Constructively Embed Quantum Error-Correcting Codes

The implementation of advanced quantum error-correcting codes (QECCs) is hindered by the mismatch between their non-local connectivity graphs and the geometric constraints of physical hardware. This work introduces a structured methodology to address this co-design challenge. We first employ manifold learning algorithms to extract the intrinsic geometric structure of QECCs, comparing Multidimensional Scaling (MDS) and Spectral Embedding. We develop an automated pipeline that translates a code's Tanner graph into a manufacturable hardware layout through three stages: continuous embedding, grid assignment, and path routing. Our flow supports both rectilinear and octilinear grid geometries, reflecting different fabrication constraints. To evaluate the results, we introduce four performance metrics–including total grid distance (W), SWAP gate overhead bounds (LB/UB), and a parallel-time bound (ParLB)–that together provide a multi-faceted assessment of hardware cost. This work establishes a foundational toolkit for assessing the implementability of quantum codes, enabling a more direct path from algorithmic design to physical realization.