  One-way measurement based quantum computations (1WQC) may describe unitary
transformations, via a composition of CPTP maps which are not all unitary
themselves. This motivates the following decision problems: Is it possible to
determine whether a ``quantum-to-quantum'' 1WQC procedure (having non-trivial
input and output subsystems) performs a unitary transformation? Is it possible
to describe precisely how such computations transform quantum states, by
translation to a quantum circuit of comparable complexity? In this article, we
present an efficient algorithm for transforming certain families of
measurement-based computations into a reasonable unitary circuit model, in
particular without employing the principle of deferred measurement.
