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([[ 0.13764463+0.65538975j,  0.74246453+0.01626838j],
       [ 0.45776527+0.58478132j, -0.6072508 +0.28236187j]])

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([[ 0.87766506, -0.47927449],
       [ 0.47927449,  0.87766506]])

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([[ 0.49527332,  0.08749933, -0.16968586,  0.84749922],
       [ 0.68834418, -0.26695275,  0.62674543, -0.24921614],
       [ 0.38614979, -0.438767  , -0.7417619 , -0.32887862],
       [ 0.36300822,  0.85355938, -0.16788735, -0.33387909]])

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([[0.34074554-0.85897194j, 0.32146645+0.20668575j],
       [0.37801036+0.05628362j, 0.30953006-0.87070745j]])

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