  Random samples of quantum states are an important resource for various tasks
in quantum information science, and samples in accordance with a
problem-specific distribution can be indispensable ingredients. Some algorithms
generate random samples by a lottery that follows certain rules and yield
samples from the set of distributions that the lottery can access. Other
algorithms, which use random walks in the state space, can be tailored to any
distribution, at the price of autocorrelations in the sample and with
restrictions to low-dimensional systems in practical implementations. In this
paper, we present a two-step algorithm for sampling from the quantum state
space that overcomes some of these limitations.
  We first produce a CPU-cheap large proposal sample, of uncorrelated entries,
by drawing from the family of complex Wishart distributions, and then reject or
accept the entries in the proposal sample such that the accepted sample is
strictly in accordance with the target distribution. We establish the explicit
form of the induced Wishart distribution for quantum states. This enables us to
generate a proposal sample that mimics the target distribution and, therefore,
the efficiency of the algorithm, measured by the acceptance rate, can be many
orders of magnitude larger than that for a uniform sample as the proposal.
  We demonstrate that this sampling algorithm is very efficient for one-qubit
and two-qubit states, and reasonably efficient for three-qubit states, while it
suffers from the "curse of dimensionality" when sampling from structured
distributions of four-qubit states.
