In this work, we present a generic approach to transform CSS codes by building upon their equivalence to phase-free ZX diagrams. Using the ZX calculus, we demonstrate diagrammatic transformations between encoding maps associated with different codes. As a motivating example, we give explicit transformations between the Steane code and the quantum Reed-Muller code, since by switching between these two codes, one can obtain a fault-tolerant universal gate set. To this end, we propose a bidirectional rewrite rule to find a (not necessarily transversal) physical implementation for any logical ZX diagram in any CSS code. Then we focus on two code transformation techniques: code morphing, a procedure that transforms a code while retaining its fault-tolerant gates, and gauge fixing, where complimentary codes can be obtained from a common subsystem code (e.g., the Steane and the quantum Reed-Muller codes from the [[15,1,3,3]] code). We provide explicit graphical derivations for these techniques and show how ZX and graphical encoder maps relate several equivalent perspectives on these code-transforming operations.

翻译：本文提出一种通用方法，通过利用CSS码与无相位ZX图的等价关系对其进行变换。借助ZX演算，我们展示了不同码对应的编码映射之间的图解变换。作为激励性示例，我们给出了Steane码与量子Reed-Muller码之间的显式变换，因为通过在这两种码之间切换，可以得到容错通用门集。为此，我们提出一种双向重写规则，用于为任意CSS码中的任何逻辑ZX图找到（不一定横向的）物理实现。随后聚焦于两种码变换技术：码变形（一种在保持容错门集的同时变换码型的流程）以及规范固定（通过该技术可从公共子系统中获得互补码，例如从[[15,1,3,3]]码获得Steane码和量子Reed-Muller码）。我们为这些技术提供显式图解推导，并展示ZX图与图解编码器映射如何关联这些码变换操作的若干等价视角。