Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)

Johannes Hauschild; Frank Pollmann

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Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)

Johannes Hauschild, Frank Pollmann

arXiv:1805.00055v4Apr 30, 2018cond-mat.str-el

TL;DR

Quantum many-body simulations face exponentially large Hilbert spaces and situations where quantum Monte Carlo is ineffective. The notes review TPS algorithms and introduce TeNPy, demonstrating TEBD and DMRG while explaining Abelian-symmetry implementations. They report TeNPy approaching ITensor speed for sufficiently large MPS bond dimensions, while ITensor is generally faster for DMRG.

Problem
Exponentially growing Hilbert spaces and ineffective quantum Monte Carlo sampling motivate efficient tensor-product-state methods for many quantum many-body systems.
Method
The notes review finite and infinite MPS/TPS algorithms, introduce the TeNPy Python library, and explain TEBD, DMRG, and Abelian-symmetry implementations.
Results
For MPS with bond dimensions χmax ≳ 300−350, TeNPy reaches nearly the same speed as ITensor; ITensor is generally faster for DMRG.
Takeaways & Limitations
TeNPy provides pedagogical working codes for finite and infinite TEBD and DMRG, giving users a practical way to explore these algorithms.
Takeaways & Limitations
The notes focus exclusively on global Abelian symmetries and omit non-Abelian symmetries, which require a different computational basis and are more difficult to implement.

Abstract

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Tensor product state (TPS) based methods are powerful tools to efficiently simulate quantum many-body systems in and out of equilibrium. In particular, the one-dimensional matrix-product (MPS) formalism is by now an established tool in condensed matter theory and quantum chemistry. In these lecture notes, we combine a compact review of basic TPS concepts with the introduction of a versatile tensor library for Python (TeNPy) [https://github.com/tenpy/tenpy]. As concrete examples, we consider the MPS based time-evolving block decimation and the density matrix renormalization group algorithm. Moreover, we provide a practical guide on how to implement abelian symmetries (e.g., a particle number conservation) to accelerate tensor operations.

1 Introduction

Quantum many-body simulations are difficult because the Hilbert space grows exponentially and QMC is ineffective for many fermionic or geometrically frustrated systems. These notes review tensor-product-state methods and introduce TeNPy through TEBD, DMRG, finite and infinite systems, and Abelian-symmetry implementations.

The many-body Hilbert space grows exponentially with system size, complicating studies of emergent quantum phenomena.
Tensor-product-state methods efficiently simulate systems where quantum Monte Carlo sampling cannot be used effectively.The cited examples include systems with fermionic degrees of freedom or geometric frustration.
DMRG exploits the area-law entanglement of many quantum ground states, which permits efficient matrix-product-state representations.DMRG was originally developed for ground-state properties of one-dimensional systems and can also address quasi-two-dimensional geometries.
The notes combine a pedagogical review of MPS and TPS algorithms with an introduction to the TeNPy Python tensor library.They cover finite and infinite systems and include example code for calling algorithms from TeNPy.
The presentation covers TEBD, DMRG, thermodynamic-limit generalizations, and Abelian symmetries for accelerating tensor operations.Particle-number conservation is given as an example of an Abelian symmetry.

2 Entanglement in quantum many-body systems

Entanglement links subsystems and provides both physical information and a route to efficient numerical representations. In one dimension, area-law ground states have limited entanglement, enabling compression through Schmidt decompositions and MPS.

Entanglement prevents independent descriptions of different degrees of freedom and helps extract universal properties of quantum states.It also underlies numerical methods for efficiently simulating quantum many-body systems.
A Schmidt decomposition bipartitions a one-dimensional system into left and right Hilbert spaces and exposes the coefficients governing bipartite entanglement.The Schmidt states form orthonormal bases of the relevant subsystem spaces.
The entanglement entropy is the von Neumann entropy of a subsystem’s reduced density matrix.The reduced density matrix has Schmidt states as eigenstates, with eigenvalues given by squared Schmidt coefficients.
2.1 Area law: Typical random states obey a volume law, with entropy proportional to subsystem volume and S ≈ N/2 log d − 1/2 for an equal bipartition.
2.1 Area law: Gapped local one-dimensional ground states obey an area law, so entropy becomes constant for chain lengths N ≳ ξ.Only fluctuations within the correlation length ξ significantly contribute to entanglement near the cut.
2.1 Area law: Area-law states have relatively few significant Schmidt values, allowing quantum states to be compressed by truncating the Schmidt decomposition.For the transverse-field Ising ground state at g = 1.5, nearly all weight is contained in a few Schmidt states.
2.1 Area law: Random volume-law states distribute weight across roughly 2^(N/2) Schmidt values, so the dominant values capture little weight.

3 Finite systems in one dimension

This section introduces matrix-product states for finite one-dimensional systems, including their matrix structure, canonical form, and applications in TEBD and DMRG. It also illustrates product, singlet, and AKLT states as MPS examples.

3.1 Matrix Product States (MPS): An MPS decomposes each wave-function coefficient into a product of site-dependent matrices connected by virtual indices.Boundary bond dimensions are trivial, while each site has a set of matrices indexed by the local physical basis.
3.1 Matrix Product States (MPS): Product states have bond dimension χ = 1, whereas entangled singlet and AKLT states require nontrivial MPS bond structure.The AKLT state provides an example with bond dimension χ = 2 and edge-mode degeneracy under open boundaries.
3.1 Matrix Product States (MPS): Every finite-system state admits an exact MPS representation, but generic volume-law states may require a bond dimension that grows exponentially with system size.Area-law states can instead be approximated well using finite maximal bond dimension.
3.2 Canonical form: Gauge freedom in the MPS representation enables a canonical form whose diagonal matrices contain Schmidt values and expose Schmidt decompositions on bonds.The canonical form organizes left and right Schmidt states and their orthonormality relations.
3.3 Time Evolving Block Decimation (TEBD) / 3.5 Density Matrix Renormalization Group (DMRG): TEBD evolves an MPS through two-site time-evolution operators, while DMRG variationally optimizes two neighboring tensors to minimize the ground-state energy.The section also notes that TEBD can be slow for finding ground states and describes iterative solution of the effective DMRG Hamiltonian.
3.2 Canonical form: Canonical-form orthogonality reduces local expectation-value calculations to contractions involving only a few local tensors.For ground-state optimization, DMRG instead constructs an effective Hamiltonian and iteratively optimizes neighboring two-site tensors.

4 Infinite systems in one dimension

Infinite MPS exploit translation-invariant unit cells and transfer matrices to evaluate observables and correlations directly in the thermodynamic limit. Infinite-system TEBD and DMRG adapt finite-system updates while treating unit-cell boundaries and growing environments appropriately.

Infinite MPS: Translation-invariant infinite MPS use either a uniform tensor or an L-site repeating unit cell compatible with the state's translation symmetry.Choosing a larger-than-necessary unit cell can explicitly test translation invariance.
Transfer matrices: The transfer matrix T reduces infinite expectation-value networks to local contractions by projecting repeated applications onto dominant left and right eigenvectors.The construction assumes a pure iMPS, with a unique largest eigenvalue normalized to 1 and a canonical form selected through gauge freedom.
iTEBD: iTEBD applies the same two-site unitary update as finite TEBD, with translation invariance applying that update across all translated unit cells.The unit cell adds the boundary Hamiltonian term h[L,L+1] ≡ h[L,1].
iTEBD: iTEBD is not equivalent to time evolution of an L-site finite system with periodic boundary conditions because the canonical form is not generally well defined on closed loops.The two-site iTEBD update implicitly relies on the canonical form.
iDMRG: Unlike iTEBD, iDMRG grows a finite system between sweeps by inserting unit cells into the environments while updating the central unit cell.The sweep includes the boundary pair (L, L+1) ≡ (L, 1), and full translation invariance is recovered only in the infinite-size limit.

5 Charge conservation

Abelian symmetries impose block structures on tensor indices through charge rules, allowing tensor-network operations to exploit conserved sectors. The notes focus on globally acting abelian symmetries and describe charge-aware tensor manipulations, contractions, and leg grouping.

Motivation and scope: Block-diagonal tensor structures exploit conserved abelian charges to accelerate matrix, singular-value, and tensor operations.The speedup does not change total-dimension scaling but can enable larger bond dimensions at the same computational cost.
Motivation and scope: The treatment focuses on global abelian symmetries acting locally in the computational basis, while non-abelian symmetries require a basis change and are excluded.Particle-number conservation is given as an example, with SU(2) identified as a more difficult alternative.
Charge assignments: For spin systems, assigning integer charges q = 2S_z/ℏ groups basis states into conserved magnetization sectors, producing block-diagonal operators.For two spins, the charge values are 2, 0, and −2, with the two zero-charge states forming one block.
Charge assignments: A tensor charge rule combines each leg's charge q and orientation sign ζ with a total charge Q to determine which tensor entries vanish.Ket legs use ζ = +1 and bra legs use ζ = −1; the rule can also determine an unknown virtual-leg charge.
Basic operations on tensors: Charge-aware tensor operations preserve block structure when permuting or truncating legs, grouping legs into pipes, and contracting compatible legs.Contractions require matching charges and opposite orientations on joined legs, while the output total charge is the sum of input total charges.

6 Conclusion

The notes pair pedagogical MPS/TPS algorithms with TeNPy’s working examples and discuss its scope, extensibility, and performance relative to ITensor. TeNPy approaches ITensor speed for sufficiently large MPS bond dimensions, while ITensor is generally faster for DMRG.

Conclusion: The TeNPy package includes minimal working codes for finite and infinite TEBD and DMRG as a pedagogical introduction.These toy codes use standard Python libraries and are intended to illustrate how the algorithms work.
Conclusion: TeNPy does not cover genuine 2D tensor-product-state methods, although its tensor tools support implementing higher-dimensional tensor networks.The abelian-symmetry approach discussed in the notes also carries over to 2D tensor-product states.
Conclusion: For simulations with MPS bond dimensions χmax ≳300−350, TeNPy reaches nearly the same speed as ITensor.At smaller bond dimensions, the Python implementation incurs substantial overhead.
Conclusion: ITensor is generally faster than the current TeNPy for DMRG, although direct comparison is difficult because the libraries use different eigensolvers.The notes invite contributions to the TeNPy library.