Quantum operations are the most widely used tool in the theory of quantum
information processing, representing elementary transformations of quantum
states that are composed to form complex quantum circuits. The class of quantum
transformations can be extended by including transformations on quantum
operations, and transformations thereof, and so on up to the construction of a
potentially infinite hierarchy of transformations. In the last decade, a
sub-hierarchy, known as quantum combs, was exhaustively studied, and
characterised as the most general class of transformations that can be achieved
by quantum circuits with open slots hosting variable input elements, to form a
complete output quantum circuit. The theory of quantum combs proved to be
successful for the optimisation of information processing tasks otherwise
untreatable. In more recent years the study of maps from combs to combs has
increased, thanks to interesting examples showing how this next order of maps
requires entanglement of the causal order of operations with the state of a
control quantum system, or, even more radically, superpositions of alternate
causal orderings. Some of these non-circuital transformations are known to be
achievable and have even been achieved experimentally, and were proved to
provide some computational advantage in various information-processing tasks
with respect to quantum combs. Here we provide a formal language to form all
possible types of transformations, and use it to prove general structure
theorems for transformations in the hierarchy. We then provide a mathematical
characterisation of the set of maps from combs to combs, hinting at a route for
the complete characterisation of maps in the hierarchy. The classification is
strictly related to the way in which the maps manipulate the causal structure
of input circuits.Comment: 12 pages, revtex styl
