Encoding Subsystem Codes

In this paper we investigate the encoding of operator quantum error correcting codes i.e. subsystem codes. We show that encoding of subsystem codes can be reduced to encoding of a related stabilizer code making it possible to use all the known results on encoding of stabilizer codes. Along the way we also show how Clifford codes can be encoded. We also show that gauge qubits can be exploited to reduce the encoding complexity. Introduction. In this paper we investigate encoding of subsystem codes. Our main result is that encoding of a subsystem code can be reduced to the encoding of a related stabilizer code, thereby making use of the previous theory on encoding stabilizer codes [2–4]. We shall prove this in two steps. First, we shall show that Clifford codes can be encoded using the same methods used for stabilizer codes. Secondly, we shall show how these methods can be adapted to encode Clifford subsystem codes. Since subsystem codes subsume stabilizer codes, noiseless subsystems and decoherence free subspaces, these results imply that we can essentially use the same methods to encode all these codes. In fact, while the exact details were not provided, it was suggested in [10] that encoding of subsystem codes can be achieved by Clifford unitaries. Our treatment is comprehensive and gives proofs for all the claims. Subsystem codes can potentially lead to simpler error recovery schemes. In a similar vein, they can also simplify the encoding process, though perhaps not as dramatically1. These simplifications have not been investigated In general, decoding is usually of greater complexity than encoding and for this reason it is often neglected in comparison. This parallels the classical case where also the decoding is studied much more extensively than encoding.