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

TL;DR

This paper explores how fundamental quantum computational mechanisms like phase kick-back are present in all physical theories with information, linking interference phenomena to computational power and proposing new avenues for understanding post-quantum computation.

Contribution

It demonstrates that key quantum computational tools exist in any operational theory with information, and connects interference behavior to computational capabilities beyond quantum theory.

Findings

Reversible controlled transformations are universal in operational theories.

Non-trivial interference is a resource for post-classical computation.

Post-quantum interference relates to higher-order interference and new particle types.

Abstract

The advent of quantum computing has challenged classical conceptions of which problems are efficiently solvable in our physical world. This motivates the general study of how physical principles bound computational power. In this paper we show that some of the essential machinery of quantum computation -- namely reversible controlled transformations and the phase kick-back mechanism -- exist in any operational-defined theory with a consistent notion of information. These results provide the tools for an exploration of the physics underpinning the structure of computational algorithms. We use these results to 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 post-quantum interference -- the higher-order interference of Sorkin -- to the existence of post-quantum particle types, potentially providing a novel experimental test for higher-order interference. Finally, we conjecture that theories with post-quantum interference can solve problems intractable even on a quantum computer.