Breakdown of AQEC Threshold in Imperfectly Prepared Topological Codes

Topological order provides a physical intuition for error correction with its definition implying Knill-Laflamme condition implicitly. Imperfect code space preparation is a fundamental noise in state preparation experimentally, but recent works imply that topological order (or at least long-range entanglement) may exists in such models. A natural question is that whether the imperfect state in such models still serves as robust approximate quantum error correction codes (AQEC) under such imperfect preparation in principle. However, with the recently raised AQEC relative entropy and by analyzing its behavior in different topological code models under thermodynamic limit, our work suggests that traditional intuition may fail in specific case, e.g. the long-range entangled state prepared by imperfect preparation fail to be a robust AQEC code. Moreover, the result also implies that this breakdown is strongly related to the choice of logical subspace, calling for more careful and subtle methods to deal with similar problems.