Equivariant Cellular Obstructions and Classical Randomness Costs in Logical Transport Between CSS Codes

This work investigates equivariant cellular obstructions and classical randomness costs in logical quantum information transfer between CSS stabilizer codes. Using finite cylindrical cell complexes with four and two sectors, we establish an obstruction to homology-preserving equivariant chain maps, including within the derived category of modules over the relevant group algebra. Despite this obstruction, we construct a completely positive trace-preserving (CPTP) quantum channel that achieves perfect logical transfer while maintaining exact physical covariance. For a specified class of Clifford-based implementations, we prove that exact covariance requires at least four components and a minimum classical selection entropy of two bits. We also derive a necessary bound for approximately covariant implementations. The results clarify the distinction between topological obstructions and operational quantum covariance, providing a finite-dimensional framework connecting algebraic topology, quantum error correction, symmetry, and information-theoretic resource costs. This study builds upon two previous works: Work 1: https://doi.org/10.5281/zenodo.22557436 Work 2: https://doi.org/10.5281/zenodo.23428851 Author: Deivison Ramos do Rosario License: Creative Commons Attribution 4.0 International (CC BY 4.0).