  In this article we use the theory of quantum channels and open quantum
systems to provide an efficient unitary characterization of a class of
stochastic generators known as quantum hidden Markov models (QHMMs). By
utilizing the unitary characterization, we demonstrate that any QHMM can be
implemented as a quantum circuit with mid-circuit measurement. We prove that
QHMMs are more efficient definitions of stochastic process languages compared
to the equivalent classical hidden Markov Models (HMMs). Starting with the
formulation of QHMMs as quantum channels, we employ Stinespring's construction
to represent these models as unitary quantum circuits with mid-circuit
measurement. By utilizing the unitary parameterization of QHMMs, we define a
formal QHMM learning model. The model formalizes the empirical distributions of
target stochastic process languages, defines hypothesis space of quantum
circuits, and introduces an empirical stochastic divergence measure -
hypothesis fitness - as a success criterion for learning. We demonstrate that
the learning model has a smooth search landscape due to the continuity of
Stinespring's dilation. The smooth mapping between the hypothesis and fitness
spaces enables the development of efficient heuristic and gradient descent
learning algorithms.
  We propose two practical learning algorithms for QHMMs. The first algorithm
is a hyperparameter-adaptive evolutionary search. The second algorithm learns
the QHMM as a quantum ansatz circuit using a multi-parameter non-linear
optimization technique.
