Continuous-variable (CV) GKP quantum error correction produces high-precision analog residual syndromes rather than binary outcomes, yet these residuals have primarily been treated only as decoding inputs. Here, we show that GKP residuals are mathematically equivalent to pilot signals in telecommunications, enabling quantum error correction (QEC) itself to function as a real-time hardware-state sensor for room-temperature photonic quantum computers. Using the wrapped Gaussian residual distribution, we derive the Fisher information for effective displacement-noise variance (V_{\mathrm{eff}}) estimation and quantitatively compare CV analog syndromes against discrete-variable (DV) binary syndromes under an equal syndrome-measurement budget. CV systems provide 4.5–17.5× greater Fisher information for (V_{\mathrm{eff}}) estimation from QEC syndromes, with the advantage increasing at higher squeezing where DV information saturates. For a (d=7) surface code with 686 modes per QEC cycle, wrapped-Gaussian maximum-likelihood estimation achieves approximately 3% (V_{\mathrm{eff}}) accuracy within only three QEC cycles ((\sim 21,\mu\mathrm{s})), whereas DV syndrome-only estimation remains near 28% error even after 100 cycles under the same measurement budget due to the intrinsic information bottleneck of sparse binary events. We further evaluate online anomaly detection using false-alarm-rate-calibrated CUSUM tests. CV systems detect subtle (+0.2,\mathrm{dB}) degradations 1.9× faster than DV systems while maintaining perfect detection across all simulated trials, whereas DV detection rates fall to 91% under identical conditions. We additionally identify a fundamental scale-invariance limitation of adaptive minimum-weight perfect matching (MWPM): uniform recalibration of decoder weights leaves the matching solution unchanged even with perfect (V_{\mathrm{eff}}) estimation. This explains why accurate global noise estimation alone yields minimal decoding improvement unless channel asymmetry is present. Across approximately 11.5 million fully reproducible simulated shots, these results establish a high-bandwidth analog-information advantage (“14-bit advantage”) of CV-QEC residuals as a telemetry channel for autonomous monitoring, diagnostics, and preventive maintenance. More broadly, they demonstrate that QEC in CV photonic systems is not only an error-correction mechanism, but also an intrinsic real-time sensor for hardware-state estimation and adaptive system control in room-temperature quantum computing.
Continuous-variable (CV) GKP quantum error correction produces high-precision analog residual syndromes rather than binary outcomes, yet these residuals have primarily been treated only as decoding inputs. Here, we show that GKP residuals are mathematically equivalent to pilot signals in telecommunications, enabling quantum error correction (QEC) itself to function as a real-time hardware-state sensor for room-temperature photonic quantum computers. Using the wrapped Gaussian residual distribution, we derive the Fisher information for effective displacement-noise variance (V_{\mathrm{eff}}) estimation and quantitatively compare CV analog syndromes against discrete-variable (DV) binary syndromes under an equal syndrome-measurement budget. CV systems provide 4.5–17.5× greater Fisher information for (V_{\mathrm{eff}}) estimation from QEC syndromes, with the advantage increasing at higher squeezing where DV information saturates. For a (d=7) surface code with 686 modes per QEC cycle, wrapped-Gaussian maximum-likelihood estimation achieves approximately 3% (V_{\mathrm{eff}}) accuracy within only three QEC cycles ((\sim 21,\mu\mathrm{s})), whereas DV syndrome-only estimation remains near 28% error even after 100 cycles under the same measurement budget due to the intrinsic information bottleneck of sparse binary events. We further evaluate online anomaly detection using false-alarm-rate-calibrated CUSUM tests. CV systems detect subtle (+0.2,\mathrm{dB}) degradations 1.9× faster than DV systems while maintaining perfect detection across all simulated trials, whereas DV detection rates fall to 91% under identical conditions. We additionally identify a fundamental scale-invariance limitation of adaptive minimum-weight perfect matching (MWPM): uniform recalibration of decoder weights leaves the matching solution unchanged even with perfect (V_{\mathrm{eff}}) estimation. This explains why accurate global noise estimation alone yields minimal decoding improvement unless channel asymmetry is present. Across approximately 11.5 million fully reproducible simulated shots, these results establish a high-bandwidth analog-information advantage (“14-bit advantage”) of CV-QEC residuals as a telemetry channel for autonomous monitoring, diagnostics, and preventive maintenance. More broadly, they demonstrate that QEC in CV photonic systems is not only an error-correction mechanism, but also an intrinsic real-time sensor for hardware-state estimation and adaptive system control in room-temperature quantum computing.
Room-temperature continuous-variable (CV) photonic quantum computers are affected by inter-mode correlated displacement noise arising from a shared pump laser, violating the independent-noise assumptions underlying standard quantum error correction decoders. Here, we systematically evaluate four decoding strategies under correlated noise and establish a decoder hierarchy across code distances d = 3, 5, and 7. We first show that V_eff-recalibration MWPM is exactly equivalent to vanilla MWPM, confirmed by 100% agreement over 170,000 shots, reflecting the intrinsic scale invariance of minimum-weight perfect matching. To overcome this limitation, we introduce residual-subtraction MWPM, which estimates and removes the common-mode component from residual syndromes, adapting classical telecom common-mode rejection to quantum error correction. This method achieves up to 41.8× improvement at d = 5 and 19.0× at d = 7. We further develop a lightweight graph neural network (GNN) decoder with only 4,033 parameters (~2,500× smaller than AlphaQubit), which learns the correlation structure directly and achieves up to 33.4× improvement. A single GNN trained on mixed correlation strengths generalizes to unseen noise conditions, retaining a 20.8× advantage at out-of-distribution ρ = 0.20. Under independent noise (ρ = 0), both correlation-aware approaches degrade relative to MWPM, confirming that their advantage derives specifically from exploiting shared-noise structure. Across ~920,000 fully reproducible simulations, our results identify practical decoder strategies for correlated CV photonic hardware and demonstrate that explicitly modeling shared-noise structure can substantially improve fault-tolerant performance.
Quantum computers can solve selected problems more efficiently than classical computers, but current devices are limited by high physical error rates. Quantum Error Correction (QEC) mitigates this by encoding many physical qubits into a logical qubit, with the surface code among the most widely studied approaches. Since syndrome measurements are produced continuously, the decoder must process them fast enough to avoid becoming a system bottleneck, making real-time decoding essential for fault-tolerant quantum computing. While most state-of-the-art real-time decoders rely on Minimum-Weight Perfect Matching (MWPM), we instead use a high-accuracy Graph Neural Network (GNN) that trades higher computational cost for lower logical error rates. To make this GNN practical for real-time decoding, we apply algorithm-hardware co-design. We reduce complexity through hardware-guided pruning and retraining, producing two hardware-friendly models that reduce parameter count by 3.1× and 6.5×. These target, respectively, an average decoding latency of one syndrome cycle and a worst-case latency within one syndrome cycle. We further reduce hardware cost using input-graph filtering and post-training quantization. Based on these optimized models, we propose an FPGA-based architecture for low-latency inference and real-time decoding. Evaluated on surface codes up to distance 7 under circuit-level noise at physical error rate p=10−3, our decoder outperforms MWPM in decoding accuracy for both average-latency and max-latency settings. It reduces logical error rate by 40% at 1μs average latency, and by 13% under a strict 1μs deadline.
Quantum error correction (QEC) plays an essential role in fault-tolerantly realizing quantum algorithms of practical interest.Among different approaches to QEC, encoding logical quantum information in harmonic oscillator modes has been shown to be promising and hardware efficient.In this work, we study multimode Gottesman-Kitaev-Preskill (GKP) codes, encoding a qubit in many oscillators, through a lattice perspective.In particular, we implement a closest point decoding strategy for correcting random Gaussian shift errors.For decoding a generic multimode GKP code, we first identify its corresponding lattice followed by finding the closest lattice point in its symplectic dual lattice to a candidate shift error compatible with the error syndrome.We use this method to characterize the error-correction capabilities of several known multimode GKP codes, including their code distances and fidelities.We also perform numerical optimization of multimode GKP codes up to ten modes and find three instances (with three, seven, and nine modes) with better code distances and fidelities compared to the known GKP codes with the same number of modes.While exact closest point decoding incurs exponential time cost in the number of modes for general unstructured GKP codes, we give several examples of structured GKP codes (i.e., of the repetition-rectangular GKP code types) where the closest point decoding can be performed exactly in linear time.For the surface-GKP code, we show that the closest point decoding can be performed exactly in polynomial time with the help of a minimum-weight-perfect-matching algorithm (MWPM).We show that this MWPM closest point decoder improves both the fidelity and the noise threshold of the surface-GKP code to 0.602 compared to the previously studied MWPM decoder assisted by log-likelihood analog information, which yields a noise threshold of 0.599.
Robust continuous-variable (CV) quantum information processing requires correcting realistic errors in bosonic systems, but all existing schemes rely on auxiliary Gottesman-Kitaev-Preskill (GKP) states which the preparation and operation are demanding in many platforms. In this work, we propose a novel CV quantum error correction (QEC) scheme that utilizes a broadly accessible resource: discrete-variable (DV) ancilla. Our scheme extracts information about CV displacement to the DV ancilla, measuring that allows counteracting the unwanted displacement error. We show that a simple single-qubit ancilla can already suppress CV infidelity by more than 20%. By concatenating with DV QEC codes, our scheme is robust against the physical errors in hybrid CV-DV systems, and yields a new class of oscillator-in-oscillator code that does not involve GKP states. Our work facilitates the implementation of CV QEC on realistic platforms.
Fault-tolerant quantum error correction is essential for implementing quantum algorithms of significant practical importance. In this work, we propose a highly effective use of the surface-GKP code, i.e., the surface code consisting of bosonic GKP qubits instead of bare two-dimensional qubits. In our proposal, we use error-corrected two-qubit gates between GKP qubits and introduce a maximum likelihood decoding strategy for correcting shift errors in the two-GKP-qubit gates. Our proposed decoding reduces the total CNOT failure rate of the GKP qubits, e.g., from 0.87% to 0.36% at a GKP squeezing of 12dB, compared to the case where the simple closest-integer decoding is used. Then, by concatenating the GKP code with the surface code, we find that the threshold GKP squeezing is given by 9.9dB under the the assumption that finite-squeezing of the GKP states is the dominant noise source. More importantly, we show that a low logical failure rate pL<10−7 can be achieved with moderate hardware requirements, e.g., 291 modes and 97 qubits at a GKP squeezing of 12dB as opposed to 1457 bare qubits for the standard rotated surface code at an equivalent noise level (i.e., p=0.36%). Such a low failure rate of our surface-GKP code is possible through the use of space-time correlated edges in the matching graphs of the surface code decoder. Further, all edge weights in the matching graphs are computed dynamically based on analog information from the GKP error correction using the full history of all syndrome measurement rounds. We also show that a highly-squeezed GKP state of GKP squeezing ≳12dB can be experimentally realized by using a dissipative stabilization method, namely, the Big-small-Big method, with fairly conservative experimental parameters. Lastly, we introduce a three-level ancilla scheme to mitigate ancilla decay errors during a GKP state preparation.
The continuous-variable (CV) Gaussian no-go theorem fundamentally limits the suppression of Gaussian displacement errors using only Gaussian gates and states. Prior studies have employed Gottesman-Kitaev-Preskill (GKP) states as ancillary qumodes to suppress small Gaussian displacement errors, but when the displacement magnitude becomes large, lattice-crossing events arise beyond the correctable range of the GKP state. To address this issue, we concatenate a Gaussian-noise-suppression circuit with an outer analog Steane code that corrects such occasional lattice-crossing events as well as other abrupt displacement errors. Unlike conventional concatenation, which primarily aims to reduce logical error rates, the Steane-GKP duality in encoding provides complementary protection against both large and small displacement errors, enabling CV error correction within the continuous encoding space and contrasting with earlier approaches that concatenate GKP states with repetition codes for discrete qubit or qudit encodings. Analytical results show that, under infinite squeezing, the concatenated code suppresses the variance of Gaussian displacement errors across all qumodes by up to 50 percent while enabling unbiased correction of lattice-crossing events, with a success probability determined by the ratio between the residual Gaussian error standard deviation and the lattice-crossing magnitude. Even with finite squeezing, the proposed architecture continues to provide Gaussian-error suppression together with lattice-crossing correction, and the presence of the outer analog Steane code relaxes the squeezing requirement of the inner GKP states, indicating near-term experimental feasibility. This work establishes a viable route toward fault-tolerant continuous-variable quantum computation and provides new insight into the design of concatenated CV error-correcting architectures.