  We address the problem of inferring the three-dimensional matter distribution
from a sparse set of one-dimensional quasar absorption spectra of the
Lyman-$\alpha$ forest. Using a Bayesian forward modelling approach, we focus on
extending the dynamical model to a fully self-consistent hierarchical
field-level prediction of redshift-space quasar absorption sightlines. Our
field-level approach rests on a recently developed semiclassical analogue to
Lagrangian perturbation theory (LPT), which improves over noise problems and
interpolation requirements of LPT. It furthermore allows for a manifestly
conservative mapping of the optical depth to redshift space. In addition, this
new dynamical model naturally introduces a coarse-graining scale, which we
exploited to accelerate the Markov chain Monte-Carlo (MCMC) sampler using
simulated annealing. By gradually reducing the effective temperature of the
forward model, we were able to allow it to first converge on large spatial
scales before the sampler became sensitive to the increasingly larger space of
smaller scales. We demonstrate the advantages, in terms of speed and noise
properties, of this field-level approach over using LPT as a forward model,
and, using mock data, we validated its performance to reconstruct
three-dimensional primordial perturbations and matter distribution from sparse
quasar sightlines.
