximate ground states.
  • Methods to obtain 1d states: TTN algorithms exploit loop-free structures to study one-dimensional gapped and critical systems, typically costing O(χ4).The cost depends on the specific tree and its bond dimension.
  • Methods to obtain 1d states: MERA variational optimization handles gapped and critical one-dimensional systems but is more difficult because tensors must be unitaries and isometries.For binary MERA, the stated computational cost is O(χ9).
  • Methods to obtain 1d states: Tangent-space methods provide real- and imaginary-time evolution without Trotter decomposition while naturally preserving spatial symmetries.The same framework supports low-energy excitations and variational uniform MPS optimization.
  • Methods to contract 2d tensor networks: Two-dimensional tensor-network calculations use boundary MPS, corner transfer matrices, tensor coarse-graining, and nested-network projections to approximate contractions.Nested networks project a three-dimensional network onto a two-dimensional plane before applying standard contraction strategies.

C. Methods to obtain 2d states

Methods for obtaining two-dimensional tensor-network states include adapting one-dimensional approaches, using PEPS, and applying 2d MERA. PEPS naturally encode two-dimensional correlations, while MPS- and tree-based methods face transverse entanglement limitations.

  • 2d DMRG: 2d DMRG wraps a system into a stripe or cylinder and uses MPS, recovering two-dimensional properties through finite-size scaling.The approach has successfully studied the 2d t−J model and the Kagome Heisenberg Antiferromagnet.
  • 2d DMRG: 2d DMRG eventually faces an exponential transverse entanglement wall that finite-bond-dimension MPS cannot handle for large systems.
  • 2d TTNs: 2d TTNs adapt tree structures to two-dimensional geometries but retain inherently one-dimensional correlations, producing exponential entanglement walls and regime-dependent accuracy.
  • PEPS: PEPS are well-suited to two-dimensional systems because they encode genuinely two-dimensional correlations.Finite- and infinite-PEPS can be optimized through variational updates or imaginary-time evolution; environment-aware updates are slower but more accurate.
  • 2d MERA: 2d MERA provides a variational ansatz for ground states, with computational cost depending on lattice geometry and the chosen unitaries and disentanglers.For an infinite square lattice, one approach has cost O(χ^16).

D. Combined methods

Tensor networks are combined with Monte Carlo, density-functional, dynamical-mean-field, wavelet, and flow-based methods. These combinations extend tensor-network optimization, environment calculation, electronic-structure modeling, impurity solving, and analytic state construction.

  • Monte Carlo TNs: Monte Carlo methods support variational tensor-network optimization, approximate environment calculations, and sampling for string-bond and plaquette-entangled states.
  • TNs for DFT: MPS have been used to construct systematic approximations to exchange-correlation potentials in density functional theory.
  • TNs for DMFT: MPS techniques can serve as high-accuracy, low-cost impurity solvers for dynamical mean-field theory, including nonequilibrium applications.
  • TNs and wavelets: Daubechies wavelets can construct analytic approximations to the 1d critical Ising ground state that correspond to instances of 1d MERA.
  • More synergies: Entanglement continuous unitary transformations combine continuous unitary flows with tensor networks by truncating flow equations according to operator entanglement.
  • Symmetries: Global symmetries can provide computational advantages in DMRG, 1d MERA, and 2d PEPS, including for U(1) and SU(2) symmetries.
  • Symmetries: Schur’s lemma implies that symmetric tensors separate symmetry-fixed structural components from degrees of freedom on degeneracy subspaces.

B. Fermionic systems

Fermionic tensor-network methods simulate fermionic systems in any dimension and directly in second-quantization language. They enforce parity symmetry and replace diagrammatic crossings with fermionic SWAP gates to represent anticommutation.

  • Fermionic systems: Fermionic tensor networks can simulate fermionic systems in any dimension directly in second quantization.
  • Fermionization rules: Fermionization uses parity-symmetric tensors and replaces planar crossings with fermionic SWAP gates.Parity is treated as a Z2 symmetry, while fermionic SWAP gates account for second-quantized operator anticommutation.
  • Fermionization rules: Fermionic tensor-network algorithms can be programmed like bosonic counterparts with only a subleading increase in computational cost.
  • Gauge symmetries: Gauge-invariant tensor networks have been developed for local symmetries, including applications to Z2 lattice gauge theories and the Schwinger model.

D. Topological order and classification of phases

Tensor networks describe topologically ordered systems through gauge-symmetric structures and connect two-dimensional bulk states to boundary entanglement Hamiltonians. For PEPS, the entanglement Hamiltonian’s interaction structure reflects whether the underlying system is gapped, critical, or topologically ordered.

  • Topological order: Eigenstates of string-net models admit exact tensor-network descriptions with tensors carrying specific gauge symmetries.This supports tensor networks as a natural language for topologically ordered systems.
  • Entanglement Hamiltonians: For a PEPS on a cylinder, bipartitioning the system and contracting one subsystem yields a reduced density matrix whose eigenvalues define the entanglement Hamiltonian spectrum.The spectrum can be grouped by quantum numbers such as vertical momentum.
  • Entanglement Hamiltonians: For gapped, non-topologically ordered 2d systems, the entanglement Hamiltonian is usually a 1d short-range Hamiltonian.
  • Entanglement Hamiltonians: For 2d critical systems, the entanglement Hamiltonian is a 1d long-range Hamiltonian, whereas for gapped topologically ordered systems it is essentially a projector.
  • Entanglement Hamiltonians: This correspondence supports the observation that boundary-MPS environment calculations for infinite 2d PEPS can converge quickly in gapped systems without topological order.The paper contrasts this with generic PEPS, which need not be efficiently contractible.

B. Emergent geometry

Tensor networks connect entanglement structure with emergent geometry and with structures used in machine learning and language models. In 1d MERA, logarithmic entanglement scaling provides a lattice connection to AdS/CFT, while neural and language models can exhibit tensor-network structures.

  • Emergent geometry: MERA provides a suggestive lattice realization of geometry in which curvature is linked to entanglement.This motivates its proposed connection to AdS/CFT and gauge/gravity duality.
  • Machine learning: Convolutional networks correspond to specific TTNs, recurrent networks to MPS, and restricted Boltzmann machines to TN states.The paper cautions that neural networks are generally nonlinear whereas TNs are linear.
  • Emergent geometry: For a 1d MERA block, S_L is bounded by log χ times the number of tensor-network links crossing its boundary, yielding S_L = O(log L).The scaling matches (1+1)d CFT behavior and corresponds to a lattice Ryu–Takayanagi prescription.
  • Language models: Probabilistic language models used for speech and text recognition have TTN or MPS structures because Chomsky’s MERGE operation acts as physical coarse-graining.Their loop-free tensor-network form matches an observed suitability of convolutional networks for language processing.

VII. FURTHER TOPICS

Tensor-network methods have been applied to several further problems, including the 2d Hubbard model, frustrated antiferromagnets, conformal field theory, and quantum chemistry. These applications provide variational results, alternative descriptions, and numerical tools across quantum many-body settings.

  • 2d Hubbard model: The 2d Hubbard model remains unresolved in its phase diagram, while the best variational ground-state energies in the strongly correlated regime have been obtained with fermionic iPEPS.The model is studied partly because of its believed relation to high-temperature superconductivity.
  • 2d quantum antiferromagnetism: Tensor-network approaches to the Kagome Heisenberg model include 2d MERA, PESS, and iPEPS, with some results compatible with a gapless quantum spin liquid.These simulations have not produced better energies than 2d DMRG.
  • Conformal field theory: MERA has been used to transform vacuum representations into thermal states and to coarse-grain partition functions in (1+1)d CFT settings.These applications target CFT properties without requiring holographic duality.
  • Quantum chemistry: Tensor-network numerical methods have long been used in quantum chemistry, including DMRG and MPS-based fermionic-orbital reordering.The paper points to dedicated reviews for recent approaches.

E. Disorder and many-body localization

Tensor-network methods have been developed for disordered systems and many-body localized phases, including representations and variational procedures targeting entire spectra or Hamiltonian diagonalization.

  • E. Disorder and many-body localization: Tensor-network methods for many-body localization include spectral networks representing all eigenstates and variational unitary MPOs for diagonalizing MBL Hamiltonians.Alternative tensor-network encodings of all 1d MBL eigenstates have also been proposed.

VIII. OUTLOOK

The paper presents itself as an overview of tensor-network developments while acknowledging that it cannot cover the entire field. It emphasizes applications beyond quantum science, where tensor networks encode generalized correlations rather than necessarily quantum entanglement, and anticipates further uses.

  • VIII. OUTLOOK: The paper surveys tensor-network states and methods across multiple research directions while acknowledging that a complete account is impossible.It aims to collect basic notions and elements of the field’s broader perspective.
  • VIII. OUTLOOK: In applications such as artificial intelligence, tensor networks may encode generalized correlations rather than quantum entanglement, and typical quantum properties such as unitarity are lost.This marks a scope boundary for interpreting cross-disciplinary tensor-network applications.
  • VIII. OUTLOOK: The authors expect tensor networks to continue finding numerical and theoretical applications in established and newly emerging directions.They associate this prospect with the presence of tensor-network structures wherever correlations occur.
  • Key concepts: An area law describes entanglement entropy scaling proportionally to a region’s boundary size.
  • Key concepts: A parent Hamiltonian is a Hamiltonian for which a given PEPS or MPS is the unique ground state.
  • Key concepts: A gap is the energy difference between a Hamiltonian’s lowest-energy eigenstate and first excited state.
  • Key concepts: A correlation length is the scale at which correlations remain sizeable in a many-body system.
  • Key concepts: Global symmetry acts equally across the whole system, whereas gauge symmetry can act differently at every point.
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