Scalable fault-tolerant quantum error correction based on Gottesman-Kitaev-Preskill (GKP) codes requires finite-rate outer codes that can exploit analog GKP information under realistic circuit noise. However, quantum low-density parity-check (qLDPC) outer codes have not been systematically compared as candidates for circuit-level GKP concatenation. Here, we benchmark BB and tricycle codes using a unified simulation framework that progresses from code-capacity noise to repeated noisy syndrome extraction and schedule-resolved displacement propagation. Across all three noise models, analog-informed belief-propagation with ordered-statistics decoding (BP-OSD) yields higher finite-size crossing estimates than hard-decision decoding. In the full circuit model, the analog-informed crossings are \(σ_{\mathrm{th}}\simeq0.212\) for the selected BB sequence and \(0.142\) for the tricycle sequence, corresponding to squeezing requirements of approximately \(10.46\,\mathrm{dB}\) and \(13.94\,\mathrm{dB}\), respectively. These results show that analog GKP information continues to improve decoding under circuit-level noise. Under the common simulation assumptions used here, the selected BB sequence also yields a higher finite-size crossing estimate than the selected tricycle sequence.
Scalable fault-tolerant quantum error correction based on Gottesman-Kitaev-Preskill (GKP) codes requires finite-rate outer codes that can exploit analog GKP information under realistic circuit noise. However, quantum low-density parity-check (qLDPC) outer codes have not been systematically compared as candidates for circuit-level GKP concatenation. Here, we benchmark BB and tricycle codes using a unified simulation framework that progresses from code-capacity noise to repeated noisy syndrome extraction and schedule-resolved displacement propagation. Across all three noise models, analog-informed belief-propagation with ordered-statistics decoding (BP-OSD) yields higher finite-size crossing estimates than hard-decision decoding. In the full circuit model, the analog-informed crossings are (σ_{\mathrm{th}}\simeq0.212) for the selected BB sequence and (0.142) for the tricycle sequence, corresponding to squeezing requirements of approximately (10.46,\mathrm{dB}) and (13.94,\mathrm{dB}), respectively. These results show that analog GKP information continues to improve decoding under circuit-level noise. Under the common simulation assumptions used here, the selected BB sequence also yields a higher finite-size crossing estimate than the selected tricycle sequence.
Concatenated bosonic-stabilizer codes have recently gained prominence as promising candidates for achieving lowoverhead fault-tolerant quantum computing in the long term. In such a system, analog information obtained from the syndrome measurements of an inner bosonic code is used to inform decoding for an outer code layer consisting of a discrete-variable stabilizer code such as a surface code. The use of Quantum Low Density Parity Check (QLDPC) codes as an outer code is of particular interest in view of the significantly higher encoding rates offered by these code families, leading to a further reduction in the overhead requirements for large-scale quantum computing. Recent works have investigated the performance of QLDPC-GKP codes in detail, and indeed, the use of analog information obtained from the inner code significantly boosts the decoder performance. However, the noise models assumed in these works are typically limited to the depolarizing or phenomenological noise models. In this paper, we investigate the performance of QLDPC-GKP concatenated codes under circuit-level noise, based on a noise model introduced by Noh et al. in relation to the surfaceGKP code. To demonstrate the performance boost provided by the use of analog information, we investigate three different scenarios, namely (a) decoding performance in the absence of soft information, (b) decoding performance with precomputed probabilities of error but without real-time soft information, and (c) decoding performance with real-time soft information obtained from round-to-round decoding of the inner GKP code. Results show minimal performance improvement between (a) and (b), but a significant boost in (c), indicating that real-time soft information is critical for concatenated decoding under circuitlevel noise. We also investigate the effect of using measurement schedules of varying depths and show that using a schedule with minimum depth is critical for obtaining reliable soft information from the inner code.
Photonic quantum networks require error-correction architectures that remain useful with finite-energy bosonic states, pure-loss fiber transmission, and explicit resource accounting. In this light, we study a concatenated architecture in which each physical rail is a finitely squeezed Gottesman--Kitaev--Preskill (GKP) qubit transmitted through a pure-loss fiber segment, corrected by teleportation-based GKP error correction with finitely squeezed ancillae, and decoded by an outer quantum parity code (QPC). The GKP layer converts continuous homodyne syndromes into effective rail-level Pauli marginals, while the QPC layer suppresses the residual qubit-level errors. For the concatenated code family considered here, we find a finite-squeezing threshold of 5.06dB at zero propagation loss. In the memory setting, the QPC layer lowers the squeezing at which repeated error correction becomes beneficial from 6.7dB for bare GKP correction to 5.2dB for QPC(3,3) and 4.3dB for QPC(5,5), and improves the average-fidelity ratio by up to 75--90% in the relevant intermediate-noise regime. In the repeater setting, avoiding pre-amplification gives larger secret-key fractions at moderate squeezing, but also produces an optimal squeezing because highly squeezed GKP peaks become sensitive to loss-induced inward displacement. Resource-normalized rates show that QPC concatenation can exceed the repeaterless PLOB benchmark by orders of magnitude and extend the communication reach, at short repeater spacing, to distances of order 104km with 14dB squeezing. However, QPC concatenation becomes detrimental when each elementary hop is too lossy. These results provide quantitative design rules for finite-squeezing GKP--QPC quantum memories and repeaters.
Quantum error correction has recently been shown to benefit greatly from specific physical encodings of the code qubits. In particular, several researchers have considered the individual code qubits being encoded with the continuous variable GottesmanKitaev-Preskill (GKP) code, and then imposed an outer discrete-variable code such as the surface code on these GKP qubits. Under such a concatenation scheme, the analog information from the inner GKP error correction improves the noise threshold of the outer code. However, the surface code has vanishing rate and demands a lot of resources with growing distance. In this work, we concatenate the GKP code with generic quantum low-density parity-check (QLDPC) codes and demonstrate a natural way to exploit the GKP analog information in iterative decoding algorithms. We first show the noise thresholds for two lifted product QLDPC code families, and then show the improvements of noise thresholds when the iterative decoder – a hardware-friendly min-sum algorithm (MSA) – utilizes the GKP analog information. We also show that, when the GKP analog information is combined with a sequential update schedule for MSA, the scheme surpasses the well-known CSS Hamming bound for these code families. Furthermore, we observe that the GKP analog information helps the iterative decoder in escaping harmful trapping sets in the Tanner graph of the QLDPC code, thereby eliminating or significantly lowering the error floor of the logical error rate curves. Finally, we discuss new fundamental and practical questions that arise from this work on channel capacity under GKP analog information, and on improving decoder design and analysis.
Fault-tolerant quantum computers will depend crucially on the performance of the classical decoding algorithm which takes in the results of measurements and outputs corrections to the errors inferred to have occurred. Machine learning models have shown great promise as decoders for the surface code; however, this promise has not yet been substantiated for the more challenging task of decoding quantum low-density parity-check (QLDPC) codes. In this paper, we present a recurrent, transformer-based neural network designed to decode circuit-level noise on Bivariate Bicycle (BB) codes. For the [ [ 72 , 12 , 6 ] ] BB code, at a physical error rate of p = 0.1 % , our model achieves logical error rates almost 5 times lower than belief propagation with ordered statistics decoding (BP-OSD), and roughly 5 times larger than a most-likely error decoder. Moreover, while BP-OSD has a wide distribution of runtimes with significant outliers, our model has a consistent runtime and is an order-of-magnitude faster than the worst-case times from a benchmark BP-OSD implementation. On the [ [ 144 , 12 , 12 ] ] BB code, our model obtains worse logical error rates but maintains the speed advantage. These results provide initial evidence that machine learning decoders can out-perform conventional decoders on small QLDPC codes, but suggest more complex architectures and/or training procedures are necessary to scale to larger code sizes.
Fault-tolerant quantum error correction requires repeated syndrome extraction to address errors induced by the syndrome-extraction circuit itself. However, repeated syndrome extraction incurs significant overhead in terms of gate count and ancilla consumption (e.g., Gottesman-Kitaev-Preskill (GKP) states). Moreover, noisy syndrome extraction can itself inject additional errors into the data qubits. To address these issues, we propose a concrete adaptive skipping scheme for the surface-GKP code, a representative GKP-concatenated architecture, that uses analog information naturally generated during inner GKP correction. At each round, the scheme selects one of four actions: measuring both Z-type and X-type surface-code stabilizers, measuring only one type, or skipping both types and reusing previous syndromes. The decision is based on a reliability comparison between reusing the previous syndrome value and performing a new noisy syndrome extraction. Using circuit-level simulations, we show that the adaptive skipping scheme can reduce the number of surface-code stabilizer measurements while maintaining logical error rates comparable to or lower than those of the full-measurement baseline. The improvement is most pronounced when gate and measurement noise are larger than idle noise, so that avoiding unnecessary syndrome extraction reduces the noise injected into the code. These results indicate that analog information from inner GKP correction can be used not only to improve decoding but also to reduce the measurement overhead of outer-code syndrome extraction.
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 2×10−3 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.