Unified Framework for Quantum Code Embedding

Abstract: Given a Calderbank-Shor-Steane (CSS) code, it is sometimes necessary to modify the stabilizer code by adding an arbitrary number of physical qubits and parity checks. Motivations may include embedding a high-dimensional quantum low-density parity check (LDPC) code into finite-dimensional Euclidean space, or reducing the weights of an arbitrary CSS code. During this embedding, it is essential that the modified CSS code possesses an isomorphic set of logical qubits as the original CSS code. However, despite numerous explicit constructions, the conditions of when such a property holds true is 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.