MATRÓIDES E CÓDIGOS QUÂNTICOS

Abstract

Whitney identified the fundamental properties of dependence, which are common among graphs and matrices giving rise to Matroid Theory in 1935. In this work will be presented the construction of new families of Matroid and the resolution of theorems in a comprehensive and easy to understand, since the theme is defined in purely abstract mathematical form. Thus, to obtain a new Matroid one uses an already given one, being defined in terms of its independent sets. It should be noted that Matroid is found in the following ranges: in vector spaces, cycles in graphs, related functions, circuits, bases, rank, closure and duality. Thus, to construct a quantum code, one must understand quantum information and coding theory such as the Postulates of Quantum Mechanics, single- or multi-qubit states, unit operators, logic gates, quantum state measures, stabilizing codes, and the class of codes of Calderbank-Shor-Steane (CSS). The linear codes are the codes of Linear Algebra, vector subspaces defined on finite bodies. Quantum codes are intended to protect potential errors from a channel to detect and correct such errors. With the acquired theoretical foundation, the possibility of constructing the connection of the Matroid theory and the theory of quantum codification can be verified by means of the parity check matrix of a CSS code and the matrix that generates a given vector matroid.