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 over non-unitary gates or am I missing something more fundamental?

Cross-posted on physics.SE

  • $\begingroup$ it's a good question, but I'm not sure there are good non-opinion-based answers to it. One the one hand, I would say it's natural to consider unitary dynamics, as non-unitary maps always come with some noise attached, and if you want to perform controlled computations, you ideally want as little noise as possible. On the other hand, people do essentially consider "open system dynamics" every time noise and error in a circuit are considered. Also, there are algorithms involving non-unitary operations (e.g. IIRC HHL2009 by Harrow et al. does). $\endgroup$
    – glS
    Oct 10, 2019 at 11:48
  • $\begingroup$ there are also protocols in which dissipation is used to generate states (by having the target as fixed orbit of a dissipative operation), although I can't find a reference for these works atm $\endgroup$
    – glS
    Oct 10, 2019 at 11:49
  • $\begingroup$ Thank you very much! Yes, I know about non-unitary approaches to universal quantum computing and especially those involving noise on purpose via engineered dissipation with encoding of the computation outcome in the steady state of the evolution (cf. nature.com/articles/nphys1342) as I studied it for some time - but that is exactly the point of my question: those models are hardly ever talked about, almost nobody in the community and certainly industry seems to care $\endgroup$ Oct 10, 2019 at 16:50

1 Answer 1


I know this isn't really what you're thinking of with your question, but measurement-based quantum computing is pretty well studied. Under the many-worlds interpretation, the system counts as open for that protocol, because every time you perform a measurement you're becoming entangled with the system.

  • $\begingroup$ from this point of view, any quantum computing protocol counts as an "open system", as you always measure something at some point $\endgroup$
    – glS
    Oct 13, 2019 at 11:38
  • $\begingroup$ @glS Fair point, but without getting into the philosophical/semantic question of whether the final readout counts as part of a calculation, it seems to me that there's a qualitative difference between externally entangling the system once at the very end vs. repeatedly throughout the calculation as key intermediate steps. $\endgroup$
    – tparker
    Oct 13, 2019 at 13:53

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