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.
Tensor networks and Matrix Product States (MPS) are commonly used to efficiently perform classical simulations of specific quantum circuits with low entanglement. In this work, we propose an approach to use MPS as an efficient framework for quantum kernel simulation for binary classification via SVMs. To this end, classical data are first embedded into higher-dimensional quantum states using shallow ZZ feature maps. We employ ZZ feature maps with low connectivity that limits entanglement, enabling us to assess whether effective classical data encoding can be achieved while retaining efficient classical simulation. This way, quantum kernels are efficiently computed through MPS overlap. We study the computational cost and accuracy as a function of the depth of the feature maps. Experiments on different classical datasets show that low entanglement quantum feature maps can achieve competitive classification performance at the same time as being efficiently simulable on classical computers.