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An optimal quantum error-correcting procedure using quantifier elimination

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Abstract

Quantum communication channels suffer from various noises, which are mathematically modeled by error super-operators. To combat these errors, it is necessary to design recovery super-operators. We aim to construct the optimal recovery that maximizes the minimum fidelity through the noisy channel. It is typically a MAX–MIN problem, out of the scope of convex optimization. Compared to existing methods, our method is exact and complete by a reduction to quantifier elimination over real closed fields in a fragment of two alternative quantifier blocks. Finally, the complexity is shown to be in EXP.

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Notes

  1. It refers to polynomial with rational coefficients. As all entries in the input \(\mathscr {E}\) are supposed to be rational, we would simply write ‘polynomial’ for ‘\(\mathbb {Q}\)-polynomial,’ unless it is specified otherwise.

  2. We prefer to encode the square of the fidelity here, rather than the fidelity, since the latter is generally expressed as a square root of an SOS polynomial. The order between fidelities could be correspondingly replaced with the order between their squares.

  3. Encoding complex entries amounts to encoding their real and imaginary parts, which are clearly real. So it is generic to tackle real algebraic numbers \(\Omega \) only.

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Acknowledgements

The authors thank the anonymous reviewers whose insightful comments resolve inconsistencies. The authors are also grateful to Yuan Feng for his valuable suggestions and Jianling Fu for her careful proof-reading. M. Xu is supported by the National Natural Science Foundation of China (No. 11871221), the National Key R&D Program of China (No. 2018YFA0306704) and the Research Funds of Happiness Flower ECNU (No. 2020ECNU-XFZH005). Y. Deng is supported by the National Natural Science Foundation of China (Nos. 61832015, 62072176) and the Inria-CAS joint project Quasar.

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  1. Shanghai Key Lab of Trustworthy Computing, East China Normal University, Shanghai, 200062, China

    Ying-Ji Sun, Ming Xu & Yuxin Deng

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  1. Ying-Ji Sun
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Correspondence to Ming Xu.

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Sun, YJ., Xu, M. & Deng, Y. An optimal quantum error-correcting procedure using quantifier elimination. Quantum Inf Process 20, 170 (2021). https://doi.org/10.1007/s11128-021-03109-w

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