The physical Church-Turing thesis and the principles of quantum theory

TL;DR

This paper explores the conditions under which quantum theory could violate the physical Church-Turing thesis, extending Gandy's classical framework to quantum physics and highlighting the relationship between physical principles and computability.

Contribution

It provides a quantum version of Gandy's theorem, establishing when quantum physics aligns with or breaches the physical Church-Turing thesis based on physical postulates.

Findings

Quantum theory can breach the physical Church-Turing thesis under certain conditions.

A quantum analogue of Gandy's theorem is formulated and proved.

The work reveals a formal link between physical symmetries and computability assumptions.

Abstract

Notoriously, quantum computation shatters complexity theory, but is innocuous to computability theory. Yet several works have shown how quantum theory as it stands could breach the physical Church-Turing thesis. We draw a clear line as to when this is the case, in a way that is inspired by Gandy. Gandy formulates postulates about physics, such as homogeneity of space and time, bounded density and velocity of information --- and proves that the physical Church-Turing thesis is a consequence of these postulates. We provide a quantum version of the theorem. Thus this approach exhibits a formal non-trivial interplay between theoretical physics symmetries and computability assumptions.