TensorLabXD.jl

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

Package summary

TensorLabXD.jl aims to be a generic package for working with tensors as they appear throughout the physical sciences. TensorLabXD 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 TensorContractionsXS.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. To deal with generic symmetries, we employ notations and concepts from category theory all the way down to the definition of a tensor.

At the same time, TensorLabXD.jl focuses on computational efficiency and performance. The underlying storage of a tensor's data can be any DenseArray. Currently, certain operations are already multithreaded, either by distributing the different blocks in case of a structured tensor (i.e. with symmetries) or by using multithreading provided by the package StridedTensorXD.jl. In the future, we also plan to investigate using CuArrays as underlying storage for the tensors data, so as to leverage GPUs for the different operations defined on tensors.

Contents of the manual

• Introduction
• • What is a tensor?
  • Symmetries and block sparsity
• 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
• Vector spaces
• • • Types
    • Properties
    • Constructors
    • Others structures
    • Details about dual, conj, flip
  • VectorSpace type
  • Fields
  • Elementary spaces
  • Composite spaces
  • Space of morphisms
  • Partial order among vector spaces
• Sectors, representation spaces and fusion trees
• • • Types
    • Properties
    • Others structures
  • General arguments
  • 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
    • Planar manipulations on a splitting tree
    • Planar manipulations on a fusion-splitting tree
    • Braiding manipulations on a splitting tree
    • Braiding manipulations on a fusion-splitting tree
    • Inspecting fusion trees as tensors
  • Fermions
  • Anyons
• Tensors and the TensorMap type
• • • Types
    • Properties
    • Constructors
    • Linear Operations
    • Index manipulations
  • General arguments
  • Storage of tensor data
  • Constructing tensor maps and accessing tensor data
  • • Random and uninitialized tensor maps
    • Tensor maps from existing data
    • Constructing similar tensors
    • Special purpose constructors
  • Tensor properties
  • Reading and writing tensors: Dict conversion
  • Vector space and linear algebra operations
  • Planar index manipulations
  • Braiding 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

• TensorLabXD.jl
• • Package summary
  • Contents of the manual
  • Library outline
• Index
• 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
  • Bibliography
• Introduction
• • What is a tensor?
  • Symmetries and block sparsity
• Sectors, representation spaces and fusion trees
• • General arguments
  • Representation theory and unitary fusion categories
  • Sectors
  • Graded spaces
  • Fusion trees
  • Fermions
  • Anyons
• Vector spaces
• • VectorSpace type
  • Fields
  • Elementary spaces
  • Composite spaces
  • Space of morphisms
  • Partial order among vector spaces
• Tensors and the TensorMap type
• • General arguments
  • Storage of tensor data
  • Constructing tensor maps and accessing tensor data
  • Tensor properties
  • Reading and writing tensors: Dict conversion
  • Vector space and linear algebra operations
  • Planar index manipulations
  • Braiding index manipulations
  • Tensor factorizations
  • Bosonic tensor contractions and tensor networks
  • Fermionic tensor contractions
  • Anyonic tensor contractions
• Tutorial
• • Cartesian tensors
  • Complex tensors
  • Symmetries