Lectura Tesi Doctoral

Títol: [QUANTUM ANALOGS OF CLASSICAL CODES] (https://doctorat.upc.edu/en/doctoral-thesis/deposit-and-assessment/theses-authorised-for-defence# "‌")

Doctorand:  RICARD VILAR ALGUERÓ

Director:      SIMEON MICHAEL BALL MARKS

The main focus of this thesis are stabilizer codes, a type of error-correcting code used to correct quantum information that has been corrupted by noise. We introduce several new general constructions of stabilizer codes. In particular we use one of the constructions to construct quantum cyclic redundancy check codes, an error-correcting code which is used in classical information to correct burst errors. We show how to use a quantum version of such codes to correct burst errors on systems of quantum bits. We include a geometric description of stabilizer codes, extending previous constructions which work only for the qubit case to quantum systems in which the quantum particles have local dimension p, where p is any prime number. Finally, we reduce the problem of ascertaining when a generalised Reed-Solomon code is contained in its Hermitian dual and therefore can be used to construct a stabilizer code. This reduction allows us to determine the shortest and longest length of generalised Reed-Solomon codes which are contained in their Hermitian dual, verifying a conjecture of Grassl and Rotteler.