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A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
Roman Orus
TL;DR
Quantum many-body wave functions are difficult to represent because the Hilbert space grows exponentially, motivating tensor-network descriptions based on entanglement. The paper provides a practical introduction to MPS and PEPS and selected numerical methods for one- and two-dimensional systems. It explains why these ansätze are useful for relevant many-body states while also identifying limitations such as finite MPS correlation length and the computational hardness of exact PEPS contraction.
Problem
For N spins 1/2, the Hilbert-space dimension is 2^N, making direct wave-function representations inefficient for many-body systems.
Method
The paper presents tensor-network concepts, MPS and PEPS, and selected numerical procedures for evaluating and updating these states in one and two dimensions.
Results
MPS and PEPS provide useful numerical ansätze for ground states and low-energy excitations, with built-in area-law entanglement appropriate for many gapped local systems.
Takeaways & Limitations
MPS can efficiently approximate low-energy states of gapped one-dimensional local Hamiltonians, while tensor-network methods offer practical tools for selected quantum-lattice calculations.
Takeaways & Limitations
MPS have finite correlation length and cannot reproduce critical or scale-invariant systems, while exact scalar-product contraction for arbitrary PEPS is exponentially hard.
Abstract
from arXiv · showhide
This is a partly non-technical introduction to selected topics on tensor network methods, based on several lectures and introductory seminars given on the subject. It should be a good place for newcomers to get familiarized with some of the key ideas in the field, specially regarding the numerics. After a very general introduction we motivate the concept of tensor network and provide several examples. We then move on to explain some basics about Matrix Product States (MPS) and Projected Entangled Pair States (PEPS). Selected details on some of the associated numerical methods for 1d and 2d quantum lattice systems are also discussed.
1 Introduction
Tensor Networks is a rapidly expanding field spanning quantum many-body theory, numerical methods, and connections to quantum gravity. This paper offers a practical, beginner-oriented introduction to selected Tensor Network topics, especially MPS and PEPS algorithms.
- Tensor Networks has produced results in quantum many-body systems and connections to holography and the AdS/CFT correspondence.
- The paper introduces selected Tensor Network topics through practical algorithmic applications of Matrix Product States and Projected Entangled Pair States.
- The presentation is based on introductory seminars and lectures and is designed to familiarize inexperienced readers with common field concepts.
- The paper is not a complete review; it presents intuitive, comprehensible information and references for readers seeking further material.
- The paper covers Tensor Network background, contractions and diagrams, MPS and PEPS basics, and numerical strategies for finite systems.
2 A bit of background
Tensor Network methods address the difficulty of understanding strongly correlated quantum systems, whose models often require faithful numerical treatment. They describe wave functions with interconnected tensors, using tensor structure and entanglement to support flexible simulations across systems and settings.
- Strongly correlated phenomena such as high-Tc superconductivity, topological order, quantum spin liquids, and deconfined quantum criticality remain difficult to understand.
- After proposing a simplified model for a quantum system, researchers generally need faithful numerical methods because most such models are not exactly solvable.
- Tensor Network methods represent a system’s wave function as a network of interconnected tensors, with entanglement acting as glue between components.
- The tensor-network pattern and tensor parameter count determine the amount and structure of represented entanglement.
- Tensor Networks include methods such as DMRG, TEBD, Folding Algorithms, and Projected Entangled Pair States.
- These methods support studies of finite and infinite systems, dimensions, boundary conditions, symmetries, particle types, phase transitions, chemistry, gauge theories, and quantum gravity.
3 Why Tensor Networks?
Tensor Networks offer an entanglement-aware representation and numerical framework for quantum many-body states, addressing limitations of conventional simulation methods and the enormous Hilbert space. Their usefulness is tied to area-law states, while their main limitation is the amount and structure of entanglement.
- 3.1 New boundaries for classical simulations: Conventional numerical methods face distinct limits, including small system sizes, perturbative assumptions, mean-field inaccuracies, sign problems, and modeling dependence.These limitations motivate alternative methods for strongly correlated systems.
- 3.1 New boundaries for classical simulations: Tensor Network methods shift the main computational limitation to the amount and structure of entanglement, extending the range of models accessible to classical computers.This limitation differs from those associated with several conventional numerical techniques.
- 3.2 New language for (condensed matter) physics: Tensor Network methods represent quantum states with interconnected tensors that capture relevant entanglement properties rather than listing wave-function coefficients directly.Their diagrams provide a visual language for quantum many-body states.
- 3.4 Hilbert space is far too large: MPS and PEPS are useful numerical ansatzes because their built-in area law matches the entanglement behavior of many ground states of gapped local systems.They target a restricted but physically relevant region of the many-body Hilbert space.
- 3.4 Hilbert space is far too large: Area-law states form a tiny corner of the exponentially large Hilbert space, and most states are unreachable in practice under local time evolution.The paper notes that reaching most states would require O(exp(N)) time.
4 Tensor Network theory
Tensor networks represent many-body wave functions by contracting smaller tensors, with diagrams making contractions and open indices explicit. This representation can reduce exponentially many wave-function coefficients to polynomially many parameters while encoding area-law entanglement and supporting efficient computation, although contraction cost depends on ordering.
- Tensor networks and contractions: A tensor network is a set of tensors whose contracted indices produce another tensor, with uncontracted indices remaining open.Contractions sum over repeated indices; the result may be a scalar, matrix, or higher-rank tensor.
- Tensor networks and contractions: Tensor network diagrams represent tensors by shapes and indices by lines, making contracted and open indices visually explicit.The diagrams are used to represent matrix products, scalar products, traces, and contractions with or without open indices.
- Contraction cost: Contraction order changes computational cost: the same network can require O(D4) or O(D5) operations.Choosing the contraction order is therefore important in practical tensor-network calculations.
- Breaking the wave function into small pieces: A rank-N wave-function tensor contains O(pN) coefficients, making direct specification exponentially large in system size.For p-level particles, the coefficients C_i1i2...iN form a rank-N tensor.
- Breaking the wave function into small pieces: Replacing the large coefficient tensor with smaller tensors typically yields a polynomial-parameter representation of the wave function.A practical network requires a sub-exponential, often polynomial, number of tensors and parameters.
- Entanglement and tensor networks: Tensor networks can encode area-law entanglement: each broken bond contributes at most log D to the entropy bound.For D = 1 the state is unentangled and becomes a product state; for D > 1 the ansatz can represent area-law scaling.
- Entanglement and tensor networks: Tensor-network states have been proven to include ground and thermal states of local, gapped Hamiltonians.The paper presents these states as relevant area-law states that can also be described efficiently using tensor language.
5 MPS and PEPS: generalities
MPS and PEPS are tensor-network state families organized for one- and two-dimensional systems, with bond dimension controlling representational capacity and entanglement. MPS support efficient one-dimensional calculations but have finite correlation length, whereas PEPS can represent critical correlations while exact contraction is exponentially hard.
- MPS: MPS place one tensor at each site in a one-dimensional array, with bond indices of dimension up to D and physical indices of dimension up to p.The supplied figure distinguishes open and periodic boundary conditions.
- MPS: MPS can represent any many-body state with sufficiently large D, while low-energy states of gapped one-dimensional local Hamiltonians can be efficiently approximated with finite D.For one-dimensional critical systems, D tends to grow polynomially with system size.
- MPS: MPS obey the one-dimensional area law, with block entropy bounded as S(L) = O(log D), matching the approximately constant entropy of large blocks in gapped one-dimensional systems.The bond dimension changes the multiplicative bound rather than the qualitative area-law behavior.
- MPS: MPS correlations decay exponentially with separation, so finite-D MPS cannot reproduce critical or scale-invariant systems with diverging correlation length.This exponential decay is typical of ground states of gapped non-critical one-dimensional systems.
- MPS examples: Examples show that MPS can exactly represent highly entangled states, including the GHZ state with D = 2, the AKLT ground state with D = 2, and the translationally invariant Majumdar-Ghosh ground state with D = 3.The AKLT construction uses projected spin-1/2 singlets, while the Majumdar-Ghosh state superposes nearest-neighbor singlet coverings related by translation.
- PEPS: PEPS are dense two-dimensional tensor-network states that can use finite D for low-energy states and can represent polynomially decaying correlations, including critical behavior, already at D = 2.A two-dimensional PEPS at β = βc provides an example with polynomially decaying correlations and infinite correlation length.
- PEPS: Exact scalar products of arbitrary PEPS are exponentially hard and belong to the #P-Hard complexity class, unlike exact scalar products for MPS.The paper states that exact PEPS contraction is therefore inefficient in principle.
6 Extracting information: computing expectation values
Tensor-network expectation values are extracted exactly for MPS and approximately for PEPS. PEPS contractions are handled by reducing two-dimensional networks to one-dimensional MPS problems, while infinite systems use boundary-MPS, CTM, or coarse-graining approaches.
- Local-observable expectation values provide an efficient way to extract physical information from tensor-network states.MPS expectation values can be computed exactly, whereas PEPS expectation values generally require approximation.
- MPS expectation values: MPS expectation values with open boundaries require O(NpD3) time for finite systems and O(pD3) for infinite systems.
- MPS expectation values: Periodic-boundary MPS contractions require O(NpD5) time, making them less efficient than open-boundary contractions.
- PEPS expectation values: Finite PEPS expectation values are approximated because exact contraction is a #P-Hard problem.
- PEPS expectation values: Finite-PEPS contraction reduces a two-dimensional lattice to successive one-dimensional MPS problems, truncating intermediate bond dimension to χ before exact final contractions.Adding each row acts as an MPO on an MPS, initially producing bond dimension D4; DMRG or TEBD performs the truncation.
- Infinite PEPS: For infinite PEPS, boundary-MPS methods approximate dominant left and right eigenvectors of an infinite MPO, while CTM and coarse-graining methods provide alternatives.After boundary-MPS convergence, the remaining network is an infinite one-dimensional stripe.
7 Determining the tensors: finding ground states
Ground-state tensors are determined by variationally minimizing energy within fixed-bond-dimension tensor-network families or by imaginary-time evolution with repeated truncation. MPS optimization is exact in the relevant environment calculations, whereas PEPS optimization is efficient but approximate, and simplified updates trade accuracy for speed.
- MPS and PEPS approximate ground states and low-energy excitations because their structure targets the relevant area-law corner of Hilbert space.
- Variational optimization: The variational principle minimizes energy within a fixed-bond-dimension tensor-network family while approaching the ground-state energy from above.A Lagrange multiplier enforces unit norm.
- Variational optimization: Tensor-by-tensor sweeping replaces simultaneous optimization of all tensor coefficients with successive local minimizations.
- Variational optimization: Each local optimization forms an effective Hamiltonian and normalization matrix, then solves a generalized eigenvalue problem.The effective objects are environments of the optimized tensor and its conjugate, represented as matrices.
- Variational optimization: Effective Hamiltonians and normalization matrices are computed exactly for MPS but efficiently and approximately for PEPS.
- Imaginary-time evolution: Imaginary-time evolution applies infinitesimal evolution, enlarges bond dimension, and truncates back to the target bond dimension.The initial state must have non-zero overlap with the ground state.
- Imaginary-time evolution: The first-order Suzuki-Trotter expansion decomposes imaginary-time evolution into repeated two-body gates.
- Imaginary-time evolution: Simple-update methods reduce computational cost by avoiding full environment calculations, but their accuracy drops substantially near quantum critical points in two dimensions.They are usually acceptable for gapped phases and can reach large D quickly.
8 Final remarks
The paper offers newcomers an introduction to key tensor-network ideas and numerical approaches for one- and two-dimensional quantum many-body systems. Its coverage is intentionally selective, leaving detailed algorithms and several important topics for future treatment.
- The paper introduces key tensor-network methods for numerically studying one-dimensional systems with MPS and two-dimensional systems with PEPS.It is aimed at non-experienced readers.
- The discussion provides only an introductory treatment rather than detailed implementations of most algorithms.Examples of omitted implementations include TEBD and iPEPS.
- Topics left uncovered include MERA, symmetry implementation, fermionic two-dimensional systems, and contraction or update schemes.These omissions are described as beyond the paper’s purpose.