Twisted Fiber Bundle Codes over Group Algebras

TL;DR

This paper introduces a new twisted fiber-bundle construction for quantum CSS codes over group algebras, extending existing codes and demonstrating potential improvements in logical qubits.

Contribution

It develops a novel twisted fiber-bundle framework for quantum CSS codes over group algebras, generalizing lifted product codes and enabling performance enhancements.

Findings

Invertible twists preserve code parameters and are chain-isomorphic to untwisted codes.

Singular twists can reduce boundary ranks and increase logical qubits.

Examples over _3 show improved logical qubits without changing blocklength or minimum distance.

Abstract

We introduce a twisted fiber-bundle construction of quantum CSS codes over group algebras \(R=\mathbb F_2[G]\), where each base generator carries a generator-dependent \(R\)-linear fiber twist satisfying a flatness condition. This construction extends the untwisted lifted product code, recovered when all twists are identities. We show that invertible twists (satisfying a flatness condition) give a complex chain-isomorphic to the untwisted one, so the resulting binary CSS codes have the same blocklength \(n\) and encoded dimension \(k\). In contrast, singular chain-compatible twists can lower boundary ranks and increase the number of logical qubits. Examples over \(R=\mathbb F_2[D_3]\) show that the twisted fiber bundle code can outperform the corresponding untwisted lifted-product code in \(k\) while keeping the same \(n\) and, in our examples, the same minimum distance \(d\).