  We revisit the Pseudo-Bayesian approach to the problem of estimating density
matrix in quantum state tomography in this paper. Pseudo-Bayesian inference has
been shown to offer a powerful paradign for quantum tomography with attractive
theoretical and empirical results. However, the computation of
(Pseudo-)Bayesian estimators, due to sampling from complex and high-dimensional
distribution, pose significant challenges that hampers their usages in
practical settings. To overcome this problem, we present an efficient adaptive
MCMC sampling method for the Pseudo-Bayesian estimator. We show in simulations
that our approach is substantially faster than the previous implementation by
at least two orders of magnitude which is significant for practical quantum
tomography.
