PaperlayerView exact revision on arXiv

Source-linked AI summary

Tensor Network Algorithms: a Route Map

Mari Carmen Bañuls

arXiv:2205.10345v2quant-ph

TL;DR

Tensor network methods address the computational difficulty of many-body problems through compact state representations and algorithmic approximations. This review organizes their basic framework, established methods, and recent developments, highlighting both their broad capabilities and limits such as entanglement growth and contraction cost. It also connects tensor networks with symmetry methods, fermionic systems, dynamics, renormalization, and other techniques.

  • Problem

    Many-body methods face exponentially growing complexity, motivating numerical approaches that efficiently capture relevant physical questions across diverse quantum systems.

  • Method

    The review synthesizes tensor network representations, algorithms, recent developments, and connections to other techniques, including truncation, higher-dimensional methods, symmetries, fermions, and dynamics.

  • Results

    Tensor network methods provide state-of-the-art results across higher-dimensional, fermionic, symmetric, dynamical, and renormalization-based applications.

  • Takeaways & Limitations

    Tensor networks offer a broad algorithmic framework whose potential is expanding through improved methods and connections with other techniques.

Abstract

from arXiv · show

Tensor networks provide extremely powerful tools for the study of complex classical and quantum many-body problems. Over the last two decades, the increment in the number of techniques and applications has been relentless, and especially the last ten years have seen an explosion of new ideas and results that may be overwhelming for the newcomer. This short review introduces the basic ideas, the best established methods and some of the most significant algorithmic developments that are expanding the boundaries of the tensor network potential. The goal is to help the reader not only appreciate the many possibilities offered by tensor networks, but also find their way through state-of-the-art codes, their applicability and some avenues of ongoing progress.

I. INTRODUCTION

Tensor networks address the exponential complexity of quantum many-body calculations by representing relevant states compactly and enabling increasingly broad numerical methods. This review surveys their foundations, established algorithms, applications, and continuing developments.

  • Quantum many-body calculations become exponentially more complex as system size grows, limiting exact numerical methods.
  • Tensor network techniques were introduced in the 1990s to efficiently address relevant physical questions and became standard numerical tools alongside exact diagonalization and Quantum Monte Carlo.
  • Connections among valence bond solids, matrix product states, DMRG, and quantum information established the conceptual basis for modern tensor network algorithms.
  • The review focuses on the general framework of tensor network algorithms and selected advances important for cutting-edge applications.
  • Tensor network states encode many-body coefficients with polynomially many parameters, exploiting the low entanglement of important physical states.

II. BASIC CONCEPTS

Tensor networks represent many-body states through interconnected tensors whose graph structure determines their expressive power, contractions, and computational properties. The main families include MPS, PEPS, TTN, and MERA, with extensions to infinite systems and mixed states.

  • A tensor network is an interconnected set of multidimensional arrays, with open legs determining the resulting tensor rank after contraction.
  • MPS: MPS use a one-dimensional chain of site tensors, support an entanglement area law, and can be prepared and contracted efficiently.
  • PEPS: PEPS generalize MPS to arbitrary graphs and higher dimensions, but their contractions and preparations are generally inefficient despite supporting critical correlations.
  • TTN and MERA: TTN are loop-free tree networks that contract efficiently, while MERA alternates disentanglers and isometries to represent renormalization and critical states.
  • Gauge freedom permits equivalent tensor descriptions, while canonical forms for loop-free networks encode bipartite entanglement through Schmidt bases.
  • Uniform MPS and periodic iPEPS extend tensor network families to the thermodynamic limit, while MPO, PEPO, and purification represent mixed states.

B. Fundamental primitives

Tensor network algorithms rely on three fundamental operations: contracting network parts, locally updating tensors, and truncating tensor dimensions.

  • Virtually all tensor network algorithms combine contraction, local tensor updates, and truncation into higher-level optimization procedures.

1. Contracting TN

Contraction evaluates the products and sums represented by tensor connections, but its cost depends on contraction order and network structure. Because exact contraction is generally hard, multidimensional algorithms typically use approximations and local environments to update tensors.

  • Contraction explicitly evaluates tensor products and sums, producing an object whose dimensionality is set by the remaining open indices.
  • Contraction order: Finding the optimal contraction order is NP-complete for general tensor networks, although regular networks often have known efficient sequences.
  • Computational cost: Exact contraction is #P-complete for arbitrary tensor networks, so algorithms involving more than one dimension usually approximate contractions through truncation.
  • Local updates: Local tensor updates require evaluating the complementary environment, often through an approximate contraction, to determine the appropriate modification.
  • Truncation: Truncation reduces tensor dimensions to approximate a global network or state while controlling computational cost and discarded correlations.

C. Classic algorithms

Classic tensor-network algorithms optimize or evolve low-dimensional tensor states through local updates, enabling efficient treatment of ground states, dynamics, thermal states, and open systems. Their effectiveness depends strongly on entanglement growth, which fundamentally limits MPS methods for far-from-equilibrium evolution.

  • DMRG: DMRG variationally minimizes an energy or cost function by optimizing one tensor at a time while keeping the others fixed.Writing the Hamiltonian as an MPO simplifies implementation and permits efficient contraction for short-range one-dimensional systems.
  • TEBD: TEBD applies two-body nearest-neighbor gates and truncates the resulting bond dimension to maintain an efficient MPS description.For one-dimensional nearest-neighbor Hamiltonians, Trotter-Suzuki decomposes time evolution into gates exp(−iδh_i), with cost O(D^3).
  • tMPS: tMPS instead optimizes tensors sequentially to minimize the distance to the state after one or more gates, solving local linear systems at cost O(D^3).This strategy can apply an MPO operator, including Trotterized time evolution, to an MPS.
  • Applications: The same tensor-network strategies address ground states, finite-temperature states, open-system dynamics, and real-time evolution.Examples include imaginary-time projection, thermofield-state purification, master-equation evolution, and METTS sampling.
  • Limitations: Linear entanglement growth during far-from-equilibrium evolution can require exponentially growing MPS bond dimension, limiting these methods to near-equilibrium dynamics or moderate times.Thermal equilibrium states satisfy an area law, whereas genuinely out-of-equilibrium scenarios can produce this fundamental limitation.

III. ADVANCED TNS METHODS

Advanced tensor-network methods extend calculations beyond standard one-dimensional applications, especially through PEPS and improved contraction, update, and variational-optimization strategies. These developments enable more accurate higher-dimensional results while retaining substantial computational challenges.

  • Higher-dimensional ansatzes: PEPS generalize MPS to higher-dimensional graphs and provide an area-law ansatz suitable for equilibrium states, but their contraction and algorithms are substantially more complex.Approximate contractions use boundary MPO-MPS methods, coarse-graining, tensor renormalization, or corner transfer matrices.
  • PEPS updates: Simple PEPS updates approximate the environment by a product of diagonal link matrices, while full updates require correlated environment approximations.Discarding environmental correlations improves efficiency but can prevent reaching the best fixed-bond-dimension PEPS.
  • PEPS updates: O(D^10) is the computational cost scaling for most PEPS algorithms, because accurate observable evaluation remains necessary even when updates use less precise environments.The efficiency gain is particularly useful for imaginary-time ground-state searches targeting a fixed point.
  • PEPS limitations: The absence of a PEPS canonical form makes effective norm matrices harder to solve and can reduce stability, although gauge optimization can improve their conditioning.Unlike MPS, PEPS generally require inversion of the effective norm term in local problems.
  • Recent developments: PEPS already outperform MPS for moderate two-dimensional Heisenberg and Hubbard problems, and stable iPEPS optimization has produced more accurate results.Advanced methods have also yielded the most accurate Hubbard-model result reported in the review and initial studies of three-dimensional problems.
  • Applications: Beyond ground states, higher-dimensional time evolution methods address finite-temperature equilibrium, open-system steady states, and real-time dynamics.Restricted PEPS families, including sequentially generated, isometric, and Gaussian fermionic PEPS, trade generality for more favorable computational properties.
  • Alternative tensor families: Tree tensor networks and augmented trees provide alternative higher-dimensional ansatzes that can treat systems of certain sizes without a full area law.These families are useful when the target states or computational setting favor restricted connectivity.

B. Symmetries

Symmetries can restrict tensor-network searches to quantum-number sectors and improve algorithmic performance. Tensor invariance yields structured tensor blocks, supports higher-dimensional Abelian and non-Abelian implementations, and can be extended to gauge symmetries.

  • Global symmetries: A conserved operator lets the tensor-network search be restricted to subspaces labeled by its quantum numbers, potentially boosting performance.Abelian symmetries such as particle-number and total-magnetization conservation are standard in DMRG.
  • Tensor implementation: Higher-dimensional tensor networks can handle both Abelian and non-Abelian global symmetries through invariant tensors unchanged by the symmetry action.The construction requires well-defined transformation properties for tensor indices and suitable virtual-index bases.
  • Tensor structure: For Abelian symmetries, invariant tensors decompose into blocks obeying quantum-number conservation; non-Abelian blocks further factor into symmetry-determined and remaining components.This internal structure organizes tensor entries according to irreducible representations.
  • Gauge symmetries: Additional link tensors can promote a global symmetry of symmetric tensors to a gauge symmetry.These link tensors are analogous to link variables in lattice gauge theories.
  • Theory: Tensor symmetries also provide a theoretical framework for characterizing MPS and PEPS, yielding fundamental results and remaining an active research area.This formal perspective complements practical symmetry exploitation in numerical algorithms.

C. Fermions

Tensor networks can treat fermionic and spin systems within a common framework, avoiding the sign-problem obstacle that often affects quantum Monte Carlo. In higher dimensions, fermionic tensor networks preserve locality by working directly with fermionic spaces and parity-symmetric tensors.

  • Motivation: Tensor-network calculations can treat fermionic systems without the quantum Monte Carlo sign problem, whose convergence cost can grow exponentially with system size.This makes fermionic tensor networks relevant to condensed-matter and fundamental-physics problems.
  • One dimension: In one dimension, the Jordan-Wigner transformation maps local fermionic models to local spin Hamiltonians, allowing standard tensor-network algorithms.In higher dimensions, the analogous transformation does not preserve locality.
  • Direct fermionic formulation: Direct fermionic tensor networks use fermionic virtual and physical spaces with well-defined parity, making the tensors symmetric under parity transformations.This construction encodes fermionic statistics directly rather than relying on a higher-dimensional spin mapping.
  • Fermionic signs: A graphical ordering of fermionic modes requires accounting for every leg crossing because crossings represent commutations of fermionic operators.The formalism can also be combined with additional symmetries.
  • Applications: iPEPS combined with fermionic tensor-network techniques has outperformed other computational methods in some Hubbard-model parameter regimes.This result illustrates the practical reach of direct fermionic tensor networks in higher-dimensional settings.

D. Dynamics

Tensor-network dynamics methods approximate time evolution through operator approximations, variational projections, or tangent-space evolution. Their central limitation is rapid entanglement growth, motivating methods that target long-time local observables directly.

  • Time evolution: Time-evolution algorithms apply an approximation to U(δ) = e−iδH and truncate the resulting bond-dimension growth.The standard approach becomes increasingly costly for longer-range interactions because approximating the Hamiltonian exponential is expensive.
  • Alternative methods: Krylov methods approximate the evolved state through Krylov vectors, while Chebyshev methods expand the exponential operator.These approaches avoid explicitly approximating the evolution operator in the full space, although the Krylov vectors still require tensor-network approximations.
  • Variational dynamics: TDVP projects Schrödinger evolution onto the local tangent plane so that MPS evolution remains within the MPS manifold.Tangent-space methods extend this geometric strategy to variational optimization, elementary excitations, and other tensor-network families.
  • Long-time limitation: Entanglement can grow exponentially with simulated time, creating an entanglement barrier that makes long-time state evolution unfeasible.This conflicts with the fact that local observables may approach values described by efficiently approximated statistical ensembles.
  • Beyond state evolution: New strategies target Heisenberg-picture operators or tensor networks representing time-dependent local observables to address long-time dynamics.These approaches focus on experimentally accessible quantities rather than requiring the full time-evolved state.

E. Excitations

Tensor-network methods target excitations by orthogonalizing against known states or representing excitations as localized, momentum-resolved tangent-space perturbations. Generic highly excited states remain difficult because they generally do not satisfy the same low-entanglement structure as low-energy states.

  • Low-energy excitations: Low excited states can be obtained by orthogonalizing a variational ground-state search against previously computed states.This approach is especially useful for finite systems.
  • Elementary excitations: Tangent-space methods represent elementary excitations as localized perturbations with position-dependent momentum factors and variationally optimized energies.The same framework can capture topologically non-trivial excitations such as domain walls, especially in the thermodynamic limit.
  • High-energy states: Generic highly excited states do not generally obey the approximate area law observed for many low-energy excitations.Many-body localized Hamiltonians are identified as an exception, motivating specialized high-energy eigenstate algorithms.
  • Broader toolbox: The broader tensor-network toolbox extends beyond standard state representations to additional approaches for exploring complex systems.This section places excitation methods within that wider methodological landscape.

A. Network renormalization approaches

Network-renormalization methods contract classical and quantum tensor networks by coarse-graining tensors while truncating bonds. Their development addresses truncation errors and connects tensor networks with Monte Carlo, machine learning, and field theory.

  • TRG and HOTRG: TRG coarse-grains a two-dimensional tensor network by replacing local tensor groups with approximate contractions and truncated bonds.Each step reduces the network size by a constant; the original truncation uses singular value decompositions.
  • Truncation limitations: TRG cannot remove some short-range entanglement structures, particularly corner double line tensors.TNR and related methods introduce disentanglers or optimized local truncations to address these internal correlations.
  • Applications: TRG contractions can support PEPS optimization and Grassmann tensor-network treatments of fermionic and bosonic systems.Grassmann tensor networks have been used for discretized field theories with fermionic degrees of freedom.
  • Connections to other techniques: Tensor-network methods connect with other techniques to produce improved algorithms and address new classes of problems.The review highlights connections with Monte Carlo, machine learning, and field theory.

V. OUTLOOK

The tensor-network field continues to expand through formal mathematical work, applied numerical-method development, and synergies with related techniques. The review directs readers toward resources and software for navigating this active area.

  • Research directions: Current tensor-network research combines formal studies of ansatz mathematics with applied development of numerical methods.The field remains a vibrant research area with multiple directions of progress.
  • Resources: Online resources include the DMRG Homepage and The Tensor Network site for introductory and community information.These resources are listed alongside software libraries and tensor-network tool collections.
  • Software: ITensor and TeNPy provide open-source tensor-network software libraries for practical calculations.The outlook also lists Tensors.net as a tensor-network resource.
Loading 2205.10345v2…