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Quantum processes with indefinite causal structure emerge when we wonderwhich are the most general evolutions, allowed by quantum theory, of a set oflocal systems which are not assumed to be in any particular causal order. Theseprocesses can be described within the framework of higher-order quantum theorywhich, starting from considering maps from quantum transformations to quantumtransformations, recursively constructs a hierarchy of quantum maps ofincreasingly higher order. In this work, we develop a formalism for quantumcomputation with indefinite causal structures; namely, we characterize thecomputational structure of higher order quantum maps. Taking an axiomaticapproach, the rules of this computation are identified as the most generalcompositions of higher order maps which are compatible with the mathematicalstructure of quantum theory. We provide a mathematical characterization of theadmissible composition for arbitrary higher order quantum maps. We prove thatthese rules, which have a computational and information-theoretic nature, aredetermined by the more physical notion of the signalling relations between thequantum systems of the higher order quantum maps.

No-signalling constrains quantum computation with indefinite causal structure