Quantum Process Realization of LDPC Code Dualities and Product Constructions

We present a framework that elevates transformations between low-density parity-check (LDPC) codes—including Kramers–Wannier duality and code-product constructions—into explicit quantum processes connecting distinct physical phases. Using ZX-calculus, we construct a gauging map that implements the Kramers–Wannier duality directly from a code's Tanner graph as a quantum process composed of ancilla initialization, unitary circuit, and projective measurements. The ancilla and measurement overhead are dictated by code symmetries and redundancies. Extending this perspective, we relate tensor and check-product constructions to coupled-layer constructions, revealing a unified physical interpretation of LDPC code operations as structured entangling transformations between quantum phases.