  Quantum circuits are the preferred formalism for expressing quantum
information processing tasks. Quantum circuit design automation methods mostly
use a waterfall approach and consider that high level circuit descriptions are
hardware agnostic. This assumption has lead to a static circuit perspective:
the number of quantum bits and quantum gates is determined before circuit
execution and everything is considered reliable with zero probability of
failure. Many different schemes for achieving reliable fault-tolerant quantum
computation exist, with different schemes suitable for different architectures.
A number of large experimental groups are developing architectures well suited
to being protected by surface quantum error correcting codes. Such circuits
could include unreliable logical elements, such as state distillation, whose
failure can be determined only after their actual execution. Therefore,
practical logical circuits, as envisaged by many groups, are likely to have a
dynamic structure. This requires an online scheduling of their execution: one
knows for sure what needs to be executed only after previous elements have
finished executing. This work shows that scheduling shares similarities with
place and route methods. The work also introduces the first online schedulers
of quantum circuits protected by surface codes. The work also highlights
scheduling efficiency by comparing the new methods with state of the art static
scheduling of surface code protected fault-tolerant circuits.
