Discrete Quantum Theories

The idea of computable numbers is of fundamental significance in computer science and has had a significant impact on logic.  While this concept has been put forward a number of times, e.g., by Feynman, Landauer, etc.  most mathematical models and hence most physical
models, including models of quantum mechanics, depend on uncomputable numbers, namely the continuum of real (or complex) numbers, which is a non-trivial philosophical issue.

We have accordingly started a foundational study of quantum mechanics from a computational perspective, studying variants of quantum mechanics formulated with computable number systems, analyzing their computational power, and drawing comparisons with conventional
models based on uncomputable numbers, isolating and teasing apart various important aspects of quantum mechanics (order, inner product, metric, angles, geometry, probability, etc.) that are "encoded" in the usual fields of real and complex numbers.

We are able to employ a framework with computable numbers to exhibit several novel results that illuminate the mechanisms from which the power and capacity of quantum theory emerges, including, for example, the possibility of distinguishing the computational cost of the representation of states from the cost of measurement of observables.