  A general approach to obtain reduced models for a wide class of discrete-time
quantum systems is proposed. The obtained models not only reproduce exactly the
output of a given quantum model, but are also guaranteed to satisfy physical
constraints, namely complete positivity and preservation of total probability.
A fundamental framework for exact model reduction of quantum systems is
constructed leveraging on algebraic methods, as well as novel results on
quantum conditional expectations in finite-dimensions. The proposed reduction
algorithm is illustrated and tested on prototypical examples, including the
quantum walk realizing Grover's algorithm.
