Quantum Process Realization of LDPC Code Dualities and Product Constructions
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.
Cite
@article{arxiv.2603.13538,
title = {Quantum Process Realization of LDPC Code Dualities and Product Constructions},
author = {Shuhan Zhang and Deepak Aryal and Yi-Zhuang You},
journal= {arXiv preprint arXiv:2603.13538},
year = {2026}
}