Fault-tolerant syndrome extraction in [[n,1,3]] non-CSS code family generated using measurements on graph states

The reliability of quantum computation critically depends on the performance of quantum error-correcting codes (QECCs), which can be severely degraded by hook errors that reduce the effective code distance. In this work, we construct a family of $[[n,1,3]]$ non-CSS QECCs to achieve fault-tolerant (FT) syndrome measurement, where $ 6 \leq n \leq 10$. We employ the bare-ancilla method of Muyuan Li \emph{et al.} to demonstrate fault tolerance in the presence of hook errors during syndrome extraction. We present a systematic protocol for generating these QECCs using graph codes. Using a custom lookup-table decoder, we simulate the code's performance under both anisotropic and circuit-level depolarizing noise. Our results reveal a trade-off in performance with respect to the code rate and identify optimized codes under these noise models. We benchmark our results against the infamous flag-qubit method of Chao \emph{et al.}. Notably, we introduce a code with improved code rate while maintaining the same distance as the work of Muyuan Li \emph{et al.} Our approach facilitates the identification and construction of a family of distance three FT non-CSS QECCs.