Some Results about Quantum Codes
by
David G Glynn
University of Canterbury, Te Whare Wananga o Waitaha

The theory of quantum codes is a recent branch of coding theory that appears to be an essential prerequisite for operational quantum computers and communication systems. These codes are subspaces of certain Hilbert spaces over the complex numbers and they are designed to correct various qubit errors.

A construction of Calderbank, Rains, Shor and Sloane from several years ago obtained quantum codes from classical additive codes over GF(4) that are self-orthogonal with respect to the Hermitian product. Such classical codes that are also linear are precisely those which are even: all their words have even weight. Now we consider a correspondence between such codes and subsets of projective spaces over GF(4) having all hyperplanes intersecting in always an odd, or always an even number of points. These subsets define geometric codes C (over GF(2)), every codeword of which corresponds to an even linear code. Using the algebraic classification of all codewords of C we obtain a description of all these codes, as well as a formula for the number of non-equivalent even linear codes, depending upon the number of words in the code. This leads to the interesting question of whether the more general additive quantum codes can also be described algebraically.

Date received: November 10, 2000