Rapid Gaussian Boson Sampling Circuit Screening for GKP States Creation via a Two-Stage Machine Learning Surrogate

Abstract: 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 $R^2 = 0.837$, reducing the total simulation burden by approximately 90%.

• The paper introduces a two-stage machine learning surrogate that filters GBS circuits to rapidly screen for high-fidelity GKP state generation.
• It employs HistGradientBoosting estimators with physics-informed features to predict heralding patterns and regress both fidelity and post-selection probability.
• The surrogate achieves a 90% reduction in simulation time for five-mode circuits while highlighting challenges like sign-convention errors and out-of-distribution performance.

Two-Stage Machine Learning Surrogate for Rapid GBS Screening of GKP State Generation

Introduction

The preparation of Gottesman–Kitaev–Preskill (GKP) states is a central requirement for scalable, fault-tolerant photonic quantum computing. GKP states serve as non-Gaussian resources that enable robust bosonic error correction, being naturally compatible with Gaussian operations but requiring significant non-Gaussianity for their generation. Gaussian Boson Sampling (GBS) provides an all-photonic route, leveraging measurement-induced nonlinearity without matter-based ancilla, but is fundamentally bottlenecked by the computational complexity of hafnian evaluation for photon-number probabilities—a process scaling exponentially with system size. The paper "Rapid Gaussian Boson Sampling Circuit Screening for GKP States Creation via a Two-Stage Machine Learning Surrogate" (2606.05992) introduces a practical framework to circumvent this bottleneck through a ML surrogate model, drastically reducing the computational resources needed to screen photonic circuits for GKP state preparation.

Problem Statement and Context

GKP state generation on optical platforms involves the design of multimode Gaussian circuits, including squeezed-state sources, a programmable interferometer, and photon-number-resolving (PNR) detection with post-selected heralding. The post-selected heralding pattern determines the non-Gaussian projected state in the unmeasured mode. The critical parameters for each configuration are the detailed squeezing profile, interferometer angles/phases, and the choice of heralding pattern.

The global optimization task is to find (1) circuit parameters and (2) a heralding pattern that jointly maximize the fidelity to a finite-energy GKP codeword, subject to a threshold (typically $F \ge 0.90$), while also ensuring that the post-selection probability is experimentally viable (as high as possible, given the exponential suppression with circuit size and target squeezing). The evaluation of each circuit/pattern pair requires exact computation of the probability and output state (and thus fidelity) via hafnian evaluation—a process that is $\#P$-hard.

For realistic circuit sizes—e.g. five-mode GBS circuits with Fock basis truncation $n_{\text{max}} = 12$—a single evaluation of a candidate configuration takes several minutes, with thousands of configurations needed in global optimizations. This computational scaling severely impedes interactive design, exhaustive searches, or even statistical explorations for photonic GKP circuits.

Surrogate Model Architecture

To alleviate this bottleneck, the authors present a surrogate model pipeline, built as a two-stage cascade of histogram gradient boosting (HistGradientBoosting) estimators. This surrogate model is designed and validated as follows:

• Stage 1 (Pattern Classifier): A multiclass classifier predicts the optimal heralding pattern for a GBS circuit, based only on the circuit parameters and configuration metadata.
• Stage 2 (Regressors): Two regressors—one for fidelity, one for post-selection log-probability—receive as input both the circuit parameters and summary statistics of the predicted heralding pattern.

Key architectural decisions include:

• Augmenting raw circuit parameters with a suite of physics-informed aggregate features (e.g., total squeezing power, asymmetry metrics, beamsplitter phase coherence), which enables generalization across circuit topologies.
• The use of a fixed-dimensional feature vector (by zero-padding parameters for small circuits), supporting circuit mode counts in $\{3,4,5\}$ without separate retraining.
• A conditional validation pipeline: all circuits passing the $F \ge 0.90$ threshold by the surrogate are subjected to full quantum simulation using Strawberry Fields and thewalrus for the final fidelity, probability, Wigner function, and Wigner log-negativity.

The training set comprises 689 optimized GBS configurations across varying mode counts and squeezing parameters, with evaluation on a 170-sample holdout and targeted failure case studies.

Quantitative Performance

On the held-out test set, the surrogate pipeline achieves:

• GKP-detection accuracy of 90.0%, a 23.7 percentage-point improvement over the majority-class baseline (66.3%).
• Fidelity mean absolute error (MAE) of 0.032, corresponding to an average 3.2% error compared to exact quantum simulation.
• Post-selection log-probability $R^2 = 0.837$ (log-space), with MAE of 0.432, providing reliable order-of-magnitude probability estimation.

Notably, pattern classification accuracy is only 64%, introducing cascade errors in about a third of the samples and highlighting a key source of regression performance degradation.

For the most demanding five-mode configurations, this surrogate-based pipeline reduces the simulation burden by approximately 90%: exhaustive screening of 10,000 candidates (requiring ≈12,500 CPU-hours) is reduced to about 1,250 CPU-hours when only the surrogate-endorsed 10% are simulated. This renders previously intractable search or exploration tasks feasible within practical timeframes.

Limitations and Failure Analysis

Detailed analysis reveals that the surrogate model exhibits systematic failures in out-of-distribution regions, particularly for squeezing sign conventions inadequately represented in the training set. The underlying architecture combines sign-invariant and sign-sensitive features, leading to errors as high as MAE ≈ 0.93 in the worst case (cases F1, F2).

Because the surrogate is strictly a filter—and all GKP-capable predictions are verified via exact simulation—such misclassifications are contained and do not affect final characterizations. However, the risk remains for false negatives (GKP-capable circuits discarded by the surrogate but not simulated) which could impact completeness in high-precision tasks. The model also does not predict Wigner log-negativity, so some circuits that meet the fidelity threshold are "classically simulable" and not actually non-Gaussian enough for fault tolerance.

The holdout $R^2$ for fidelity regression is 0.76, but cross-validation variability is pronounced—$0.689 \pm 0.294$—implying poor reliability in some parameter subspaces. These findings suggest that the model's extrapolation capacity is weak and that any application outside the training domain would require retraining and architectural revision.

Implications and Future Directions

The surrogate framework presented in this study enables tractable, high-throughput screening for GKP state generation in GBS photonic circuits. This capability directly addresses the key computational bottleneck in the search and design of non-Gaussian resource states for quantum information processing.

From a practical standpoint, the surrogate drastically accelerates large-scale circuit design, permitting broad sweeps over parameter space and focusing expensive quantum simulation only where warranted. The framework’s limitations—most notably sign dependence, lack of direct non-classicality screening (WLN), and generalization gaps—define immediate research avenues. Promising mitigation strategies include:

• Symmetry-augmented training and feature engineering for sign-invariance.
• Incorporation of WLN as an additional regression target.
• Adoption of active learning protocols to concentrate simulation resources in high-uncertainty or underrepresented regions of parameter space.
• Potential migration to architectures capable of joint pattern-fidelity-probability modeling without cascade vulnerabilities.

The approach outlined here is extensible to more complex quantum resource engineering scenarios, including higher mode counts, high Fock truncation, and the full design of universal gate sets where non-Gaussianity and heralded state quality are jointly optimized. Such surrogates are also potentially valuable in closed-loop experimental optimization environments, where real-time feedback from surrogate models can guide on-hardware reconfiguration.

Conclusion

This paper introduces a two-stage ML surrogate pipeline for rapid GBS circuit screening targeting GKP state preparation, achieving a significant reduction in classical simulation resources while maintaining strong performance within its training domain. The approach leverages conditional validation, physics-informed feature construction, and gradient-boosted models to filter candidate circuits prior to expensive quantum simulation. While the surrogate’s effectiveness is limited by sign-convention sensitivity, training set coverage, and fidelity-only screening, the pipeline represents a substantial methodological advancement for scalable photonic quantum circuit design. Future improvement avenues include sign-invariant architectures, WLN regression, and active learning strategies to further enhance robustness and accuracy for universal resource state generation (2606.05992).