Simple quantum error-correcting codes

Abstract

Methods of finding good quantum error correcting codes are discussed, and many example codes are presented. The recipe C ⊥ 2 ⊆ C 1 , where C 1 and C 2 are classical codes, is used to obtain codes for up to 16 information qubits with correction of small numbers of errors. The results are tabulated. More efficient codes are obtained by allowing C 1 to have reduced distance, and introducing sign changes among the code words in a systematic manner. This systematic approach leads to single-error correcting codes for 3, 4 and 5 information qubits with block lengths of 8, 10 and 11 qubits respectively.

Entanglement purification and quantum error correction

“…Refs. [10,11,12] for reviews, and also [2,13,14,15,16,17,18,19,20,21]), quantum information is protected by encoding one logical qubit of quantum information into several physical qubits, or more generally k logical qubits into n physical qubits. The basic idea of such a redundant encoding is borrowed from classical coding and error correction, although additional requirements have to be met for quantum error correction, e.g.…”

“…For a more comprehensive treatment of quantum error correction we refer the reader for example to Refs. [10,11,12] (see also [2,13,14,15,16,17,18,19,20,21]). We concentrate on theoretical aspects of entanglement purification and its applications, where our main focus lies on the work which has been carried out by our group in the last couple of years.…”

Quantum serial turbo-codes

“…Lemma 3: Given a seed transformation , let (44) For any element of , there exists a unique element in , such that there is an edge of physical weight from to .…”

“…Quantum information and quantum error correction [4], [17], [23], [41], [44] are much younger theories and differ from their classical cousins in many aspects. For instance, there exists a quantum analogue of the Shannon channel capacity called the quantum channel capacity [12], [26], [40], which sets the maximum rate at which quantum information can be sent over a noisy quantum channel.…”

Asymptotically good quantum codes

“…Several methods were proposed to construct quantum codes, see, e.g. [4], [5], [6], [8], [10], [15], [17], [18], [19], [20]. However, when n grows for a fixed R > 0, the relative minimum distance δ of all these codes tends to zero.…”