Several notions of code products are known in quantum error correction, such as hyper-graph 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 generalisation 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 on 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 specialise 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 depolarising noise. We compare the results with other codes of comparable block length and rate, including a code from the family of asymptotically good quantum Tanner codes. We observe that our reference product CSS code outperforms all other examined codes.
翻译:在量子错误校正中,人们知道一些代码产品的概念,例如高造产品、同质产品、脱装产品、平衡产品等等。在本文中,我们引入了一种新的产品编码结构,将古典产品编码自然地概括为量子编码:从一套组件Calderbank-Shor-Steane(CSS)代码开始,获得一个更大的CSS代码,其中美元对等检查和美元对等检查都与古典产品编码有关。我们从成分编码的属性(包括代码距离的界限)中推断出CSS产品编码的若干特性,并表明在衡平价检查中的内在冗余导致所谓的元校验,可以用来纠正综合读出错误。然后我们专门处理单价检查(SPC)产品编码的情况,在古典域中,单价是构建产品编码的共同选择。SPC的3美元倍产品CSS编码的逻辑误率模拟,其中含有参数[512,174,8],显示在最大的可能性下,对等价检查的结果进行所谓的元检查,包括将Cmock-crodecodeal 用于我们测试其他的Crocodex的系统,并进行其他标准的系统,以比较其他的系统标准,并进行。