Qiskit: How did they use Haar measure in random_statevector?

Question

I have read this article to understand sampling state vectors from Haar measure: Understanding the Haar Measure. Qiskit uses Haar measure to sample state vectors, but when I looked at the source code I couldn't see how did they use Haar measure.

Source

[docs]def random_statevector(dims, seed=None):
    """Generator a random Statevector.

    The statevector is sampled from the uniform (Haar) measure.

    Args:
        dims (int or tuple): the dimensions of the state.
        seed (int or np.random.Generator): Optional. Set a fixed seed or
                                           generator for RNG.

    Returns:
        Statevector: the random statevector.
    """
    if seed is None:
        rng = np.random.default_rng()
    elif isinstance(seed, np.random.Generator):
        rng = seed
    else:
        rng = default_rng(seed)

    dim = np.product(dims)

    # Random array over interval (0, 1]
    x = rng.random(dim)
    x += x == 0
    x = -np.log(x)
    sumx = sum(x)
    phases = rng.random(dim) * 2.0 * np.pi
    return Statevector(np.sqrt(x / sumx) * np.exp(1j * phases), dims=dims)

As far as I understood, they applied a unitary matrix to a zero state $|0\rangle$, but where did this factor come from np.sqrt(x / sumx)?

Answer 1

The Haar measure is associated with a uniform probability distribution. In the above code, x is a list of randomly chosen squared amplitudes from the probability distribution. And exp(1j*phases) are the phases. Together they construct the statevector. sumx within the sqrt is the normalization coefficient.