Chapter 1 Clifford Codes

Abstract:  Abstract 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. 1 2 1. CLIFFORD CODES 1.1 Introduction A quantum computer takes advantage of entangled states stored in the state of atoms, nuclear spins, photons, or other quantum systems. The interaction of the quantum computer with its environment leads to decoherence errors which alter the state of the memory. The protection of the memory against these errors is a crucial part in the construction of a resilient quantum computer. We describe in this chapter a generalization of stabilizer codes that allows to protect the encoded states.