Introduction

1 Introduction

A quantum error control code is a subspace $Q$ of a the ambient state space $Cd$ . We usually want more structure to simplify the search for good quantum codes and to ease the design of encoding algorithms. The first conceptual simplification is done by introducing an abstract error group $E$ , a finite group with a representation of unitary $d ×d$ matrices that allows one to express an arbitrary error as a linear combination of the representing matrices of $E$ .

We exploit the group structure of an abstract error group $E$ to define quantum codes. Let $N$ be a normal subgroup of $E$ and $χ$ an irreducible character of $N$ . A Clifford code (or better Knill code) for $(N,χ)$ is a subspace of $Cd$ that is given by the image of an orthogonal projector

χ(1)∑ -1 P = |N | χ(n )ρ(n). n∈N

(1)

The definition allows one to elegantly describe the detectable errors in representation theoretic terms, as was noted by Knill. In principle, it is thus possible to compute the most important parameters of the code, but it is computationally too hard in many cases.

A special case of this definition is of particular interest. If $N$ is an abelian normal subgroup of $E$ and $χ$ an irreducible character, then (1) is said to define a stabilizer code. A nonzero stabilizer code is then a joint eigenspace $Fix(N ) = {v ∈ Cd |ρ(n )v = χ(n)v for all n ∈ N }$ .

Efficient Decoherence Control Algorithms

1 Introduction