Minimal-Memory, Noncatastrophic, Polynomial-Depth Quantum Convolutional Encoders

Abstract

Abstract-Quantum convolutional coding is a technique for encoding a stream of quantum information before transmitting it over a noisy quantum channel. Two important goals in the design of quantum convolutional encoders are to minimize the memory required by them and to avoid the catastrophic propagation of errors. In a previous paper, we determined minimal-memory, non-catastrophic, polynomial-depth encoders for a few exemplary quantum convolutional codes. In this paper, we elucidate a general technique for finding an encoder of an arbitrary quantum convolutional code such that the encoder possesses these desirable properties. We also provide an elementary proof that these encoders are non-recursive. Finally, we apply our technique to many quantum convolutional codes from the literature.

New optimal quantum convolutional codes

“…However, we acknowledge the fact that the notion of memory of quantum convolutional code has been rede¯ned recently, in order to make the concept more natural. 23 The degree of Q is de¯ned as…”

“…In Ref. 23, the authors have constructed minimalmemory encoders for quantum convolutional codes. They have introduced a new de¯nition of quantum memory for quantum convolutional codes, which is more natural than the previous de¯nitions.…”

The Construction and Viterbi Decoding of New (2k, k, l) Convolutional Codes

“…1 with (4) and (5), we construct five (6, 3, l) convolutional codes and four (8,4, l) convolutional codes. Their BER performances are shown in the real line of Fig.…”

“…Currently, convolutional LDPC codes [6,7] have become the new hot topic, as they can obtain a good cost performance when implementing the Belief Propagation (BP) algorithm. In addition, the quantum convolutional codes used in quantum communication have also become absorbing quantum error correcting codes [8].…”

“…However, even if k is not too large, as a result of the multiplier effect, the growth of memory length of the (2k, k, l) convolutional codes is more significant than those of the (2, 1, l) convolutional codes. So, we will pick out (6,3) and (8,4), which are two double loop cyclic codes, as embedded codes in [10] to construct (6, 3, l) and (8, 4, l …”