In almost all research on (universal) quantum computation the common models assumed from the outset are either the quantum circuit model with unitary gates, the measurement-based one-way model or the adiabatic model - all of which are (polynomially) equivalent in computational power and employ the idealization of pure states.
But as an alternative, one can consider quantum computation models based on mixed states and general superoperators (quantum operations/channels given by CPTP maps, not necessarily unitary) as gates. This possibility had for example been studied by Tarasov around 2000. Of course any mixed state can be purified in a larger state space (additional qubits) and any CPTP map obtained via a unitary acting on the larger space by tracing out these ancilla qubits, but the generalized view might have its own merits for scalable quantum information processing due to the present obstacles with decoherence/noise for the standard (unitary) approaches.
Theoretical considerations based on “open quantum computation” might lead to new quantum algorithms or insights into other physical implementations. Why is most of science and industry focused on the strict model with unitary gates and delicate pure states - while the obvious alternative seems to be entirely neglected? Does the problem with the second approach merely boil down to insufficient control of non-unitary gates or am I missing something more fundamental?