QFGN: A Quantum Approach to High-Fidelity Implicit Neural Representations

TL;DR

This work tackles the challenge of high-frequency fidelity in implicit neural representations (INRs) by introducing QFGN, a hybrid classical-quantum model that combines a Fourier-Gaussian feature scaling (FGFS) layer with a data-encoding quantum circuit to realize a Fourier-series-like function with an expanded frequency spectrum. The quantum circuit encodes inputs via $S(\mathbf{x}) = e^{-i x H}$ and yields outputs that form a Fourier series with spectrum $\Omega_{KJ} = \{\Lambda_K - \Lambda_J\}$, while the FGFS layer expands and balances frequencies so that the overall function captures both low- and high-frequency content. Empirical results on medical image reconstruction and super-resolution show consistent PSNR/SSIM gains over state-of-the-art INR baselines, and hardware experiments on IBM devices demonstrate viability of QML in the Noisy Intermediate-Scale Quantum (NISQ) regime with some mitigations. The work provides theoretical and practical insights into quantum advantages for continuous function fitting and points to broader QML applications in imaging.

Abstract

Implicit neural representations have shown potential in various applications. However, accurately reconstructing the image or providing clear details via image super-resolution remains challenging. This paper introduces Quantum Fourier Gaussian Network (QFGN), a quantum-based machine learning model for better signal representations. The frequency spectrum is well balanced by penalizing the low-frequency components, leading to the improved expressivity of quantum circuits. The results demonstrate that with minimal parameters, QFGN outperforms the current state-of-the-art (SOTA) models. Despite noise on hardware, the model achieves accuracy comparable to that of SIREN, highlighting the potential applications of quantum machine learning in this field.

Figures (6)

• Figure 1: The overall framework of QFGN. In the Fourier part $h_1$, only the bias $b$ is trainable
• Figure 2: The combination of low frequency and high frequency with different amplitudes. $\phi_1 (x)$ is a low-frequency function with high amplitude, $\phi_2 (x)$ is a high-frequency function with low amplitude, $\phi_3 (x)$ is a combination of $\phi_1 (x)$ and $\phi_2 (x)$ to mimic the spectral bias and $\phi_4 (x)$ is a combination of low-frequency and high-frequency with the same amplitude, an expected output of $h_2$.
• Figure 3: Image representation of each model and the corresponding frequency error with regard to the ground truth. The dark color indicates a large error.
• Figure 4: Image super- resolution of each model.
• Figure 5: The output from IBM-Sherbrooke.