Cobordism Constraints for Topological Quantum Memories via Partial Framings

This project links partially framed manifolds and quaternionic cobordism theory with topological quantum error-correcting (TQEC) codes. It models stabilizer codes as partially framed surface bundles encoding boundary types, twists, and defect framings, and uses cobordism invariants—such as signature divisibility, Stiefel–Whitney constraints, and quaternionic refinements—to prove universal trade-offs between logical qubits (k), code distance (d), and resources like area, genus, and crosscaps. The novelty is a cobordism-theoretic obstruction theory for code parameters and defect patterns that extends existing homological bounds by incorporating framing and non-orientability data. Constructively, it proposes new code families derived from quaternionic cobordism classes, predicting parameter regimes unattainable by standard toric or surface codes. This framework unifies manifold topology, partial framings, and code design, potentially guiding hardware-specific layouts on exotic substrates.

References:

1. Algebraic Topology Principles behind Topological Quantum Error Correction. Xiang Zou, Hoi-Kwong Lo (2025).