Quantum Process Realization of LDPC Code Dualities and Product Constructions

TL;DR

This paper develops a unified quantum framework for realizing classical LDPC code dualities and product constructions using quantum processes, ZX-calculus, and operator algebra mappings, linking code theory with quantum phases.

Contribution

It introduces a systematic quantum circuit construction for classical code transformations, connecting them to quantum phases via operator algebra mappings and ZX-calculus.

Findings

Quantum processes realize code dualities and products.

ZX-calculus provides a diagrammatic representation and systematic algorithm.

Code transformations correspond to algebraic maps and phase transitions.

Abstract

We realize a broad class of code constructions, including Kramers-Wannier duality, tensor product, and check product, as quantum processes consisting of ancilla initialization, local unitaries, and projective measurements. Using ZX-calculus, we represent these transformations diagrammatically and provide a systematic algorithm for extracting quantum circuits. Central to our framework is the observation that the physical content of a classical LDPC code is captured by the operator algebra associated with its Tanner graph, and that code transformations correspond to maps between such algebras. Kramers-Wannier duality then admits a natural interpretation as gauging, while tensor and check products correspond to coupled-layer constructions in which interlayer coupling and projection implement a quotient on stacked operator algebras. Together, these results establish a unified framework connecting code transformations, quantum circuits, and mappings between distinct quantum phases of matter.