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arXiv:2501.12072

Bare-ancilla fault-tolerant syndrome extraction: General extensions of distance-three codes and structural criteria for graph codes

Gupta, Harsh, Bhattacharyya, Mainak, Jain, Ritik, Raina, Ankur

Published Jan 21, 2025• 0 citations
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Abstract

The reliability of quantum computation critically depends on the performance of quantum error-correcting codes (QECCs). Performance of QECCs can be severely degraded by hook errors, which effectively reduce the code distance. We develop a systematic framework for bare-ancilla syndrome extraction in $[[n,k,3]]$ stabilizer codes. It provides a criterion for a specific single ancilla fault model and jointly searches for suitable stabilizer generators and data-ancilla interaction orders to correct hook errors. We call such stabilizer generator sets bare-ancilla codes (BACs). We also present two constructions that enlarge the set of valid BACs while preserving minimum distance, either by keeping the number of logical qubits fixed or by increasing it. Under the correlated-data error model, $3177$ $[[7,1,3]]$ graph codes are found to be BACs out of $3379$ and all $335415$ $[[8,1,3]]$ graph codes are BACs. We tested $500$ $[[7,1,3]]$ and $1000$ $[[8,1,3]]$ graph codes in the presence of anisotropic and depolarizing noise. The number of data-ancilla interactions is an important predictor of the pseudo-threshold of these graph codes. In the analyzed codes, the BAC, which works with the bare ancilla method, usually performs well or better than the flag method, particularly for depolarizing noise. Notably, we report new bare ancilla codes, namely $[[6,1,3]]$ and $[[7,1,3]]$ with improved code rate compared to the bare code used in the work of Muyuan Li \emph{et al.} and Maheshwari \emph{et al.} respectively.

DOI
10.48550/arxiv.2501.12072

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•Dec 31, 2025•0 citations
The reliability of quantum computation critically depends on the performance of quantum error-correcting codes (QECCs). Performance of QECCs can be severely degraded by hook errors, which effectively reduce the code distance. We develop a systematic framework for bare-ancilla syndrome extraction in [[n,k,3]][[n,k,3]][[n,k,3]] stabilizer codes. It provides a criterion for a specific single ancilla fault model and jointly searches for suitable stabilizer generators and data-ancilla interaction orders to correct hook errors. We call such stabilizer generator sets bare-ancilla codes (BACs). We also present two constructions that enlarge the set of valid BACs while preserving minimum distance, either by keeping the number of logical qubits fixed or by increasing it. Under the correlated-data error model, 317731773177 [[7,1,3]][[7,1,3]][[7,1,3]] graph codes are found to be BACs out of 337933793379 and all 335415335415335415 [[8,1,3]][[8,1,3]][[8,1,3]] graph codes are BACs. We tested 500500500 [[7,1,3]][[7,1,3]][[7,1,3]] and 100010001000 [[8,1,3]][[8,1,3]][[8,1,3]] graph codes in the presence of anisotropic and depolarizing noise. The number of data-ancilla interactions is an important predictor of the pseudo-threshold of these graph codes. In the analyzed codes, the BAC, which works with the bare ancilla method, usually performs well or better than the flag method, particularly for depolarizing noise. Notably, we report new bare ancilla codes, namely [[6,1,3]][[6,1,3]][[6,1,3]] and [[7,1,3]][[7,1,3]][[7,1,3]] with improved code rate compared to the bare code used in the work of Muyuan Li \emph{et al.} and Maheshwari \emph{et al.} respectively.
Building reliable quantum computers requires protecting fragile quantum states from inevitable environmental noise and operational errors. While quantum error correction codes like the Steane [ ⁣[7,1,3] ⁣][\![7,1,3]\!][[7,1,3]] code provide elegant theoretical solutions, their practical success hinges critically on how we measure errors - a process called syndrome extraction. The challenge lies in the ancilla qubits used for measurement: when they fail, errors can cascade across the entire quantum system, destroying the very information we're trying to protect. We address this fundamental problem by implementing and comparing three sophisticated syndrome measurement strategies: Shor's cat-state approach, which distributes measurements across multiple entangled ancillas achieving 85-92% preparation success; Steane's encoded-ancilla method using complete error-corrected logical qubits reaching 97.8% syndrome fidelity; and a flexible unified framework that adapts strategies based on hardware capabilities. Through extensive simulations using IBM's Qiskit platform spanning randomized benchmarking and T-heavy circuits, we demonstrate that intelligent ancilla management improves error suppression by up to 2.4×\times× compared to standard approaches. Our implementations achieve logical error rates as low as 5.1×10−55.1 \times 10^{-5}5.1×10−5 under realistic noise conditions with physical error rates of 10−310^{-3}10−3, while maintaining near-unity logical fidelity (0.99997) even for deep circuits. The threshold analysis reveals robust performance across distance-3 to distance-13 codes with characteristic threshold curves showing exponential error suppression below the critical physical error rate. These results provide immediately deployable tools for near-term quantum devices and establish practical design principles for scaling toward fault-tolerant quantum computers.
We present a fault-tolerant [[8, 1, 3]] non-CSS quantum error correcting code and study its logical error rates. We modify Gottesman's encoding procedure for stabilizer codes to suit a class of non-CSS quantum codes. We adopt the bare ancilla method presented by Brown et al. to reorder the measurement sequence in the syndrome extraction step and upgrade it to obtain higher pseudo-thresholds and lower leading order terms of logical error rates under the standard depolarizing and anisotropic noise models.
Reliable quantum computation requires fault-tolerant protocols to prevent errors from propagating during syndrome extraction in quantum error correction. We present a novel fault-tolerant syndrome extraction technique for CSS codes, which we refer to as the cut-cat state scheme. While each ancilla qubit interacts non-fault-tolerantly with a pair of data qubits, we introduce additional cat stabilizer measurements to identify and correct the resulting hook errors. Our approach maintains the key benefit of cat-based extraction, i.e., parallelized data qubit interactions, while reducing the number of simultaneous qubits required by more than half. Compared to flag-based state-of-the-art protocols, the cut-cat scheme offers a notable advantage in terms of two-qubit gate count as the code distance increases.
Remarkable technical progress has made high-rate, high-distance, quantum low-density parity-check codes (QLDPC) promising candidates for scalable quantum computing. However, it is hard to design low-depth syndrome extraction circuits that do not spread errors from ancilla qubits to multiple data qubits, also known as hook errors, for general QLDPC codes. Additionally, widely used decoders for these codes based on belief propagation are impaired due to short loops in the Tanner graph. Here, we investigate a hardware-aware approach to avoid these hooks and loops using biased noise ancillas. Using examples of bicycle bivariate codes and a cyclic hypergraph product code, which have been widely considered for practical application, we show that the effective fault-distance of the conventional syndrome extraction circuit can be significantly higher and the number of short loops can be significantly lower when the ancillas are subject to phase-flip errors only, compared to when they are also subject to bit-flip errors. This can result in almost an order of magnitude improvement in the logical error rate at circuit noise of 2×10−32\times 10^{-3}2×10−3 and when bit-flip errors in the ancilla are 50 times less likely than phase-flip errors. Our work demonstrates a significant and practical quantum error correction advantage with biased noise qubits in which full-bias cannot be maintained.
Hook errors are a major challenge in implementing logical operations with the surface code, because they can reduce the fault distance below the code distance. This motivates syndrome-extraction circuits that suppress hook-error effects for the stabilizer layouts that appear during logical operations. However, the existing methods either increase circuit depth or require simultaneous execution of measurements and CNOT gates, both of which introduce additional overheads and degrade the threshold. We propose the ZX interleaving syndrome extraction, which preserves the full fault distance ddd for any surface-code layout with regular stabilizer tiles at minimum depth, i.e., four layers of CNOT gates, without requiring additional circuit depth or simultaneous execution of measurements and CNOT gates. The key idea is to interleave the Z and X stabilizer tiles so that hook-error edges in the decoding graph are shortened and effectively eliminated. Numerical simulations under uniform depolarizing noise for memory and lattice-surgery experiments confirm that the proposed method achieves a full fault distance of ddd, whereas the best existing minimum-depth approach achieves d−1d-1d−1. Since the full fault distance is achievable for any regular tiling layout of the surface code, the proposed method may serve as an indispensable technique for practical fault-tolerant quantum computation.
Building reliable quantum computers requires protecting fragile quantum states from inevitable environmental noise and operational errors. While quantum error correction codes like the Steane [[7,1,3]] code provide elegant theoretical solutions, their practical success hinges critically on how we measure errors—a process called syndrome extraction. The challenge lies in the ancilla qubits used for measurement: when they fail, errors can cascade across the entire quantum system, destroying the very information we’re trying to protect. We address this fundamental problem by implementing and comparing three sophisticated syndrome measurement strategies: Shor’s cat-state approach, which distributes measurements across multiple entangled ancillas achieving 85-92% preparation success; Steane’s encoded-ancilla method using complete error-corrected logical qubits reaching 97.8% syndrome fidelity; and a flexible unified framework that adapts strategies based on hardware capabilities. Through extensive simulations using IBM’s Qiskit platform spanning randomized benchmarking and T-heavy circuits, we demonstrate that intelligent ancilla management improves error suppression by up to 2.4×compared to standard approaches. Our implementations achieve logical error rates as low as 5.1 ×10^−5 under realistic noise conditions with physical error rates of 10^−3, while maintaining near-unity logical fidelity (0.99997) even for deep circuits. The threshold analysis reveals robust performance across distance-3 to distance-13 codes with characteristic threshold curves showing exponential error suppression below the critical physical error rate. These results provide immediately deployable tools for near-term quantum devices and establish practical design principles for scaling toward fault-tolerant quantum computers.