toqito.random.random_unitary

toqito.random.random_unitary(dim: list[int] | int, is_real: bool = False) → np.ndarray[source]

Generate a random unitary or orthogonal matrix [MO09].

Calculates a random unitary matrix (if is_real = False) or a random real orthogonal matrix (if 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 \(2\)-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 is_unitary function from 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 list as opposed to an int. Here is an example of a random unitary matrix of dimension \(4\).

>>> 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
Parameters:
  • dim – The number of rows (and columns) of the unitary matrix.
  • is_real – Boolean denoting whether the returned matrix has real entries or not. Default is False.
Returns:

A dim-by-dim random unitary matrix.