Unified Framework for Quantum Code Embedding

TL;DR

This paper introduces a unified homological algebra framework for modifying CSS quantum codes while preserving logical qubits, enabling various code transformations with guaranteed isomorphism.

Contribution

It provides a general theoretical framework that unifies previous methods for code modification, ensuring logical qubit preservation.

Findings

Framework guarantees isomorphism between original and modified codes

Explicitly unifies previous code embedding techniques

Applicable to various code concatenation and embedding scenarios

Abstract

Given a Calderbank-Shor-Steane (CSS) 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 (LDPC) 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 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.