Generalized "Square roots of Not" matrices, their application to the unveiling of hidden logical operators and to the definition of fully matrix circular Euler functions

TL;DR

This paper generalizes the square root of the logical Not operator to matrices of arbitrary size, explores their applications in quantum and classical logic, and develops matrix-based Euler functions and circular functions.

Contribution

It introduces a general matrix form of the square root of Not, extending quantum logic operators to broader contexts and deriving matrix Euler and circular functions.

Findings

Derived general expressions for square roots of Not matrices.

Extended Deutsch's algorithm to non-quantum matrix domains.

Developed matrix versions of Euler expansions and circular functions.

Abstract

The square root of Not is a logical operator of importance in quantum computing theory and of interest as a mathematical object in its own right. In physics, it is a square complex matrix of dimension 2. In the present work it is a complex square matrix of arbitrary dimension. The introduction of linear algebra into logical theory has been enhanced in recent decades by the researches in the field of neural networks and quantum computing. Here we will make a brief description of the representation of logical operations through matrices and we show how general expressions for the two square roots of the Not operator are obtained. Then, we explore two topics. First, we study an extension to a non-quantum domain of a short form of Deutsch's algorithm. Then, we assume that a root of Not is a matrix extension of the imaginary unit i, and under this idea we obtain fully matrix versions for the Euler expansions and for the representations of circular functions by complex exponentials.