Spacetime Layout and Logical Compilation of Color Code

As quantum computing moves toward utility-scale fault tolerance, the gap between abstract logical algorithms and the physical hardware layout becomes a critical bottleneck. While surface codes have well-established compilation pipelines, the 2D color code—which offers distinct advantages like transversal gates—lacks a systematic, automated method for mapping logical operations onto physical spacetime volumes. This paper addresses this gap by developing a compilation framework that bridges the gap between high-level logical circuits and the topological constraints of the color code.

To solve this, the authors introduce a spacetime block-diagram representation where logical patches (prisms) and operations (pipes) are treated as geometric building blocks. These blocks are grounded in the anyon-condensation and domain-wall structure of the color code. The authors establish a formal correspondence between these geometric blocks and ZX diagrams, allowing them to treat logical compilation as a graph-transformation problem. They develop an automated pipeline that uses edge-decorated ZX diagrams and fusion-region-aware routing to convert arbitrary Clifford+T circuits into valid, optimized spacetime layouts that respect the color code's unique assembly rules.

• Universal Compilation: The framework supports a universal set of logical operations, including transversal Clifford gates and T-gate teleportation, by encoding them into the connectivity and domain-wall structure of the spacetime layout.
• Higher Connectivity: The color code's geometry allows for junctions of up to degree five, offering greater flexibility in layout design compared to the degree-four limit of surface-code architectures.
• Automated Synthesis: The authors demonstrate that their automated compilation strategy consistently reduces the logical spacetime volume required for a wide range of algorithms compared to traditional surface-code compilation techniques.