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Which quantum algorithms suit photonic continuous-variable hardware? A comparative resource and noise-robustness study

Hikaru Wakaura, Taiki Tanimae

Published May 22, 2026• 0 citations
Publisher

Abstract

Summary. A comparative, model-level study of four quantum algorithms ported to a common photonic continuous-variable (CV) model: control-free quantum phase estimation (QPE), randomized product formulas (qDRIFT), quantum Kolmogorov-Arnold networks (QKAN), and Regev's factoring algorithm. Algorithms are compared under a shared noise channel combining exact Fock-space photon loss and Gaussian displacement, with Gottesman-Kitaev-Preskill (GKP) error correction. Main findings. Control-free QPE is the most noise-robust because it needs no controlled unitaries and, for quadratic Hamiltonians, no non-Gaussian gates; the advantage survives for non-Gaussian Hamiltonians. GKP correction helps only once the physical noise exceeds the finite-squeezing floor (proved), and whether an algorithm benefits is set by a noise- versus systematic-limited dichotomy. The core GKP correction model is validated against an explicit Glancy-Knill error-correction Monte Carlo (agreement ≤0.1%). A discrete-variable surface-code baseline shows the CV advantage is structural (Gaussian operations are free), not threshold-based. Scope. This is a model-level numerical comparison, not an experiment; the GKP correction is a validated surrogate rather than a full Fock-space error-correction circuit, and instances are modest in size. Results, sensitivity analyses, and a two-dimensional noise phase diagram are included. Reproducibility. Standalone scripts and released JSON results reproduce every figure and table.

DOI
10.5281/zenodo.20341677

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Summary. A comparative, model-level study of four quantum algorithms ported to a common photonic continuous-variable (CV) model: control-free quantum phase estimation (QPE), randomized product formulas (qDRIFT), quantum Kolmogorov-Arnold networks (QKAN), and Regev's factoring algorithm. Algorithms are compared under a shared noise channel combining exact Fock-space photon loss and Gaussian displacement, with Gottesman-Kitaev-Preskill (GKP) error correction. Main findings. Control-free QPE is the most noise-robust because it needs no controlled unitaries and, for quadratic Hamiltonians, no non-Gaussian gates; the advantage survives for non-Gaussian Hamiltonians. GKP correction helps only once the physical noise exceeds the finite-squeezing floor (proved), and whether an algorithm benefits is set by a noise- versus systematic-limited dichotomy. The core GKP correction model is validated against an explicit Glancy-Knill error-correction Monte Carlo (agreement ≤0.1%). A discrete-variable surface-code baseline shows the CV advantage is structural (Gaussian operations are free), not threshold-based. Scope. This is a model-level numerical comparison, not an experiment; the GKP correction is a validated surrogate rather than a full Fock-space error-correction circuit, and instances are modest in size. Results, sensitivity analyses, and a two-dimensional noise phase diagram are included. Reproducibility. Standalone scripts and released JSON results reproduce every figure and table.
This thesis develops a theoretical framework for hybrid continuous-variable (CV) and discrete-variable (DV) quantum systems, with emphasis on quantum control, state preparation, and error correction. A central contribution is non-abelian quantum signal processing (NA-QSP), a generalization of quantum signal processing to settings where control parameters are non-commuting operators. Within this framework, we introduce the Gaussian-Controlled-Rotation (GCR) protocol, which enables high-fidelity control of CV states using DV ancillae. This approach allows for deterministic preparation of squeezed, cat, and Gottesman-Kitaev-Preskill (GKP) states without numerical optimization. Two previously unpublished contributions are included: (i) Chapter 2.3 introduces the Gaussian hierarchy, a classification of CV operations analogous to the Clifford hierarchy, offering a new lens for understanding CV gate sets; (ii) Chapter 5 presents an analytical framework for correcting photon loss in finite-energy GKP codes, introducing the notion of probabilistic error correction, providing insight into recent GKP experiments surpassing break-even thresholds. Overall, this work lays the groundwork for scalable, fault-tolerant quantum computation in hybrid CV-DV architectures, with applications to logical gate synthesis, readout, and hybrid algorithm design using ancilla oscillators.
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.
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.
In order to solve practical problems on a quantum computer, it is necessary to use fault-tolerant quantum error correction schemes to overcome noise: the errors arising from imperfections in physical components. The Gottesman-Kitaev-Preskill (GKP) code aims at achieving this in a hardware efficient manner by encoding finite-dimensional logical subspaces in the Hilbert space of one or more continuous variable modes. In near term implementations, however, it is not feasible to eliminate errors entirely, so it is natural to also employ alternative error mitigation techniques together with error correction. In this work, we study a quantum error mitigation method known as probabilistic error cancellation in the context of the GKP code. We compare Steane-type and teleportation-based GKP error correction, and calculate the sampling overheads associated with the technique for square and hexagonal GKP codes. We employ the stabilizer subsystem decomposition for GKP codes to obtain an effective logical channel for noisy operations that is used to express the ideal one of a target logical unitary. We consider noise from finite squeezing on the data and the two ancilla modes, needed for error correction, and examine the relationship between the sampling overhead and the noise for different decoding methods. Our results are calculated for single- and two-qubit GKP Clifford gates and one round of error correction, and show the nontrivial combination of error correction and mitigation in continuous variable codes.
The Gottesman-Kitaev-Preskill (gkp) quantum error-correcting code attracts much attention in continuous variable (CV) quantum computation and CV quantum communication due to the simplicity of error-correcting routines and the high tolerance against Gaussian errors. Since the gkp code state should be regarded as a limit of physically meaningful approximate ones, various approximations have been developed until today, but explicit relations among them are still unclear. In this paper, we rigorously prove the equivalence of these approximate gkp codes with an explicit correspondence of the parameters. We also propose a standard form of the approximate code states in the position representation, which enables us to derive closed-form expressions for the Wigner function, inner products, and the average photon number in terms of the theta functions. Our results serve as fundamental tools for further analyses of fault-tolerant quantum computation and channel coding using approximate gkp codes.
Hybrid continuous-variable--discrete-variable (CV--DV) architectures process quantum information in bosonic modes and qubits, but noise limits their performance. To reduce the noise, existing DV error correction must be complemented by CV noise reduction. Existing CV noise-reduction schemes -- such as GKP-stabilizer codes -- can reduce CV noise, but only for Gaussian gates. Therefore, no current noise-reduction scheme can correct arbitrary CV--DV gates, including non-Gaussian ones. Here, we develop noise reduction for a universal CV--DV gate set, making it applicable to arbitrary CV--DV gates. We do so by introducing an ancilla qubit into a GKP-stabilizer code, allowing us to reduce the standard deviation of Gaussian displacement noise from σσσ to O~(σ2)\tilde O(σ^2)O~(σ2). To demonstrate the scheme, we show that it significantly reduces noise and improves fidelity in the preparation of non-Gaussian cat and Fock states.