  We provide a systematic way of constructing entanglement-assisted quantum
error-correcting codes via graph states in the scenario of preexisting
perfectly protected qubits. It turns out that the preexisting entanglement can
help beat the quantum Hamming bound and can enhance (not only behave as an
assistance) the performance of the quantum error correction. Furthermore we
generalize the error models to the case of not-so-perfectly-protected qubits
and introduce the quantity infidelity as a figure of merit and show that our
code outperforms also the ordinary quantum error-correcting codes.
