Unified framework for quantum code embedding
Phys. Rev. A 113, 022438 – Published 23 February, 2026
DOI: https://doi.org/10.1103/gtlz-3q9d
Abstract
Given a Calderbank-Shor-Steane code, it is sometimes necessary to modify the code by adding an arbitrary number of physical qubits and parity checks. Motivations may include concatenating codes, embedding low-density parity check codes into finite-dimensional Euclidean space, or reducing the weights of parity checks. During this embedding, it is essential that the modified code possesses an isomorphic set of logical qubits as the original code. However, despite numerous explicit constructions, the conditions under which such a property holds true are not known in general. Therefore, using the language of homological algebra, we provide a unified framework that guarantees a natural isomorphism between the output and input codes. In particular, we explicitly show how previous works fit into our framework.
