  The problem of estimating certain distributions over $\{0,1\}^d$ is
considered here. The distribution represents a quantum system of $d$ qubits,
where there are non-trivial dependencies between the qubits. A maximum entropy
approach is adopted to reconstruct the distribution from exact moments or
observed empirical moments. The Robbins Monro algorithm is used to solve the
intractable maximum entropy problem, by constructing an unbiased estimator of
the un-normalized target with a sequential Monte Carlo sampler at each
iteration. In the case of empirical moments, this coincides with a maximum
likelihood estimator. A Bayesian formulation is also considered in order to
quantify posterior uncertainty. Several approaches are proposed in order to
tackle this challenging problem, based on recently developed methodologies. In
particular, unbiased estimators of the gradient of the log posterior are
constructed and used within a provably convergent Langevin-based Markov chain
Monte Carlo method. The methods are illustrated on classically simulated output
from quantum simulators.
