This is a question motivated from quantum computing but is also relevant to math.
I would like an algorithm that generates a random quantum state (that is, an element of the complex projective space of some dimension), sampled according to some probability distribution which is not uniform in Hilbert space. Assume though that I do have at my disposal a uniform (i.e., Haar) random state generator. How do I do that?
As a toy problem consider the case of a single qubit --- a 2 level system with basis states $|0\rangle,|1\rangle$. The Hilbert space can be understood as the Bloch sphere.
In such a case a Haar random state is a point on the Bloch sphere which is distributed according to the Haar measure $d\psi$ on the sphere. One way to generate such states on a computer is to create a column vector with real and imaginary parts i.i.d. according to $\mathcal{N}(0,1)$, then normalize it. This method generalizes to multiple qubits.
But suppose I want to generate states $|\psi\rangle $ sampled not according to $d\psi$, but according to $$ p( \psi) d\psi = 2 \langle \psi|0\rangle \langle 0|\psi \rangle d\psi, $$ where $|0\rangle$ is the state corresponding to the North Pole on the Bloch sphere. One can check that $p(\psi) \geq 0$ and $\int d\psi p(\psi) d\psi = 1$ so $p(\psi$) is a valid probability density function. This distribution, is such that states near the North Pole occur more likely than states near the South Pole.
How would I write a simple computer program to do this? (I would actually like to understand how to do this more generally than the specific case of the qubit model above)
Note: this is similar to the standard case of real numbers where if we have a uniform r.n.g. in $[0,1]$ we can use this to generate random numbers sampled from any other arbitrary distribution on the real line, e.g. using Box-Muller, inverse transform, ziggurat, rejection sampling. Presumably some variant of the above methods generalizes but since the sample space is different I am finding it difficult to think about it.