Quantum Error Correction and Orthogonal Geometry

A quantum error-correcting code is a way of encoding quantum states into qubits (two-state quantum systems) so that error or decoherence in a small number of individual qubits has little or no effect on the encoded data. The existence of quantum error-correcting codes was discovered only recently [1]. Although the subject is relatively new, a large number of papers on quantum error correction have already appeared. Many of these describe specific examples of codes [1–9]. However, the theoretical aspects of these papers have been concentrated on properties and rates of the codes [7,10–12], rather than on recipes for constructing them. This Letter introduces a unifying framework which explains all the codes discovered to date and greatly facilitates the construction of new examples. The basis for this unifying framework is group theoretic. It rests on the structure of certain finite subgroups E , L in Os2nd and E0 , L0 in Us2nd [13]. Since the natural setting for quantum mechanics is complex space, it might appear more appropriate to focus on the complex groups E0 and L0. However, we shall begin by discussing the real groups E and L, since their structure is easier to understand and they are sufficient for the construction of the known quantum error-correcting codes. We will first construct the subgroup E of Os2nd. This group E provides a bridge between quantum error-correcting codes in Hilbert space and binary orthogonal geometry. We then construct the larger subgroup L , Os2nd as the normalizer of E. The group E is the group of tensor products 6w1 ≠ . . . ≠ wn where each wj is either the identity or one of the Pauli matrices sx , sy or sz applied to the jth qubit. Mathematically, the group E is realized as an irreducible group of 2112n orthogonal 2n 3 2n matrices. The center of E, JsEd, is h6Ij and the group E has the extraspecial property that Ē ­ EyJsEd is elementary Abelian (hence a binary vector space). Let V denote the vector space Z2 (where Z2 ­ h0, 1j) and label the standard basis of R2 n