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.