Unitary Evolution on a Discrete Phase Space

TL;DR

This paper constructs unitary operators on a discretized phase space that realize the metaplectic representation of the modular group, with applications in non-commutative geometry, quantum field theories, and quantum algorithms.

Contribution

It introduces a new class of unitary evolution operators on power-of-two discretized phase spaces, connecting them to the metaplectic representation of SL(2,Z_{2^n}).

Findings

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

Potential applications in non-commutative field theories.

Useful for efficient quantum algorithm implementation.

Abstract

We construct unitary evolution operators on a phase space with power of two discretization. These operators realize the metaplectic representation of the modular group SL(2,Z_{2^n}). It acts in a natural way on the coordinates of the non-commutative 2-torus, T_{2^n}^2$ and thus is relevant for non-commutative field theories as well as theories of quantum space-time. The class of operators may also be useful for the efficient realization of new quantum algorithms.