Quantum error-correcting code for higher dimensional systems can, in general, be directly constructed from the codes for qubit systems. What remains unknown is whether there exist efficient code design techniques for higher dimensional systems. In this paper, we propose a 7-qutrit error-correcting code for the ternary quantum system and show that this design formulation has no equivalence in qubit systems. This code is optimum in the number of qutrits required to correct a single error while maintaining the CSS structure. This degenerate CSS code can (i) correct up to seven simultaneous phase errors and a single bit error, (ii) correct two simultaneous bit errors on pre-defined pairs of qutrits on eighteen out of twenty-one possible pairs, and (iii) in terms of the cost of implementation, the depth of the circuit of this code is only two more than that of the ternary Steane code. Our proposed code shows that it is possible to design better codes explicitly for ternary quantum systems instead of simply carrying over codes from binary quantum systems.

中文翻译：

---

利用简并性构造良好的三元量子纠错码

高维系统的量子纠错码通常可以直接从量子位系统的码中构造出来。仍然未知的是是否存在用于更高维系统的有效代码设计技术。在本文中，我们为三元量子系统提出了一个 7-qutrit 纠错码，并表明该设计公式在量子位系统中没有等效性。此代码在保持 CSS 结构的同时纠正单个错误所需的 qutrit 数量是最佳的。这种退化的 CSS 代码可以 (i) 纠正多达 7 个同时相位错误和一个单比特错误，(ii) 在 21 个可能的对中的 18 个上纠正预定义的 qutrit 对上的两个同时比特错误，以及 (iii)在实施成本方面，该码的电路深度仅比三元Steane码多两倍。我们提出的代码表明，可以为三元量子系统明确设计更好的代码，而不是简单地从二进制量子系统继承代码。