  Quantum supermaps are transformations that map quantum operations to quantum
operations. It is known that quantum supermaps which respect a definite,
predefined causal order between their input operations correspond to
fixed-order quantum circuits. A systematic understanding of the physical
interpretation of more general types of quantum supermaps--in particular, those
incompatible with a definite causal structure--is however lacking. Here we
identify two new types of circuits that naturally generalise the fixed-order
case and that likewise correspond to distinct classes of quantum supermaps,
which we fully characterise. We first introduce "quantum circuits with
classical control of causal order", in which the order of operations is still
well-defined, but not necessarily fixed in advance: it can in particular be
established dynamically, in a classically-controlled manner, as the circuit is
being used. We then consider "quantum circuits with quantum control of causal
order", in which the order of operations is controlled coherently. The
supermaps described by these classes of circuits are physically realisable, and
the latter encompasses all known examples of physically realisable processes
with indefinite causal order, including the celebrated "quantum switch".
Interestingly, it also contains new examples arising from the combination of
dynamical and coherent control of causal order, and we detail explicitly one
such process. Nevertheless, we show that quantum circuits with quantum control
of causal order can only generate "causal" correlations, compatible with a
well-defined causal order. We furthermore extend our considerations to
probabilistic circuits that produce also classical outcomes, and we demonstrate
by an example how our characterisations allow us to identify new advantages for
quantum information processing tasks that could be demonstrated in practice.
