ALGEBRAIC AND TOPOLOGICAL PERSPECTIVES ON POST-QUANTUM CRYPTOGRAPHY AND QUANTUM ERROR CORRECTION

Abstract

Post-quantum cryptography (PQC) and fault-tolerant quantum computing are conventionally treated as disjoint engineering disciplines. This paper establishes a disciplined algebraic and topological comparison between them and reports completed, reproducible results. We address three research questions: (i) which algebraic structures genuinely recur across both domains; (ii) where formally similar expressions serve divergent operational purposes; and (iii) what concrete, actionable conclusions follow. We provide a transparent literature-selection account and a comparison table contrasting prior reviews with the specific gap addressed here. Two fully worked examples are developed—a Module-LWE instance over and a homological surface code with explicit maps, kernels, quotient spaces, and distance metrics—and we prove a structural correspondence demonstrating that the two quotient constructions constitute the same inverse problem with inverted design objectives. A synthesis table separates exact isomorphisms, useful structural analogies, and relationships that do not hold. We then apply the framework to a concrete co-design decision: a fault-tolerant cryptographic coprocessor in which PQC polynomial multipliers and QEC decoders must be decoupled. We report measured ML-KEM benchmarks, measured surface-code Monte Carlo results, and a measured coprocessor decoupling experiment, all generated by released code. The framework correctly predicts the observed decoupling requirement. We do not claim that the two domains are the same mathematical problem; rather, we identify precisely where their abstractions intersect and where they diverge.