  In this work, the efficient quantum error-correction protocol against the
general independent noise is constructed with the three-qubit quantum
error-correction codes. The rules of concatenation are summarized according to
the error-correcting capability of the codes. The codes not only play the role
of correcting errors, but the role of polarizing the effective channel. The
performance of the protocol is dependent on the type of noise and the level of
concatenation. For the general Pauli noise, the threshold of the protocol is
0.91518 when against the depolarizing noise, while for the amplitude damping
noise it is about 0.849. For the general independent noise, the threshold of
the protocol is about 0.932, which is obtained through the numerical
simulations. If there is no limit on the level of concatenation, the
error-correction threshold can be much lower. The physical resources costed by
this protocol is multiple of the physical resources costed by realizing the
three-qubit quantum error-correction, with no increase in complexity. We
believe it will be helpful for realizing quantum error-correction in the
physical system.

Quantum Physics