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Quantum Implicit Neural Representations
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Proposes QIREN, a hybrid quantum-classical generalization of Fourier neural networks built from data re-uploading quantum circuits, claimed to represent Fourier series more parameter-efficiently and to outperform SIREN and RFF-MLPs on signal representation, super-resolution, and INR-based image generation in simulation.
Problem
Classical ReLU-based MLPs struggle to represent high-frequency signal components, and classical Fourier neural networks (RFF, SIREN) need growing parameter counts; the paper asks whether quantum circuits can represent signals with fewer parameters in a practical INR setting.
Approach
Stack hybrid layers, each a linear layer with BatchNorm feeding a data re-uploading quantum circuit (simulated with PennyLane/TorchQuantum) whose expectation values form the next layer's input; the authors argue the circuit's output is a Fourier series whose spectrum can grow exponentially with circuit size, and that a learnable linear encoding expands the spectrum.
Key results
- On image representation, QIREN reaches MSE 4.0/1.1/1.5 (x1e-3) on Astronaut/Camera/Coffee with 657 parameters vs SIREN 9.0/1.5/2.3 with 701 parameters.
- On image generation at 32x32, QIREN achieves FID 11.53 on FFHQ and 11.78 on CelebA-HQ, versus SIREN 22.31 and 20.97 and ReLU+RFF 15.01 and 13.91.
- The paper reports a reduction in fitting error of up to 35% relative to the SOTA model on signal representation; the StyleGAN2 baseline still achieves better FID (5.14) than the QIREN-based generator in the ablation.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use QIREN instead of SIREN | implicit neural representation · Cello sound representation | MSE (× 10^-3): 5.5 -> 5.5 | No clear effect |
| use QIREN instead of SIREN | implicit neural representation · Astronaut, Camera, and Coffee image representation | MSE (× 10^-3): 9.0 -> 4.0; 1.5 -> 1.1; 2.3 -> 1.5 | Helped |
| use QIREN instead of Bilinear | implicit neural representation · Astronaut, Camera, and Coffee image superresolution | MSE (× 10^-3): 25.2 -> 24.3; 9.2 -> 7.9; 12.2 -> 9.4 | Helped |
| use QIREN instead of StyleGAN2 | image generator · FFHQ image generation | FID: 5.14 -> 11.53 | Hurt |
| use QIREN instead of ReLU+RFF | image generator · CelebA-HQ image generation | FID: 13.91 -> 11.78 | Helped |
| add BatchNorm (ablation) | QIREN · Cello sound representation | loss | Helped |
| increase circuit depth (number of data re-uploadings) (ablation) | QIREN · Cello sound representation | representation capability | Helped |
| add noise with RX rotations before measurement (ablation) | QIREN · Cello sound representation | signal representation capability | Hurt |
| replace CNOT gates with CZ gates (ablation) | QIREN · Cello sound representation | signal representation performance | No clear effect |
| add Linear layers (ablation) | pure quantum circuit · Astronaut, Camera, and Coffee image representation | MSE (× 10^-3): 71.1 -> 4.0; 51.9 -> 1.1; 73.2 -> 1.5 | Helped |
| add factorized multiplicative modulation (ablation) | QIREN generator · FFHQ image generation | FID: 11.77 -> 11.53 | Helped |
| expand encoding dimension from 8 to 32 (ablation) | QIREN generator · FFHQ image generation | FID: 11.53 -> 7.67 | Helped |
Surprising findings
- Pure quantum circuit ablation with 72 parameters performs far worse (MSE 71.1 on Astronaut) than the hybrid QIREN, indicating hybrid structure is necessary under current hardware limits.
- On image generation, the hypernetwork could not generate parameters for quantum components directly, so an extra ReLU MLP was added after QIREN.
Limitations
- All experiments run on quantum simulators (PennyLane, TorchQuantum), not real quantum hardware.
- Quantum advantage claim relies on optimal-coefficient assumptions and a Hamiltonian locality assumption that is not verified empirically.
- On image generation, QIREN-based generator underperforms StyleGAN2 in FID, and experiments are limited to 32x32 resolution.
Experimental setup
Datasets: Bach Cello Suite excerpt Astronaut Camera Coffee FFHQ CelebA-HQ
Benchmarks: Bach Cello Suite sound representation Astronaut/Camera/Coffee image representation image superresolution 32->64 FFHQ FID CelebA-HQ FID
Metrics: MSE FID
Baselines: ReLU-based MLP Tanh-based MLP ReLU+RFF SIREN Nearest-neighbor interpolation Bilinear interpolation StyleGAN2
Builds on: SIREN Random Fourier Features data re-uploading quantum circuits StyleGAN2 exponential encoding scheme
Closest papers by meaning
- Quantum Visual Fields with Neural Amplitude Encoding (NeurIPS 2025 · similarity 0.86)
- Learning Spatially Collaged Fourier Bases for Implicit Neural Representation (AAAI 2024 · similarity 0.83)
- A sampling theory perspective on activations for implicit neural representations (ICML 2024 · similarity 0.83)
- A Unified Theory of Quantum Neural Network Loss Landscapes (ICLR 2025 · similarity 0.83)
- Implicit Neural Representations and the Algebra of Complex Wavelets (ICLR 2024 · similarity 0.83)
- Representing Sounds as Neural Amplitude Fields: A Benchmark of Coordinate-MLPs and a Fourier Kolmogorov-Arnold Framework (AAAI 2025 · similarity 0.83)
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