Title: Realizing efficient quantum error-correction with three-qubit codes

Abstract: 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.