Códigos correctores de errores cuánticos

Abstract

We could start our story talking about the revolution of quantum mechanics and the need to decipher the mystery that it creates, or we could start by talking about the birth of modern computation, which changed the world we live in in ways beyond repair. In any way, the gestation of quantum computation, unavoidable consequence of those events, led to a new and unsettling question: Is quantum computation the last link in the evolution towards the construction of efficient algorithms? In 1985, David Deutsch - the procurer in this confusing story - presented his idea of Universal Quantum Computer and showed the world the first quantum algorithm, which seemed to insinuate a greater efficiency of quantum computers over classical computers. In 1994, Peter Shor climbed on the back of Deutsch to find efficient algorithms for the problems of factorization and discrete logarithm in quantum computation, and in 1995 Lov Grover did the same with search algorithms. At the same time, and following the steps of Richard Feynman, it was shown that a quantum computer can efficiently simulate any classical computer. All the ingredients seemed to be on the table, but no one was able to cook the cake, and to this day, no one has been: despite all evidences there is no proof that quantum computation is more efficient than classical computation in all of its aspects. The subject we will develop in this memoir, although connected, will be slightly different from what we have already discussed. Rather than focusing on computation, we will follow another young branch of mathematics: information theory. The adventure of quantum information theory is short but intense. In 1995, Ben Schumacher defined the qubit, and announced a similar result to that of Claude Shannon in 1948 for noiseless channels. However, a result for the coding of channels in the presence of noise has not been found, although in the way we have found some interesting classes of quantum correcting codes which allow quantum computers to work in the presence of noise. Despite of its youth, many results have been found in the field of quantum information, such as superdense coding - which allows to send two bits of classical information using only one qubit -, distributed quantum computing -which shows that a network of computers need exponentially less communication to solve problems than classic computers - or the fact that two quantum channels with zero capacity may transmit information. All these results move away from our path, which will take us, however, from the bases of quantum computation to the theory of quantum correction codes, walking by the formalisation of quantum noise and the introduction of metrics between quantum states. By doing this, we will try to answer two questions. In the first place, is it possible to construct quantum correction codes which protect us against the action of noise when we broadcast quantum information? And secondly, do these codes offer any advantages over our well known classical codes?