Performance of five-qubit code for arbitrary error channels

TL;DR

This paper demonstrates that the five-qubit quantum error correction code remains effective even when errors on different qubits are arbitrary and not identical, simplifying practical implementation by relying on initial channel fidelity.

Contribution

It shows the robustness of the five-qubit code against arbitrary, non-uniform errors and proposes a fidelity-based approach for error correction without full channel knowledge.

Findings

Five-qubit code works efficiently with arbitrary qubit errors.

Error correction can be based on initial channel fidelity.

Fidelity threshold is approximately 0.992 for effective correction.

Abstract

In quantum error correction, it is an important assumption that errors on different qubits are independent. In our previous work [Phys. Rev. A {\bf 92}, 052320 (2015)], the generality of the concatenated five-qubit code has been investgated when noises of the principal system and the auxiliary environment are assumed to be the same. In the error correction with concatenated code, tiny differences (in fidelity or independent errors) between initial quantum channels may introduce different effective channels in the next level, and therefore, it is necessary to study a meaningful question: Does five-qubit code still work efficiently when errors on different qubits are different? In the present work, it is discovered that even errors for different qubits are arbitrary, the five-qubit code still works efficiently. Since it is much easier and more accurate to measure the fidelity, one can construct quantum error correction with five-qubit code according to the initial channel fidelity, and it is not necessary to know the complete information of the initial channel. Moreover, when the initial channel fidelity is below $0.992$, the fidelity threshold for five-qubit code is the fidelity of the effective channel after error correction in bit-flip channels.