# Source code for toqito.random.random_unitary

```
"""Generate random unitary."""
from __future__ import annotations
import numpy as np
[docs]def random_unitary(dim: list[int] | int, is_real: bool = False) -> np.ndarray:
"""
Generate a random unitary or orthogonal matrix [MO09]_.
Calculates a random unitary matrix (if :code:`is_real = False`) or a random real orthogonal
matrix (if :code:`is_real = True`), uniformly distributed according to the Haar measure.
Examples
==========
We may generate a random unitary matrix. Here is an example of how we may be able to generate a
random :math:`2`-dimensional random unitary matrix with complex entries.
>>> from toqito.random import random_unitary
>>> complex_dm = random_unitary(2)
>>> complex_dm
[[0.40563696+0.18092721j, 0.00066868+0.89594841j],
[0.4237286 +0.78941628j, 0.27157521-0.35145826j]]
We can verify that this is in fact a valid unitary matrix using the :code:`is_unitary` function
from :code:`toqito` as follows
>>> from toqito.matrix_props import is_unitary
>>> is_unitary(complex_dm)
True
We can also generate random unitary matrices that are real-valued as follows.
>>> from toqito.random import random_unitary
>>> real_dm = random_unitary(2, True)
>>> real_dm
[[ 0.01972681, -0.99980541],
[ 0.99980541, 0.01972681]]
Again, verifying that this is a valid unitary matrix can be done as follows.
>>> from toqito.matrix_props import is_unitary
>>> is_unitary(real_dm)
True
We may also generate unitaries such that the dimension argument provided is a :code:`list` as
opposed to an :code:`int`. Here is an example of a random unitary matrix of dimension :math:`4`.
>>> from toqito.random import random_unitary
>>> mat = random_unitary([4, 4], True)
>>> mat
[[ 0.48996358, -0.20978392, 0.56678587, -0.62823576],
[ 0.62909119, -0.35852051, -0.68961425, -0.01181086],
[ 0.38311399, 0.90865415, -0.1209574 , -0.11375677],
[ 0.46626562, -0.04244265, 0.4342295 , 0.76957113]]
As before, we can verify that this matrix generated is a valid unitary matrix.
>>> from toqito.matrix_props import is_unitary
>>> is_unitary(mat)
True
References
==========
.. [MO09] How to generate a random unitary matrix,
Maris Ozols
March 16, 2009,
home.lu.lv/~sd20008/papers/essays/Random%20unitary%20%5Bpaper%5D.pdf
:param dim: The number of rows (and columns) of the unitary matrix.
:param is_real: Boolean denoting whether the returned matrix has real
entries or not. Default is :code:`False`.
:return: A :code:`dim`-by-:code:`dim` random unitary matrix.
"""
if isinstance(dim, int):
dim = [dim, dim]
# Construct the Ginibre ensemble.
gin = np.random.rand(dim[0], dim[1])
if not is_real:
gin = gin + 1j * np.random.rand(dim[0], dim[1])
# QR decomposition of the Ginibre ensemble.
q_mat, r_mat = np.linalg.qr(gin)
# Compute U from QR decomposition.
r_mat = np.sign(np.diag(r_mat))
# Protect against potentially zero diagonal entries.
r_mat[r_mat == 0] = 1
return q_mat @ np.diag(r_mat)
```