Several notions of code products are known in quantum error correction, such as hypergraph products, homological products, lifted products, balanced products, to name a few. In this paper we introduce a new product code construction which is a natural generalization of classical product codes to quantum codes: starting from a set of component Calderbank-Shor-Steane (CSS) codes, a larger CSS code is obtained where both $X$ parity checks and $Z$ parity checks are associated to classical product codes. We deduce several properties of product CSS codes from the properties of the component codes, including bounds to the code distance, and show that built-in redundancies in the parity checks result in so-called meta-checks which can be exploited to correct syndrome read-out errors. We then specialize to the case of single-parity-check (SPC) product codes which in the classical domain are a common choice for constructing product codes. Logical error rate simulations of a SPC $3$-fold product CSS code having parameters $[[512,174,8]]$ are shown under both a maximum likelihood decoder for the erasure channel and belief propagation decoding for depolarizing noise. We compare the results with other codes of comparable length and dimension, including a code from the family of asymptotically good Tanner codes. We observe that our reference product CSS code outperforms all the other examined codes.

中文翻译：

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量子 Calderbank-Shor-Steane 码的经典乘积码构造


量子纠错中已知代码乘积的几种概念，例如超图乘积、同调乘积、提升乘积、平衡乘积等。在本文中，我们介绍了一种新的乘积代码构造，它是经典乘积代码到量子代码的自然推广：从一组组件 Calderbank-Shor-Steane (CSS) 代码开始，获得更大的 CSS 代码，其中 $X$奇偶校验和 $Z$ 奇偶校验与经典产品代码相关联。我们从组件代码的属性中推导出产品 CSS 代码的几个属性，包括代码距离的界限，并表明奇偶校验中的内置冗余会导致所谓的元检查，可用于纠正综合症读取-输出错误。然后，我们专门研究单奇偶校验（SPC）产品代码的情况，它在经典领域是构建产品代码的常见选择。具有参数 $[[512,174,8]]$ 的 SPC $3$ 倍产品 CSS 代码的逻辑错误率模拟在擦除通道的最大似然解码器和去极化噪声的置信传播解码下显示。我们将结果与具有可比长度和尺寸的其他代码进行比较，包括来自渐近良好 Tanner 代码系列的代码。我们观察到，我们的参考产品 CSS 代码优于所有其他检查过的代码。