On Embedding Design in Quantum Physics-Informed Neural Networks
Abstract
Quantum physics-informed neural networks (QPINNs) solve partial differential equations (PDEs) by training parameterized quantum circuits against physics-based residuals, yet the role of the embedding that maps coordinates into quantum states remains insufficiently understood. In this research, we introduce a unified embedding framework that formulates embedding as a functional transformation shaping the feature representation available to the variational circuit, and we propose two embeddings within it. The Linear Quantum Neural Network Trainable Embedding QPINN (LQNN-TE-QPINN) generates data-dependent features with an auxiliary quantum circuit and adds them to the input coordinates, instead of multiplying them as in our previous formulation. The adaptive-frequency QPINN (AdaFreq-QPINN) promotes the frequency-scaling factors of an arccos-Chebyshev encoding to trainable parameters. We evaluate both against direct and fixed analytical frequency encodings and alternative hybrid architectures on one- and two-dimensional Burgers equations. In one dimension, LQNN-TE-QPINN achieves a relative L2 error of 0.0916 with 720 trainable parameters, compared with 0.4501 for direct-encoding QPINN and 0.1233 for a locally adaptive classical physics-informed neural network with approximately eleven times more parameters, while AdaFreq-QPINN attains the lowest error among the analytical embeddings with only four additional parameters. In two dimensions, LQNN-TE-QPINN attains the lowest training objective, although its solution error is not clearly separated from that of a data re-uploading QPINN. Under simulated hardware noise, LQNN-TE-QPINN retains the lowest absolute derivative error while degrading most relative to its noiseless baseline. These results demonstrate that embedding design substantially influences the approximation and optimization behavior of QPINNs.
Related
- Accelerating noisy variational quantum algorithms with physics-informed denoising networks
- Lazy training of quantum physics informed neural networks
- Physics-Informed Classical and Quantum Neural Networks for One-Dimensional Schrodinger Eigenvalue Problems
- QPI-DeepONet-MAC: A Scalable and Stable Hybrid Classical-Quantum Architecture for Physics-Informed Deep Operator Networks
- High-fidelity quantum gate learning via physics-informed neural networks
BibTeX
@misc{tran2026embedding,
title = {{On Embedding Design in Quantum Physics-Informed Neural Networks}},
author = {Ban Q. Tran and Nahid Binandeh Dehaghani and Susan Mengel and Rafal Wisniewski and A. Pedro Aguiar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33700},
eprint = {2609.33700},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33700},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Embedding Design in Quantum Physics-Informed Neural Networks
AU - Ban Q. Tran
AU - Nahid Binandeh Dehaghani
AU - Susan Mengel
AU - Rafal Wisniewski
AU - A. Pedro Aguiar
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33700
UR - https://arxiv.org/abs/2609.33700
AB - Quantum physics-informed neural networks (QPINNs) solve partial differential equations (PDEs) by training parameterized quantum circuits against physics-based residuals, yet the role of the embedding that maps coordinates into quantum states remains insufficiently understood. In this research, we introduce a unified embedding framework that formulates embedding as a functional transformation shaping the feature representation available to the variational circuit, and we propose two embeddings within it. The Linear Quantum Neural Network Trainable Embedding QPINN (LQNN-TE-QPINN) generates data-dependent features with an auxiliary quantum circuit and adds them to the input coordinates, instead of multiplying them as in our previous formulation. The adaptive-frequency QPINN (AdaFreq-QPINN) promotes the frequency-scaling factors of an arccos-Chebyshev encoding to trainable parameters. We evaluate both against direct and fixed analytical frequency encodings and alternative hybrid architectures on one- and two-dimensional Burgers equations. In one dimension, LQNN-TE-QPINN achieves a relative L_2 error of 0.0916 with 720 trainable parameters, compared with 0.4501 for direct-encoding QPINN and 0.1233 for a locally adaptive classical physics-informed neural network with approximately eleven times more parameters, while AdaFreq-QPINN attains the lowest error among the analytical embeddings with only four additional parameters. In two dimensions, LQNN-TE-QPINN attains the lowest training objective, although its solution error is not clearly separated from that of a data re-uploading QPINN. Under simulated hardware noise, LQNN-TE-QPINN retains the lowest absolute derivative error while degrading most relative to its noiseless baseline. These results demonstrate that embedding design substantially influences the approximation and optimization behavior of QPINNs.
ER -