Twisted Fiber Bundle Codes over Group Algebras

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 singular chain-compatible twists can increase the encoded dimension \(k\) at fixed blocklength \(n\), and in these finite examples the minimum distance \(d\) remains unchanged. This provides evidence that singular twisting enlarges the design space beyond the ordinary lifted product construction.