  In this work, the efficient quantum error-correction protocol against the
general independent noise is constructed with the three-qubit 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. For any independent noise, the most
suitable error-correction protocol can be constructed based on the rules of
concatenation. The most significant aspect of using the concatenated
three-qubit codes is to realize quantum error-correction with the non-ideal
gate operations, because the error-correction schemes of the three-qubit codes
are simple and modular. For example, for the amplitude damping noise with the
initial channel fidelity 0.9, the effective channel fidelity can reach 0.94731
(or 0.961634 with the ideal quantum gate operations) when using the 4 levels
concatenated quantum error-correction with the three-qubit codes. The protocol
with the three-qubit codes needs 81 qubits and 366 quantum gate operations when
the accuracy rate is 0.9995. Meanwhile, when using the 3 levels concatenated
quantum error-correction with the five-qubit code, the effective channel
fidelity can only reach 0.922798 (or 0.975488 with the ideal quantum gate
operations). The protocol with the five-qubit code needs 125 qubits and 1147
quantum gate operations when the accuracy rate is 0.9995. The physical
resources costed is multiple of the physical resources costed by realizing the
three-qubit quantum error-correction, the protocol has a higher fault tolerance
threshold, and no increase in complexity. So, we believe it will be helpful for
realizing quantum error-correction in the physical system.
