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.
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.
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 σth≃0.212 for the selected BB sequence and 0.142 for the tricycle sequence, corresponding to squeezing requirements of approximately 10.46dB and 13.94dB, 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.
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.
Quantum error correction is crucial for scalable quantum information-processing applications. Traditional discrete-variable quantum codes that use multiple two-level systems to encode logical information can be hardware intensive. An alternative approach is provided by bosonic codes, which use the infinite-dimensional Hilbert space of harmonic oscillators to encode quantum information. Two promising features of bosonic codes are that syndrome measurements are natively analog and that they can be concatenated with discrete-variable codes. In this work, we propose novel decoding methods that explicitly exploit the analog syndrome information obtained from the bosonic qubit readout in a concatenated architecture. Our methods are versatile and can be generally applied to any bosonic code concatenated with a quantum low-density parity-check (QLDPC) code. Furthermore, we introduce the concept of quasi-single shot protocols as a novel approach that significantly reduces the number of repeated syndrome measurements required when decoding under phenomenological noise. To realize the protocol, we present the first implementation of time-domain decoding with the overlapping window method for general QLDPC codes and a novel analog single-shot decoding method. Our results lay the foundation for general decoding algorithms using analog information and demonstrate promising results in the direction of fault-tolerant quantum computation with concatenated bosonic-QLDPC codes. Published by the American Physical Society 2024
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.
(Abridged.) This thesis investigates scalable fault-tolerant quantum computation through the development of bosonic quantum codes, quantum LDPC codes, and decoding protocols that connect continuous-variable and discrete-variable error correction. We investigate superconducting microwave implementations of continuous-variable quantum computing, including the deterministic generation of cubic phase states, and introduce the dissipatively stabilized squeezed cat qubit, a noise-biased bosonic encoding with enhanced error suppression and faster gates. The performance of rotation-symmetric and GKP codes is analyzed under realistic noise and measurement models, revealing key trade-offs in measurement-based schemes. To integrate bosonic codes into larger architectures, we develop decoding methods that exploit analog syndrome information, enabling quasi-single-shot decoding in concatenated systems. On the discrete-variable side, we introduce localized statistics decoding, a highly parallelizable decoder for quantum LDPC codes, and propose quantum radial codes, a new family of single-shot LDPC codes with low overhead and strong circuit-level performance. Finally, we present fault complexes, a homological framework for analyzing faults in dynamic quantum error correction protocols. Extending the role of homology in static CSS codes, fault complexes provide a general language for the design and analysis of fault-tolerant schemes.
Bosonic codes provide an alternative option for quantum error correction. An important category of bosonic codes called the Gottesman-Kitaev-Preskill (GKP) code has aroused much interest recently. Theoretically, the error correction ability of the GKP code is limited since it can only correct small shift errors in position and momentum quadratures. A natural approach to promote the GKP error correction for large-scale, fault-tolerant quantum computation is concatenating encoded GKP states with a stabilizer code. The performance of the XZZX surface-GKP code, i.e., the single-mode GKP code concatenated with the XZZX surface code is investigated in this paper under two different noise models. Firstly, in the code-capacity noise model, the asymmetric rectangular GKP code with parameter $\ensuremath{\lambda}$ is introduced. Using the minimum weight perfect matching decoder combined with the continuous-variable GKP information, the optimal threshold of the XZZX-surface GKP code reaches $\ensuremath{\sigma}\ensuremath{\approx}0.67$ when $\ensuremath{\lambda}=2.1$, compared with the threshold $\ensuremath{\sigma}\ensuremath{\approx}0.60$ of the standard surface-GKP code. Secondly, we analyze the shift errors of two-qubit gates in the actual implementation and build the full circuit-level noise model. By setting the appropriate bias parameters, the logical error rate is reduced by several times in some cases. These results indicate the XZZX surface-GKP codes are more suitable for asymmetric concatenation under the general noise models. We also estimate the overhead of the XZZX-surface GKP code which uses about 291 GKP states with the noise parameter 18.5 dB ($\ensuremath{\kappa}/g\ensuremath{\approx}0.71%$) to encode a logical qubit with the error rate $2.53\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$, compared with the qubit-based surface code using 3041 qubits to achieve almost the same logical error rate.