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arXiv:2610.41341

Time-Domain Multiplexing for Resource-Efficient Heralded GKP State Preparation

Ke-Ming Hu, Jun-Jie He, Yu-Ze Zhu, Pei-Lin Du, Lin Yang, Hao Tang, Xian-Min Jin

quant-phSubmitted Oct 30, 2026• Announced Nov 2, 2026
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Abstract

Gottesman-Kitaev-Preskill (GKP) states are key resources for fault-tolerant photonic quantum computing, but their conditional preparation via Gaussian boson sampling involves a trade-off between source count and photon-number resolution. Distributing heralding photons across more measured modes reduces the resolution required per mode, yet increases the number of simultaneously operated squeezed sources in fully squeezed spatial architectures. Here, we show that time-domain multiplexing (TDM) can reduce these physical source requirements while preserving GKP state quality. Using an end-to-end differentiable Bargmann framework, we find that TDM circuits reusing one or two squeezed sources achieve fidelities comparable to fully squeezed spatial circuits across several configurations. Selected nine-mode configurations retain fidelities above 0.95 with one photon heralded in each measured mode, mitigating detector resolution demands without a proportional increase in source count. Loss-aware reoptimization further shows that, under the investigated spatial loss models, two-mode TDM maintains fidelities near 0.80 at 1.0 dB per channel, whereas the corresponding spatial benchmarks fall below 0.60. These results demonstrate that temporal reuse provides an effective architectural route for balancing source overhead, detection demands, and state preparation performance in the presence of optical loss.

AI Summary

The paper asks whether time-domain multiplexing (TDM) can reduce the physical squeezed-source count in photon-number-heralded GKP preparation while distributing heralding photons across more detectors. It compares single-loop circuits reusing one or two programmable squeezed sources with spatial interferometers having either all inputs squeezed or only two squeezed inputs. A differentiable Bargmann-kernel simulation optimizes fidelity, stabilizer-based effective squeezing, and heralding probability for total detected photon numbers $N=8$ and $12$. The target is a finite-energy square-lattice GKP $|0\rangle$ with $\Delta=\kappa=0.36$. For $N=8$ across nine modes, single- and two-source TDM achieve fidelities $0.9600$ and $0.9599$, versus $0.9582$ for nine spatial sources, with one photon detected in each measured mode. Corresponding success probabilities are $1.10\times10^{-6}$, $5.27\times10^{-6}$, and $2.37\times10^{-5}$. At $N=12$, thirteen-mode two-source TDM reaches $F=0.9611$ with probability $2.94\times10^{-7}$. Loss-aware reoptimization finds two-source TDM fidelities near $0.80$ at $1$ dB per specified spatial-loss channel, versus approximately $0.59$–$0.60$ for spatial benchmarks. These are not matched-loss advantages: spatial loss includes the retained output, whereas TDM spatial loss excludes the final retained loop mode; delay loss is evaluated separately. The results are numerical and configuration-dependent, with restricted TDM connectivity, programmable pulse-to-pulse settings, and reduced success probabilities for single-photon patterns. Equal physical source counts also do not imply equal numbers of squeezed input pulses. Loss-aware results undergo cutoff-convergence checks, with metric changes and conditional-state trace deficits below $10^{-7}$.

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13 pages, 5 figures

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Gottesman-Kitaev-Preskill (GKP) states are key resources for fault-tolerant photonic quantum computing, but their conditional preparation via Gaussian boson sampling involves a trade-off between source count and photon-number resolution. Distributing heralding photons across more measured modes reduces the resolution required per mode, yet increases the number of simultaneously operated squeezed sources in fully squeezed spatial architectures. Here, we show that time-domain multiplexing (TDM) can reduce these physical source requirements while preserving GKP state quality. Using an end-to-end differentiable Bargmann framework, we find that TDM circuits reusing one or two squeezed sources achieve fidelities comparable to fully squeezed spatial circuits across several configurations. Selected nine-mode configurations retain fidelities above 0.95 with one photon heralded in each measured mode, mitigating detector resolution demands without a proportional increase in source count. Loss-aware reoptimization further shows that, under the investigated spatial loss models, two-mode TDM maintains fidelities near 0.80 at 1.0 dB per channel, whereas the corresponding spatial benchmarks fall below 0.60. These results demonstrate that temporal reuse provides an effective architectural route for balancing source overhead, detection demands, and state preparation performance in the presence of optical loss.
The paper asks whether time-domain multiplexing (TDM) can reduce the physical squeezed-source count in photon-number-heralded GKP preparation while distributing heralding photons across more detectors. It compares single-loop circuits reusing one or two programmable squeezed sources with spatial interferometers having either all inputs squeezed or only two squeezed inputs.
A differentiable Bargmann-kernel simulation optimizes fidelity, stabilizer-based effective squeezing, and heralding probability for total detected photon numbers N=8N=8N=8 and 121212. The target is a finite-energy square-lattice GKP ∣0⟩|0\rangle∣0⟩ with Δ=κ=0.36\Delta=\kappa=0.36Δ=κ=0.36. For N=8N=8N=8 across nine modes, single- and two-source TDM achieve fidelities 0.96000.96000.9600 and 0.95990.95990.9599, versus 0.95820.95820.9582 for nine spatial sources, with one photon detected in each measured mode. Corresponding success probabilities are 1.10×10−61.10\times10^{-6}1.10×10−6, 5.27×10−65.27\times10^{-6}5.27×10−6, and 2.37×10−52.37\times10^{-5}2.37×10−5. At N=12N=12N=12, thirteen-mode two-source TDM reaches F=0.9611F=0.9611F=0.9611 with probability 2.94×10−72.94\times10^{-7}2.94×10−7.
Loss-aware reoptimization finds two-source TDM fidelities near 0.800.800.80 at 111 dB per specified spatial-loss channel, versus approximately 0.590.590.59–0.600.600.60 for spatial benchmarks. These are not matched-loss advantages: spatial loss includes the retained output, whereas TDM spatial loss excludes the final retained loop mode; delay loss is evaluated separately.
The results are numerical and configuration-dependent, with restricted TDM connectivity, programmable pulse-to-pulse settings, and reduced success probabilities for single-photon patterns. Equal physical source counts also do not imply equal numbers of squeezed input pulses. Loss-aware results undergo cutoff-convergence checks, with metric changes and conditional-state trace deficits below 10−710^{-7}10−7.
Gottesman-Kitaev-Preskill (GKP) states are essential non-Gaussian resources for fault-tolerant photonic quantum computing, enabling logical qubit encoding with intrinsic robustness against errors. Several approaches to GKP state preparation have been explored, including measurement-based protocols in circuit QED and trapped-ion systems, cat-state breeding, and photon-subtraction schemes. However, these methods are either restricted to specific platforms or require deep non-Gaussian resource chains with exponentially low success probabilities. Gaussian Boson Sampling (GBS) offers a compelling all-photonic alternative by generating non-Gaussian states through measurement-induced nonlinearity, without the need for matter-based ancilla or active feedforward. Nevertheless, its practical implementation is limited by the exponential computational cost of evaluating matrix hafnians-#P-complete functions that govern photon-number probabilities. To address this challenge, we introduce a two-stage Histogram Gradient Boosting surrogate pipeline that predicts, without any hafnian computation, the optimal heralding pattern, circuit fidelity, and post-selection probability for candidate GBS circuits, while reserving exact quantum simulation exclusively for surrogate-selected candidates. Trained on circuit configurations across 3-5 optical modes, the surrogate achieves 90.0% GKP-detection accuracy on a held-out set, representing a 23.7 percentage-point improvement over the baseline, with a fidelity mean absolute error of 0.032 and a log-scale post-selection probability R2=0.837R^2 = 0.837R2=0.837, reducing the total simulation burden by approximately 90%.
The scalability of photonic implementations of fault-tolerant quantum computing based on Gottesman-Kitaev-Preskill (GKP) qubits is injured by the requirements of inline squeezing and reconfigurability of the linear optical network.In this work we propose a topologically error-corrected architecture that does away with these elements at no cost-in fact, at an advantage-to state preparation overheads.Our computer consists of three modules: a two-dimensional (2D) array of probabilistic sources of GKP states; a depthfour circuit of static beam splitters, phase shifters, and short delay lines; and a 2D array of homodyne detectors.The symmetry of our proposed circuit allows us to combine the effects of finite squeezing and uniform photon loss within the noise model, resulting in more comprehensive threshold estimates.These jumps over both architectural and analytical hurdles considerably expedite the construction of a photonic quantum computer.
Gottesman-Kitaev-Preskill (GKP) qubits, known for their exceptional error-correction capabilities, are highly coveted in quantum computing. However, generating optical GKP qubits has been a significant challenge. Measurement-based methods, where a portion of entangled squeezed vacuum modes are measured with photon number resolving detectors heralding a desired state in the undetected modes, have emerged as leading candidates for optical GKP qubit generation due their minimal resource requirements. While the current measurement-based methods can produce high-quality GKP qubits, they suffer from low success probabilities limiting experimental realization. The heart of the problem lies in the duality of photon number resolving measurements, being both the source of nonlinearity needed to generate quality GKP qubits and the component driving down their probability of successful production. Our method, breeding approximate squeezed coherent state superpositions created by generalized photon subtraction, overcomes this problem by supplementing two photon number resolving measurements with a single high success probability homodyne measurement. This scheme achieves success probabilities $\ensuremath{\ge}{10}^{\ensuremath{-}5}$, two orders of magnitude higher than strictly photon number resolving measurement based methods, while still producing states with high fidelity, possessing error-correction capabilities equivalent to up to a 10 dB squeezed GKP qubit. This breakthrough significantly advances the practical use of the optical GKP qubit encoding.
Continuous-variable measurement-based quantum computation on cluster states has in recent years shown great potential for scalable, universal, and fault-tolerant quantum computation when combined with the Gottesman-Kitaev-Preskill (GKP) code and quantum error correction.However, no complete fault-tolerant architecture exists that includes everything from cluster-state generation with finite squeezing to gate implementations with realistic noise and error correction.In this work, we propose a simple architecture for the preparation of a cluster state in three dimensions in which gates can be efficiently implemented by gate teleportation.To accommodate scalability, we propose architectures that allow both spatial and temporal multiplexing, with the temporally encoded version requiring as little as two squeezed light sources.Because of its three-dimensional structure, the architecture supports topological qubit error correction, while GKP error correction is efficiently realized within the architecture by teleportation.To validate fault tolerance, the architecture is simulated using surface-GKP codes, including noise from GKP states as well as gate noise caused by finite squeezing in the cluster state.We find a fault-tolerant squeezing threshold of 12.7 dB, with room for further improvement.
Building a useful photonic quantum computer requires robust techniques to synthesize optical states that can encode qubits. Gottesman–Kitaev–Preskill (GKP) states1 offer one of the most attractive classes of such qubit encodings, as they enable the implementation of universal gate sets with straightforward, deterministic and room temperature-compatible Gaussian operations2. Existing pioneering demonstrations generating optical GKP states3 and other complex non-Gaussian states4–11 have relied on free-space optical components, hindering the scaling eventually required for a utility-scale system. Here we use an ultra-low-loss integrated photonic chip fabricated on a customized multilayer silicon nitride 300-mm wafer platform, coupled over fibre with high-efficiency photon number resolving detectors, to generate GKP qubit states. These states show critical mode-level features necessary for fault tolerance, including at least four resolvable peaks in both p and q quadratures, and a clear lattice structure of negative Wigner function regions, in this case a 3 × 3 grid. We also show that our GKP states show sufficient structure to indicate that the devices used to make them could, after further reduction in optical losses, yield states for the fault-tolerant regime. This experiment validates a key pillar of bosonic architectures for photonic quantum computing2,12, paving the way for arrays of GKP sources that will supply future fault-tolerant machines. An ultra-low-loss integrated photonic chip fabricated on a customized multilayer silicon nitride 300-mm wafer platform, coupled over fibre with high-efficiency photon number resolving detectors, is used to generate Gottesman–Kitaev–Preskill qubit states.
We present an architecture for the generation of GKP states in which quadrature squeezing operations are used to control the average photon number statistics of probabilistic photon number measurements on Gaussian resource states. Specifically, we present an architecture employing a teleportation-based squeezing protocol and polynomial-gate applications integrated into a time-multiplexed multi-mode cluster state to generate cat states with high amplitudes, which are consequently used to generate GKP states with high quadrature effective squeezing. Compared to our previous work, in addition to using squeezing as a resource, the present architecture reduces damping and noise by minimizing the number of homodyne measurements required in GKP state generation. We demonstrate the effectiveness of these improvements - including dynamic input-state resetting and an improved breeding algorithm - by achieving a fault-tolerance threshold of 11.5 dB cluster squeezing using the RHG surface code for error correction, without requiring active switching or photon-number resource states.