  Continuous-time measurements are instrumental for a multitude of tasks in
quantum engineering and quantum control, including the estimation of dynamical
parameters of open quantum systems monitored through the environment. However,
such measurements do not extract the maximum amount of information available in
the output state, so finding alternative optimal measurement strategies is a
major open problem.
  In this paper we solve this problem in the setting of discrete-time
input-output quantum Markov chains. We present an efficient algorithm for
optimal estimation of one-dimensional dynamical parameters which consists of an
iterative procedure for updating a `measurement filter' operator and
determining successive measurement bases for the output units. A key ingredient
of the scheme is the use of a coherent quantum absorber as a way to
post-process the output after the interaction with the system. This is designed
adaptively such that the joint system and absorber stationary state is pure at
a reference parameter value. The scheme offers an exciting prospect for optimal
continuous-time adaptive measurements, but more work is needed to find
realistic practical implementations.
