We introduce a high-level graphical framework for designing and analysing quantum error correcting codes, centred on what we term the coherent parity check (CPC). The graphical formulation is based on the diagrammatic tools of the ZX-calculus of quantum observables. The resulting framework leads to a construction for stabilizer codes that allows us to design and verify a broad range of quantum codes based on classical ones, and that gives a means of discovering large classes of codes using both analytical and numerical methods. We focus in particular on the smaller codes that will be the first used by near-term devices. We show how CSS codes form a subset of CPC codes and, more generally, how to compute stabilizers for a CPC code. As an explicit example of this framework, we give a method for turning almost any pair of classical [n,k,3] codes into a [[2n−k+2,k,3]] CPC code. Further, we give a simple technique for machine search which yields thousands of potential codes, and demonstrate its operation for distance 3 and 5 codes. Finally, we use the graphical tools to demonstrate how Clifford computation can be performed within CPC codes. As our framework gives a new tool for constructing small- to medium-sized codes with relatively high code rates, it provides a new source for codes that could be suitable for emerging devices, while its ZX-calculus foundations enable natural integration of error correction with graphical compiler toolchains. It also provides a powerful framework for reasoning about all stabilizer quantum error correction codes of any size.

中文翻译：

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用于量子纠错设计和验证的图形结构


我们引入了一个用于设计和分析量子纠错码的高级图形框架，其核心是我们所说的相干奇偶校验（CPC）。图形公式基于量子可观测量的 ZX 演算的图形工具。由此产生的框架导致了稳定器代码的构建，使我们能够设计和验证基于经典代码的广泛量子代码，并提供了使用分析和数值方法发现大类代码的方法。我们特别关注近期设备将首先使用的较小代码。我们展示了 CSS 代码如何形成 CPC 代码的子集，以及更一般地说，如何计算 CPC 代码的稳定器。作为该框架的一个明确示例，我们给出了一种将几乎任何一对经典 [n,k,3] 代码转换为 [[2n−k+2,k,3]] CPC 代码的方法。此外，我们给出了一种简单的机器搜索技术，可以产生数千个潜在代码，并演示了其对距离 3 和 5 代码的操作。最后，我们使用图形工具来演示如何在 CPC 代码中执行 Clifford 计算。由于我们的框架提供了一种用于构建具有相对较高码率的中小型代码的新工具，因此它为适合新兴设备的代码提供了新的来源，而其 ZX 演算基础可以将纠错与图形编译器工具链。它还提供了一个强大的框架，用于推理任何大小的所有稳定器量子纠错码。