Unitary Evolution on a Phase Space with Power of Two Discretization

TL;DR

This paper constructs quantum evolution operators on discretized phase spaces that realize the metaplectic representation of the modular group, with applications in noncommutative geometry and quantum algorithms.

Contribution

It introduces a novel construction of quantum evolution operators on discretized phase spaces that implement the metaplectic representation of SL(2,Z_2^n).

Findings

Operators act naturally on non-commutative 2-torus coordinates.

Potential applications in noncommutative field theories.

May enable more efficient quantum algorithms.

Abstract

We construct quantum evolution operators on the space of states, that realize the metaplectic representation of the modular group SL(2,Z_2^n). This representation acts in a natural way on the coordinates of the non-commutative 2-torus and thus is relevant for noncommutative field theories as well as theories of quantum space-time. The larger class of operators, thus defined, may be useful for the more efficient realization of new quantum algorithms.