Polynomial neural surrogates for designing photonic quantum experiments

Rohit Chaurasiya; Xuemei Gu

quant-phOct 5, 2026

Polynomial neural surrogates for designing photonic quantum experiments

Authors: Rohit Chaurasiya, Xuemei Gu

Organizations: Institute of Condensed Matter Theory and Optics, Friedrich Schiller University Jena, Max-Wien-Platz 1, 07743 Jena, Germany

Abstract

Physics simulators can support the discovery of quantum experiments by predicting the states generated by experimental configurations. When these simulators are computationally expensive, repeated simulator calls can limit the search for experiments that generate a desired quantum state. Here, we develop a physics-inspired polynomial neural surrogate for PyTheus, a graph-based quantum-optics simulator, to predict quantum states and use it to design quantum experiments. Its polynomial activations are motivated by the relation between graph perfect matchings and the resulting state amplitudes. We train separate surrogate models for four-, six-, and eight-photon systems and show that they achieve higher prediction accuracy with fewer trainable parameters than standard multilayer perceptrons. We then use the trained surrogates for inverse design of GHZ, W, and linear-cluster states. For the larger systems, the surrogates also enable faster inverse design than direct optimization with PyTheus. These results suggest that incorporating the underlying physics into neural surrogates can provide an efficient approach to quantum experiment design.

Figures & tables

Figure 1: Graph representation and perfect matching contributions. (a) Fully connected four-vertex graph. Blue and red colours represent the photon states $\ket{0}$ and $\ket{1}$ , respectively. An edge joining vertices $a$ and $b$ with photon states $i$ and $j$ has weight $w_{ab}^{ij}\in[-1,1]$ . The four colour combinations for each of the six vertex pairs give 24 weights, collected in the vector $\mathbf{x}$ . Diamonds indicate negative weights. (b) For $n=4$ , three perfect matchings contribute to the $\ket{1111}$ amplitude. A perfect matching is a set of edges that covers every vertex exactly once. Summing the contributions from these three matchings gives the amplitude in the unnormalised output state: $\langle 1111|\Psi_{\mathrm{post}}\rangle=w_{01}^{11}w_{23}^{11}+w_{03}^{11}w_{12}^{11}+w_{02}^{11}w_{13}^{11}$ .

Figure 2: Surrogate training and inverse design. (a) Surrogate training. A weighted graph, encoded by the graph-weight vector $\mathbf{x}$ , is passed through the neural network to predict the corresponding quantum-state amplitudes $\ket{\Psi_{\mathrm{NN}}}$ . (b) Surrogate-assisted inverse design. A randomly initialised graph-weight vector $\mathbf{x}$ is passed through the trained model, with its weights and biases frozen. The graph weights are iteratively updated to maximize the target evaluated from the predicted state until the stopping criterion is reached.

$n$	Model	Layers	Neurons	Activation	Best epoch	Epochs trained	Best val. loss	Training time	Parameters
$4$	PNN	1	400	$z^{2}$	2506	4506	$9.13\times 10^{-7}$	$0.93\,\mathrm{h}$	16,416
$4$	FNN	3	400	GELU	19819	20000	$2.05\times 10^{-5}$	$23.12\,\mathrm{h}$	337,216
$6$	PNN	1	2000	$z^{3}$	2497	4497	$3.90\times 10^{-6}$	$3.81\,\mathrm{h}$	250,064
$6$	FNN	3	2000	GELU	19975	20000	$2.21\times 10^{-4}$	$292.63\,\mathrm{h}$	8,254,064
$8$	PNN	1	15000	$z^{4}$	2984	4984	$5.93\times 10^{-5}$	$55.6\,\mathrm{h}$	5,535,256
$8$	FNN	3	8000	GELU	3119	5119	$2.85\times 10^{-1}$	$1021.08\,\mathrm{h}$	130,968,256

Table 1: Comparison of the neural-network architectures used for surrogate training. The best epoch corresponds to the minimum validation loss, while epochs trained denotes the total number of epochs before termination.

Figure 3: Training and validation loss curves for the PNN and FNN surrogate models for

n=4

6

, and

8

. The mean absolute error (MAE) between the predicted and PyTheus-generated unnormalised output states is shown on a logarithmic scale. Across all system sizes, the PNN achieves a lower validation loss than the MLP. For

n=8

, MLP training stopped after

2000

epochs without further improvement. The best validation losses for all architectures and system sizes are reported in Table 1 .

Figure 4: Test-set prediction errors of the trained PNN surrogates. Distributions of the sample-wise mean absolute error (MAE) for the (a)

n=4

, (b)

n=6

, and (c)

n=8

systems. Each histogram contains all two million samples from the corresponding held-out test set. For each sample, the MAE is the mean absolute difference between the PNN-predicted and PyTheus-generated unnormalised output-state amplitudes over all output components. The red vertical line indicates the mean sample-wise MAE for each system size. The horizontal axes are shown in units of

10^{-6}

Figure 5: Jacobian sensitivity comparison for $n=4$ . (a) Mean absolute sensitivities from the PNN (left) and analytical perfect-matching calculation (right), averaged over $1000$ sampled graphs. (b) Absolute element-wise difference between the two sensitivity matrices. The similar patterns and small differences show that the PNN reproduces the sensitivity of the output amplitudes to variations in the graph weights

Figure 6: Surrogate-assisted inverse design. Distributions of the initial and final fidelities for

1000

random graphs for each target state and system size, with each graph optimized independently using the PNN surrogate. Blue and orange indicate the initial and final fidelities, respectively, with counts shown on a logarithmic scale.

Target	$n$	$\overline{F}_{\mathrm{PNN}}$	$\overline{F}_{\mathrm{PyTheus}}$	PyTheus-verified fraction
				$F\geq 0.9999$	$F\geq 0.99999$
GHZ	4	0.999991	0.999991	100%	84.4%
	6	0.999991	0.999991	100%	85.3%
	8	0.999991	0.999991	100%	95.1%
W	4	0.999990	0.999990	100%	58.6%
	6	0.999991	0.999991	100%	83.2%

Table 2: PyTheus verification of the graphs designed using the PNN surrogate. Each row contains

1000

graphs. The last two columns report the fraction with PyTheus-verified fidelities above the indicated thresholds. Samples with fidelities slightly below

0.99999

correspond to runs that reach the maximum optimization step limit.

Target	$n$	PyTheus CPU (s)	PNN CPU (s)	PNN GPU (s)
GHZ	4	$1.44\pm 0.81$	$2.88\pm 0.27$	$1.28\pm 0.14$
	6	$56.56\pm 71.79$	$3.65\pm 1.68$	$2.87\pm 2.88$
	8	$2754.79\pm 1552.90$	$12.56\pm 11.13$	$5.86\pm 5.04$
W	4	$0.71\pm 0.38$	$2.89\pm 0.30$	$1.71\pm 0.30$
	6	$20.41\pm 5.06$	$4.96\pm 3.13$	$5.81\pm 6.38$
	8	$1523.95\pm 166.61$	$34.96\pm 27.05$	$13.13\pm 11.84$

Table 3: Inverse-design runtimes over

50

runs, reported as mean

\pm

standard error of the mean. PyTheus was evaluated on CPU, as GPU acceleration is not available in PyTheus, while the PNN was evaluated on both CPU and GPU.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Jacobian-based sensitivity comparison for the $n=6$ graph system. (a) Mean absolute sensitivities from the PNN (left) and the analytical perfect-matching calculation (right), evaluated on the same $1000$ sampled graphs. Rows represent output-state amplitudes and columns represent graph weights. (b) Absolute element-wise difference between the two sensitivity matrices. Their matching patterns and small differences show that the PNN reproduces the mean absolute sensitivities of the output amplitudes to the graph weights.

Figure 8: Jacobian-based sensitivity comparison for the $n=8$ graph system. (a) Mean absolute sensitivities from the PNN (left) and the analytical perfect-matching calculation (right), evaluated on the same $1000$ sampled graphs. Rows represent output-state amplitudes and columns represent graph weights. (b) Absolute element-wise difference between the two sensitivity matrices. Their matching patterns and small differences show that the PNN reproduces the mean absolute sensitivities of the output amplitudes to the graph weights.

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For quantum machine learning, the exact boundary between classical and quantum advantage is still poorly understood. Direct comparison between quantum neural networks (QNNs) and existing classical models, which encompass fundamentally different function classes, often fails to provide broader insight into the difference between the two. Inspired by the techniques of Neural Quantum States and Random Fourier Features, this work introduces Neural Fourier Surrogates (NFS), a stochastic classical neural network architecture for efficiently learning coefficients over the same finite Fourier series support as quantum neural networks. Testing on a selection of tabular benchmark datasets, we find that NFS is an effective classifier architecture broadly competitive with established classical baselines, including a comparable Random Fourier Features model, and possessing comparable performance to data-reuploading QNNs; combined with additional analysis comparing the learned Fourier spectra of QNNs and NFS on synthetic data, these results establish NFS as a natural classical baseline for evaluating QNN performance.

Abstract

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