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A library for Abelian symmetry preserving tensors in Python 3

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# Introduction
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abeliantensors is a Python 3 package that implements U(1) and ℤₙ symmetry
preserving tensors, as described by Singh et al. in
[arXiv: 0907.2994](https://arxiv.org/abs/0907.2994) and
[arXiv: 1008.4774](https://arxiv.org/abs/1008.4774). abeliantensors has been
designed for use in tensor network algorithms, and works well with the
[ncon function](https://github.com/mhauru/ncon).

## Installation

If you just want to use the library:
```
pip install --user abeliantensors
```

If you also want to modify and develop the library
```
git clone https://github.com/mhauru/abeliantensors
pip install --user -e abeliantensors/[tests,doc]
```

## Usage

For reference documentation see [here][reference-url].

abeliantensors exports classes `TensorU1`, `TensorZ2`, and `TensorZ3`. Other
cyclic groups ℤₙ can be implemented with one-liners, see the file
`symmetrytensors.py` for examples. abeliantensors also exports a class called
`Tensor`, that is just a wrapper around regular NumPy ndarrays, but that
implements the exact same interface as the symmetric tensor classes. This
allows for easy switching between utilizing and not utilizing the symmetry
preserving tensors by simply changing the class that is imported.

Each symmetric tensor has, in addition to its tensor elements, the following
pieces of what we call form data:
* `shape` describes the dimensions of the tensors, just like with NumPy arrays.
  The difference is that for symmetric tensors the dimension of each index
  isn't just a number, but a list of numbers, that sets how the vector space is
  partitioned by the irreducible representations (irreps) of the symmetry. So
  for instance `shape=[[2,3], [5,4]]` could be the shape of a Z2 symmetric
  matrix of dimensions 5 x 9, where the first 2 rows and 5 columns are
  associated with one of the two irreps of Z2, and the remaining 3 rows and 4
  columns with the other.
* `qhape` is like `shape`, but lists the irrep charges instead of the
  dimensions. Irrep charges are often also called quantum numbers, hence the q.
  In the above example `qhape=[[0,1], [0,1]]` would mark the first part of both
  the row and column space to belong to the trivial irrep of charge 0, and the
  second part to the irrep with charge 1. For ℤₙ the possible charges are 0, 1,
  ..., n, for U(1) they are all positive and negative integers.
* `dirs` is a list of 1s and -1s, that gives a direction to each index: either
  1 for outgoing or -1 for ingoing.
* `charge` is an integer, the irrep charge associated to the tensor. In most
  cases you want `charge=0`, which is also the default when creating new
  tensors.

Note that each element of the tensor is associated with one irrep charge for
each of the indices. The symmetry property is then that an element can only be
non-zero if the charges from each index, multiplied by the direction of that
index, add up to the charge of the tensor. Addition of charges for ℤₙ tensors
is modulo n. For instance for a `charge=0` `TensorZ2` object this means that
the charges on each leg must add up to an even number for an element to be
non-zero. The whole point of this library is to store and use such symmetric
tensors in an efficient way, where we don't waste memory or computation time on
the elements we know are zero by symmetry, and can't accidentally let them be
non-zero.

Here's a simple nonsense example of how abeliantensors can be used:
```
import numpy as np
from abeliantensors import TensorZ2

# Create a symmetric tensor from an ndarray. All elements that should be zero
# by symmetry are simply discarded, whether they are zero or not.
sigmaz = np.array([[1, 0], [0, -1]])
sigmaz = TensorZ2.from_ndarray(
    sigmaz, shape=[[1, 1], [1, 1]], qhape=[[0, 1], [0, 1]], dirs=[1, -1]
)

# Create a random symmetric tensor.
a = TensorZ2.random(
    shape=[[3, 2], [2, 4], [4, 4], [1, 1]],
    qhape=[[0, 1]] * 4,
    dirs=[-1, 1, 1, -1],
)

# Do a singular value decomposition of a tensor, thinking of it as a matrix
# with some of the indices combined to a single matrix index, like one does
# with numpy.reshape. Here we combine indices 0 and 2 to form the left matrix
# index, and 1 and 3 to form the right one. The indices are reshaped back to
# the original form after the SVD, so U and V are in this case order-3 tensors.
U, S, V = a.svd([0, 2], [1, 3])

# You can also do a truncated SVD, in this case to truncating to dimension 4.
U, S, V = a.svd([0, 2], [1, 3], chis=4)

# We can contract tensors together easily using the ncon package.
# Note that conjugation flips the direction of each index, as well as the
# charge of the tensor, which in this case though is 0.
from ncon import ncon
aadg = ncon((a, a.conjugate()), ([1, 2, -1, -2], [1, 2, -11, -12]))

# Finally, knowing that aadg is Hermitian, do an eigenvalue
# decomposition of it, this time truncating not to a specific dimension, but
# to a maximum relative truncation error of 1e-5.
E, U = aadg.eig([0, 1], [2, 3], hermitian=True, eps=1e-5)
```

There are many other user-facing methods and features, for more, see the
[reference documentation][reference-url].

## Demo and performance

The folder `demo` has an implementation of Levin and Nave's [TRG
algorithm](https://arxiv.org/abs/cond-mat/0611687), and a script that runs it
on the square lattice Ising model, using both symmetric tensors of the
`TensorZ2` class and dense `Tensors`, and compares the run times.  Below is a
plot of how long it takes to run a single TRG step at various bond dimensions
for both of them.

![Running time of a single TRG step as a function of bond dimension, compared
between using and not using symmetric tensors](figs/trg_performance.svg)

Note that both axes are logarithmic.

At low bond dimensions the simple `Tensor` class outperforms `TensorZ2`,
because keeping track of the symmetry structure imposes an overhead. The time
complexity of the overhead is subleading as a function of bond dimension, and
as one goes to higher bond dimensions the symmetric tensors become faster.
Asymptotically both have the same scaling as a function of bond dimension, but
the prefactor is smaller for `TensorZ2` by a factor of 1/4. This is because
instead of multiplying or decomposing an `m` x `m` matrix at cost `m**3`, we
are multiplying two `m/2` by `m/2` matrices, at a total cost of `2*(m/2)**3 =
(m**3)/4`.  For larger symmetry groups, the asymptotic benefit would be
greater. For instance for `TensorZ3`, we should see an approximately 9-fold
speed-up.

Similar results can be obtained for other algorithms, although the cross-over
point in bond dimension will be different.

## Design and structure

The implementation is built on top of NumPy. The block-wise sparse
structure of the symmetry preserving tensors is implemented with Python
dictionaries, the values of which are the NumPy `ndarray`s for the non-zero
blocks.

Here's a class diagram for the library:
![Class diagram](https://raw.githubusercontent.com/mhauru/abeliantensors/master/figs/classdiagram.svg)

The user-facing classes that one would instantiate are the ones at the bottom.
`TensorU1`, `TensorZ2`, and `TensorZ3` implement symmetric tensors for the
three different symmetry groups, `Tensor` is the wrapper class around NumPy
arrays.

Implementation-wise, all the fun is in `AbelianTensor`, that is the parent of
all the symmetric tensor classes. Its methods include implementations of
various common tensor operations, such as contractions and decompositions,
preserving and making use of the block-wise sparse structure these tensors
have. `TensorCommon` is a parent class of all the other classes that implements
some higher-level features using the lower-level methods.

The wrapper class `Tensor` is designed so that any call to a method of the
`AbelianTensor` class is also a valid call to a similarly named method of the
`Tensor` class. All the symmetry-related information is simply discarded and
some underlying NumPy function is called.  Even if one doesn't use symmetry
preserving tensors, the `Tensor` class provides some neat convenience
functions, such as an easy-to-read one-liner for the
transpose-reshape-decompose-reshape-transpose procedure for singular value and
eigenvalue decompositions of tensors. Note that `Tensor` is a subclass of
NumPy's `ndarray`, so anything you can do with `ndarray`s, you can also do with
`Tensor`s.

## Tests

The `tests` folder has plenty of tests for the various classes. They can be run
by calling `pytest`, provided abeliantensors was installed with the extras
option `tests`.

Most of the tests are based on generating a random instance of one of the
"fancy" tensor classes in this package, and confirming that the following
diagram commutes:
```
Fancy tensor ─── map to numpy ndarray ───> ndarray
    │                                         │
    │                                         │
Do the thing                             Do the thing
    │                                         │
    │                                         │
    V                                         V
Fancy tensor ─── map to numpy ndarray ───> ndarray
```

Two command line arguments can be provided, `--n_iters` which sets how many
times each test is run, with different random tensors each time (100 by
default), and `--tensorclass` which can be used to specify which
tensorclass(es) the tests are run on (by default all of them). Here's an
example of how one might run a specific test repeatedly:
```
pytest tests/test_tensors.py::test_to_and_from_ndarray --tensorclass TensorZ2 --n_iters 1000
```

[reference-url]: https://abeliantensors.readthedocs.io/en/latest/
[travis-img]: https://travis-ci.org/mhauru/abeliantensors.svg?branch=master
[travis-url]: https://travis-ci.org/mhauru/abeliantensors
[codecov-img]: https://codecov.io/gh/mhauru/abeliantensors/branch/master/graph/badge.svg
[codecov-url]: https://codecov.io/gh/mhauru/abeliantensors
[rtd-badge]: https://readthedocs.org/projects/abeliantensors/badge/?version=latest
[rtd-url]: https://abeliantensors.readthedocs.io/en/latest/?badge=latest

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