  Known quantum error correction schemes are typically able to take advantage
of only a limited class of classical error-correcting codes.
Entanglement-assisted quantum error correction is a partial solution which made
it possible to exploit any classical linear codes over the binary or quaternary
finite field. However, the known entanglement-assisted scheme requires
noiseless qubits that help correct quantum errors on noisy qubits, which can be
too severe an assumption. We prove that a more relaxed and realistic assumption
is sufficient by presenting encoding and decoding operations assisted by qubits
on which quantum errors of one particular kind may occur. As in entanglement
assistance, our scheme can import any binary or quaternary linear codes. If the
auxiliarly qubits are noiseless, our codes become entanglement-assisted codes,
and saturate the quantum Singleton bound when the underlying classical codes
are maximum distance separable.
