Fixed the random unitary generation from the Haar measure (#760)

[ikkoham]

Summary

To generate random unitary matrix from a Haar measure, I used scipy.stats.unitary_group.

Fix #519

[delapuente]

@ikkoham than you very much! Can you add a line to the CHANGELOG.md, under the UNRELEASED > Fixed section telling what are you fixing, please?

[ikkoham]

Thanks, I updated changelog.rst.

[nonhermitian]

It seems the scipy.stats.unitary_group does not implement the Haar measure formula from the referenced publication (https://arxiv.org/abs/math-ph/0609050). They skip the last step. Here is a correct implementation that can be used here:

import math
import numpy as np
import scipy.linalg as la

def haar_unitary(n):
    z = (np.random.randn(n,n) + 1j*np.random.randn(n,n))/math.sqrt(2.0)
    q,r = la.qr(z)
    d = np.diagonal(r)
    ph = d/np.abs(d)
    q = np.multiply(q,ph,q) #This line is missing in the SciPy version.
    return q

[ikkoham]

Thank you.

Does the following line correspond to the last step?
https://github.com/scipy/scipy/blob/master/scipy/stats/_multivariate.py#L3789

[delapuente]

I think they are the same. I'm going to merge with master. Meanwhile, @nonhermitian, if you have a couple of minutes, could you confirm @ikkoham observation, please.

[nonhermitian]

Ahh, yes it does. Good catch. This is good.