rand.random_unitary

Generates a random unitary matrix.

Functions

random_unitary(dim[, is_real, seed])

Generate a random unitary or orthogonal matrix [1].

Module Contents

rand.random_unitary.random_unitary(dim, is_real=False, seed=None)

Generate a random unitary or orthogonal matrix [1].

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.rand import random_unitary
>>> complex_dm = random_unitary(2)
>>> complex_dm
array([[-0.59597046+0.06963662j,  0.68835876+0.40759314j],
       [ 0.55431572+0.57680503j,  0.06860805+0.59608975j]])

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.rand import random_unitary
>>> real_dm = random_unitary(2, True)
>>> real_dm
array([[ 0.99999631, -0.00271622],
       [-0.00271622, -0.99999631]])

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.rand import random_unitary
>>> mat = random_unitary([4, 4], True)
>>> mat
array([[ 0.08457995,  0.02911453, -0.98921738,  0.11596361],
       [ 0.77315815, -0.49113837,  0.00461571, -0.40123343],
       [-0.50492423, -0.85772782, -0.05947552,  0.0762704 ],
       [-0.3743317 ,  0.14912557, -0.13375477, -0.90539881]])

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

It is also possible to pass a seed to this function for reproducibility.

>>> from toqito.matrix_props import is_unitary
>>> seeded = random_unitary(2, seed=42)
>>> seeded
array([[ 0.14398279-0.92188954j, -0.05864249+0.35489392j],
       [ 0.35459797+0.06040626j,  0.91839541+0.16480666j]])

And once again, we can verify that this matrix generated is a valid unitary matrix.

>>> from toqito.matrix_props import is_unitary
>>> is_unitary(seeded)
True

References

[1] (1,2)

Maris Ozols. How to generate a random unitary matrix. 2009. URL: http://home.lu.lv/~sd20008/papers/essays/Random%20unitary%20[paper].pdf.

Parameters:

  • dim (list[int] | int) – The number of rows (and columns) of the unitary matrix.

  • is_real (bool) – Boolean denoting whether the returned matrix has real entries or not. Default is False.

  • seed (int | None) – A seed used to instantiate numpy’s random number generator.

Returns:

A dim-by-dim random unitary matrix.

Return type:

numpy.ndarray