Multi-valued quantum systems can store more information than binary ones for a given number of quantum states. For reliable operation of multi-valued quantum systems, error correction is mandated. In this paper, we propose a 5-qutrit quantum error-correcting code and provide its stabilizer formulation. Since 5 qutrits are necessary to correct a single error, our proposed code is optimal in the number of qutrits. We prove that the error model considered in this paper spans the entire $(3 \times 3)$ operator space. Therefore, our proposed code can correct any single error on the codeword. This code outperforms the previous 9-qutrit code in (i) the number of qutrits required for encoding, (ii) our code can correct any arbitrary $(3 \times 3)$ error, (ii) our code can readily correct bit errors in a single step as opposed to the two-step correction used previously, and (iii) phase error correction does not require correcting individual subspaces.

中文翻译：

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三元量子系统的最佳纠错码

对于给定数量的量子态，多值量子系统可以存储比二进制系统更多的信息。为了多值量子系统的可靠运行，必须进行纠错。在本文中，我们提出了一种 5-qutrit 量子纠错码并提供了其稳定剂配方。由于纠正单个错误需要 5 个 qutrit，因此我们提出的代码在 qutrit 数量上是最佳的。我们证明了本文考虑的误差模型跨越了整个 $(3 \times 3)$ 操作符空间。因此，我们提出的代码可以纠正码字上的任何单个错误。此代码在 (i) 编码所需的 qutrit 数量方面优于之前的 9-qutrit 代码，(ii) 我们的代码可以纠正任意 $(3 \times 3)$ 错误，