  Quantum processes with indefinite causal structure emerge when we wonder
which are the most general evolutions, allowed by quantum theory, of a set of
local systems which are not assumed to be in any particular causal order. These
processes can be described within the framework of higher-order quantum theory
which, starting from considering maps from quantum transformations to quantum
transformations, recursively constructs a hierarchy of quantum maps of
increasingly higher order. In this work, we develop a formalism for quantum
computation with indefinite causal structures; namely, we characterize the
computational structure of higher order quantum maps. Taking an axiomatic
approach, the rules of this computation are identified as the most general
compositions of higher order maps which are compatible with the mathematical
structure of quantum theory. We provide a mathematical characterization of the
admissible composition for arbitrary higher order quantum maps. We prove that
these rules, which have a computational and information-theoretic nature, are
determined by the more physical notion of the signalling relations between the
quantum systems of the higher order quantum maps.
