Generalised phase kick-back: the structure of computational algorithms from physical principles

The advent of quantum computing has challenged classical conceptions of which problems are efficiently solvable in our physical world. Thismotivates the general study of howphysical principles bound computational power. In this paper we show that some of the essentialmachinery of quantum computation—namely reversible controlled transformations and the phase kick-backmechanism— exist in any operational-defined theorywith a consistent notion of information. These results provide the tools for an exploration of the physics underpinning the structure of computational algorithms. We investigate the relationship between interference behaviour and computational power, demonstrating that non-trivial interference behaviour is a general resource for post-classical computation. In proving the above, we connect higher-order interference to the existence of postquantumparticle types, potentially providing a novel experimental test for higher-order interference. Finally, we conjecture that theories with post-quantum interference—the higher-order interference of Sorkin—can solve problems intractable even on a quantum computer.