A further study on the encoding complexity of quantum stabilizer codes

In this paper, we investigate the encoding complexity of binary quantum stabilizer codes. When doing the encoding through a “standard generator matrix”, a tight upper bound of the encoding complexity is derived in this paper to indicate that the encoding complexity decreases quadratically as the number r1 of primary generators of the stabilizer group decreases. A class of equivalent transformations on stabilizer codes is explored to reduce the number r1 of primary generators. The minimum possible r1 is determined for several classes of optimal stabilizer codes of distance two or three and for some codes of length n ≤ 12. It appears that a code with large minimum distance will have large r1, reflecting high encoding complexity.