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([[ 0.6608344 +5.63273539e-01j, -0.34638359+3.55020022e-01j],
       [ 0.49600438+6.79987312e-04j,  0.05951131-8.66278012e-01j]])

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([[ 0.43486567, -0.90049534],
       [ 0.90049534,  0.43486567]])

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([[ 0.40957691, -0.7230892 , -0.11411148,  0.5443963 ],
       [-0.41107949, -0.63668853, -0.27139085, -0.59328608],
       [ 0.25113465,  0.26468702, -0.93049525, -0.03241539],
       [ 0.77472046, -0.0413582 ,  0.2179545 , -0.5921094 ]])

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

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