Clifford codes are a class of quantum error control codes that form a natural generalization of stabilizer codes, These codes were introduced in 1996 by Knill, but only a single Clifford code was k...
Clifford codes are a natural generalization of quantum stabilizer codes based primarily on representation theory. This class of codes has previously been extended to the setting of quantum subsystem codes. We formulate a two-fold generalization of Clifford codes, for both the hybrid classical and quantum information and projective representation theory settings. This leads to new classes of hybrid subspace and subsystem Clifford codes. We extend the fundamental representation theoretic quantum error correction theorem to include these codes, based on the operator algebra quantum error correction framework. We also discuss several examples throughout the presentation, of both stabilizer and non-stabilizer type.
Quantum error control codes allow to detect and correct errors that are due to decoherence effects. We review some basic properties of these codes and give some constructions. Our main focus will be on a construction of quantum error control codes that have been introduced by Knill in 1996 with the intention to generalize stabilizer codes. These so-called Clifford codes can be constructed and analyzed with tools from representation theory of finite groups. We show that a large class of Clifford codes are actually stabilizer codes. And we construct the smallest example of a Clifford code that is not a stabilizer code.