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https://github.com/mhauru/abeliantensors

A library for Abelian symmetry preserving tensors in Python 3
https://github.com/mhauru/abeliantensors

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A library for Abelian symmetry preserving tensors in Python 3

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README 

        
# Introduction

[![Readthedocs status badge][rtd-badge]][rtd-url]

[![Travis status badge][travis-img]][travis-url]

[![Codecov status badge][codecov-img]][codecov-url]



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](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