TensorKit.jl

A Julia package for large-scale tensor computations, with a hint of category theory.

Package summary

TensorKit.jl aims to be a generic package for working with tensors as they appear throughout the physical sciences. TensorKit implements a parametric type Tensor (which is actually a specific case of the type TensorMap) and defines for these types a number of vector space operations (scalar multiplication, addition, norms and inner products), index operations (permutations) and linear algebra operations (multiplication, factorizations). Finally, tensor contractions can be performed using the @tensor macro from TensorOperations.jl.

Currently, most effort is oriented towards tensors as they appear in the context of quantum many-body physics and in particular the field of tensor networks. Such tensors often have large dimensions and take on a specific structure when symmetries are present. By employing concepts from category theory, we can represent and manipulate tensors with a large variety of symmetries, including abelian and non-abelian symmetries, fermionic statistics, as well as generalized (a.k.a. non-invertible or anyonic) symmetries.

At the same time, TensorKit.jl focusses on computational efficiency and performance. The underlying storage of a tensor's data can be any DenseArray. When the data is stored in main memory (corresponding to Array), multiple CPUs can be leveraged as many operations come with multithreaded implementations, either by distributing the different blocks in case of a structured tensor (i.e. with symmetries) or by using multithreading provided by the package Strided.jl. Support for storing and manipulating tensors on NVidia and AMD GPUs is currently being developed, whereas support for distributed arrays is planned for the future.

Contents of the manual

• Introduction
• • What is a tensor?
  • Symmetries and block sparsity
• Vector spaces
• • Fields
  • Elementary spaces
  • Operations with elementary spaces
  • Composite spaces
  • More operations with vector spaces
  • Space of morphisms
• Sectors, graded spaces and fusion trees
• • Representation theory and unitary fusion categories
  • Sectors
  • • Existing group representations
    • Combining different sectors
    • Defining a new type of sector
    • Generalizations
  • Graded spaces
  • • Implementation details
    • Constructing instances
    • Methods
    • Examples
  • Fusion trees
  • • Canonical representation
    • Manipulations on a fusion tree
    • Manipulations on a splitting - fusion tree pair
    • Inspecting fusion trees as tensors
  • Fermions
  • Anyons
• Tensors and the TensorMap type
• • Storage of tensor data
  • Constructing tensor maps and accessing tensor data
  • • Random and uninitialized tensor maps
    • Tensor maps from existing data
    • Assigning block data after initialization
    • Constructing similar tensors
    • Special purpose constructors
  • Tensor properties
  • Reading and writing tensors: Dict conversion
  • Vector space and linear algebra operations
  • Index manipulations
  • Tensor factorizations
  • • Eigenvalue decomposition
    • Orthogonal factorizations
    • Factorizations for custom index bipartions
  • Bosonic tensor contractions and tensor networks
  • Fermionic tensor contractions
  • Anyonic tensor contractions

Library outline

• Symmetry sectors
• • Type hierarchy
  • Useful constants
  • Methods for characterizing and manipulating Sector objects
• Fusion trees
• Type hierarchy
• • Methods for defining and generating fusion trees
  • Methods for manipulating fusion trees
• Vector spaces
• • Type hierarchy
  • Useful constants
  • Methods
• Tensors
• • Type hierarchy
  • TensorMap constructors
  • AbstractTensorMap properties and data access
  • AbstractTensorMap operations
  • TensorMap factorizations

Appendix

• Optional introduction to category theory
• • Categories, functors and natural transformations
  • Monoidal categories
  • Duality: rigid, pivotal and spherical categories
  • Braidings, twists and ribbons
  • Adjoints and dagger categories
  • Direct sums, simple objects and fusion categories
  • Topological data of a unitary pivotal fusion category
  • • Monoidal structure
    • Duality and pivotal structure
    • Braidings and twists
  • Bibliography