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

TL;DR

The paper introduces a family of distance-3 non-CSS quantum error-correcting codes, BACs, derived from graph codes and measured via MBQC, to achieve fault-tolerant syndrome extraction with a single bare ancilla. A tailored parity-check matrix construction and a lookup-table decoder are used to correct hook errors, and the codes are analyzed under both anisotropic and depolarizing noise near circuit-level realistic conditions. The authors demonstrate that, for $6\le n\le 10$, BACs can outperform flag-qubit methods in many regimes (notably for $n\ge 8$ under depolarizing noise and $n\ge 7$ under anisotropic noise), while providing improved code-rate trade-offs in specific cases relative to prior work. These results advance resource-efficient FT quantum error correction by linking graph-code structures with measurement-based encoding and targeted syndrome permutations, with potential impact on near-term architectures.

Abstract

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.

Figures (11)

• Figure 1: A Cluster state, which is also depicted as an undirected graph $G=(V,E)$ where $V$ is the set of vertices corresponding to qubits, each initialized with $|{+}\rangle$ state and $E$ is the set of edges, each corresponding to a $\mathrm{CZ}$ gate.
• Figure 2: The message qubits ($\bullet$) are encoded into the cluster by measuring it in the $X$ basis.
• Figure 3: Due to $X$ error at various positions ($a_i$) in the ancilla qubit, every hook error $\rho_g(a_i)$ is color-coded accordingly for an ordered stabilizer $g =P_3P_4P_1P_2P_5, P \in \{X, Y, Z \}$.
• Figure 4: Comparision of Pseudo-threshold results for $[[n,1,3]]$ QECCs between bare and flag method. Note that for depolarizing noise of $[[7,1,3]]$ code, the threshold for logical error rate is always less than the physical error rate using the flag method.
• Figure 5: Pseudo-threshold results for $[[n,1,3]]$ QECCs using the bare method and flag method under depolarizing and anisotropic noise. * – threshold always less than physical error rate in that regime; $\dagger$ – threshold always greater than physical error rate in that regime.

Theorems & Definitions (8)

• Lemma 1
• proof
• Corollary 2
• proof
• Lemma 3
• proof
• Corollary 4
• proof