High-performance syndrome extraction circuits for quantum codes

TL;DR

A fast and effective framework for analysing and designing syndrome-extraction circuits (SECs) based on left-right circuits, with consistent improvements in logical performance of up to an order of magnitude compared to existing single-ancilla SEC designs.

Abstract

We present a fast and effective framework for analysing and designing syndrome-extraction circuits (SECs). Our approach is based on left-right circuits, a general design for SECs which maintain low depth by staggering $X$ and $Z$ checks without interleaving gates. Initially proposed for specific classes of codes, we generalise this construction to arbitrary CSS codes and optimise the circuit structure to achieve low qubit idling time, large effective distance, and reduced minimum-weight failure mechanisms. A key component of our framework is the formal notion of residual errors and their associated distance metrics, which form lightweight tools for capturing error propagation and quantifying the potential harm of circuit-level errors. Applying our automated framework to diverse classes of codes, we observe consistent improvements in logical performance of up to an order of magnitude compared to existing single-ancilla SEC designs. We also use these tools to prove that no non-interleaving SEC can achieve circuit distance $12$ for the gross code, and identify an explicit circuit that we conjecture achieves distance $11$, exceeding previously known constructions.

Figures (12)

• Figure 1: The Tanner graph of a general quantum CSS code can be highly irregular, with logical operators lacking simple structure. A weight-1 error can propagate through the circuit into high-weight residual error (here, qubits $7$ and $8$) that overlaps a minimum-weight logical operator, effectively reducing the code distance. This is known as a hook error.
• Figure 2: Performance of circuits designed using our left--right circuit (LRC) framework and general QEC packages QUITSkang2025quantum and LDPCRoffe_LDPC_Python_tools_2022, where applicable. We compare SECs of a quantum Tanner code $[[200, 10, 10]]$radebold2025explicit, a hypergraph product code $[[625, 25, 8]]$TremblayEtAl2022kang2025quantum, a Haah's cubic code $[[128, 14, 8]]$haah2011fractal and a novel fiber bundle code $[[126, 8, 9]]$ that we construct following methods in hastings2021fiberbundlecodes. Most of the error bars are comparable in size to the markers, and therefore, not visible. A Python package for generating high-performance left--right circuits for arbitrary CSS codes is publicly available on https://github.com/PurePhys/LR-circuits. The exact Stim circuits are available on https://doi.org/10.5281/zenodo.18853600.
• Figure 3: Left–right syndrome-extraction circuit. Data qubits are partitioned into left/right sets, inducing $H_X=[L_X\,|\,R_X]$ and $H_Z=[L_Z\,|\,R_Z]$. CNOTs for $L_X$ and $R_Z$ are executed in parallel for $t_1$ time steps, then $L_Z$ and $R_X$ for $t_2$ time steps. Ancilla preparations/measurements are staggered so all $Z$-check CNOTs precede all $X$-check CNOTs of the same round, ensuring validity and leading to the total circuit depth $t_1+t_2+2$.
• Figure 4: Each operation in the circuit involves at most one data qubit, yet an error can propagate to multiple data qubits through subsequent steps. For instance, a $Z$-error on the ancilla after a CNOT gate propagates to every data qubit involved in later CNOT operations. For the given check, the set of all residual errors is $\{\{3\}, \{2,3\}, \{1, 2, 3\}\}$. Here, we display the residual error $\{2,3\}$.
• Figure 5: A guide for the proof of Proposition \ref{['propos:circ_dist_bound']}. Regardless of the CNOT network, a non-trivial circuit-level logical operator must have a support $S \in \ker (H_X) \setminus \operatorname{span}(H_Z)$ on data qubits at the end of the circuit. This support only contains singular data qubit errors or multi-qubit residual errors. Note that they may overlap and partially cancel out each other. Finally, note that the indicated hook error (red) is equivalent to a two-qubit $ZZ$ error after the subsequent CNOT. Hence, any CNOT error on data/$Z$-ancilla pair is equivalent to a single-qubit hook error (this was also observed in shaw2025lowering).

Theorems & Definitions (18)

• Proposition 1: HGP circuit depth
• proof
• Definition 1: Residual distance
• Definition 2: Extended code
• Definition 3: Extended-code distance
• Theorem 1: Circuit distance bounds
• Proposition 2: Residual distance bound
• proof
• Proposition 3: Extended-code distance bound
• proof