Title:

Bounds on computation from physical principles

The advent of quantum computing has challenged classical conceptions of which problems are efficiently solvable in our physical world. This raises the general question of what broad relationships exist between physical principles and computation. The current thesis explores this question within the operationallydefined framework of generalised probabilistic theories. In particular, we investigate the limits on computational power imposed by simple physical principles. At present, the best known upper bound on the power of quantum computers is that BQP is contained in AWPP, where AWPP is a classical complexity class contained in PP. We define a circuitbased model of computation in the above mentioned operational framework and show that in theories where local measurements suffice for tomography, efficient computations are also contained in AWPP. Moreover, we explicitly construct a theory in which the class of efficiently solvable problems exactly equals AWPP, showing this containment to be tight. We also investigate how simple physical principles bound the power of computational paradigms which combine computation and communication in a nontrivial fashion, such as interactive proof systems. Additionally, we show how some of the essential components of computational algorithms arise from certain natural physical principles. We use these results to investigate the relationship between interference behaviour and computational power, demonstrating that nontrivial interference behaviour is a general resource for postclassical computation. We then investigate whether postquantum interference is a resource for postquantum computation. Sorkin has defined a hierarchy of possible postquantum interference behaviours where, informally, the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multislit experiment. In quantum theory, at most pairs of paths can ever interact in a fundamental way. We consider how Grover's speedup depends on the order of interference in a theory, and show that, surprisingly, the quadratic lower bound holds regardless of the order of interference.
