  Quantum error-correcting codes are used to protect quantum information from
decoherence. A raw state is mapped, by an encoding circuit, to a codeword so
that the most likely quantum errors from a noisy quantum channel can be removed
after a decoding process.
  A good encoding circuit should have some desired features, such as low depth,
few gates, and so on. In this paper, we show how to practically implement an
encoding circuit of gate complexity $O(n(n-k+c)/\log n)$ for an $[[n,k;c]]$
quantum stabilizer code with the help of $c$ pairs of maximally-entangled
states. For the special case of an $[[n,k]]$ stabilizer code with $c=0$, the
encoding complexity is $O(n(n-k)/\log n)$, which is previously known to be
$O(n^2/\log n)$. For $c>0,$ this suggests that the benefits from shared
entanglement come at an additional cost of encoding complexity.
  Finally we discuss decoding of entanglement-assisted quantum stabilizer codes
and extend previously known computational hardness results on decoding quantum
stabilizer codes.
