A Continuous-Variable Quantum Fourier Layer: Applications to Filtering and PDE Solving

Abstract

Fourier representations play a central role in operator learning methods for partial differential equations and are increasingly being explored in quantum machine learning architectures. The classical fast Fourier transform (FFT), particularly in its Cooley--Tukey decomposition, exhibits a structure that naturally matches continuous-variable quantum circuits. This correspondence establishes a direct structural isomorphism between the Cooley-Tukey butterfly network and Gaussian photonic gates, enabling the FFT to be realized as a native optical computation in continuous-variable quantum computing. Building on this observation, we introduce a continuous-variable Quantum Fourier Layer (CV--QFL) based on a bipartite Gaussian encoding and a Cooley-Tukey quantum Fourier transform, enabling exact two-dimensional spectral processing within a Gaussian photonic circuit. We test the CV--QFL on two representative tasks: spectral low-pass filtering and Fourier-domain integration of the heat equation. In both cases, the results match the classical reference to machine precision. Beyond these examples, our method naturally extends to optical-input settings in which the signal is already available as a Gaussian optical field. In such scenarios, coherent light coupled into single-mode waveguides can be processed directly by the CV--QFL, bypassing the need for an explicit classical-to-quantum encoding stage. This enables native spectral processing of light and lays the groundwork for new approaches to quantum scientific machine learning, in particular for future neural operator architectures within the CV framework.

Figures (6)

• Figure 1: Butterfly diagram of the CT algorithm for the FFT of the bit-reversed input $x\in\mathbb{R}^8$. Each cross is a radix-2 butterfly with twiddle factor $\omega_n^k = e^{-i2\pi k/n}$.
• Figure 2: CV implementation of a single CT butterfly. The phase gate $R(\varphi_k{+}\pi)$ maps $b\mapsto -\omega_n^k b$; the beam splitter $\mathrm{BS}(\pi/4,0)$ then produces $(a\pm\omega_n^k b)/\sqrt{2}$, identical to the classical output \ref{['eq:butterfly']} up to the $1/\sqrt{2}$ ortho-normalisation factor.
• Figure 3: CV--QFL circuit for an $m \times n$ input matrix on $m+n$ optical modes. Encoding: TMS gates inject the singular values of $M$; interferometers $\mathcal{U}$ and $\mathcal{V}^\top$ reconstruct the input matrix via SVD. Fourier Layer: the CT QFT is applied independently to both registers, yielding the 2D optical Fourier transform.
• Figure 4: Comparison of the spectral masks used in the filtering experiment: a circular mask with $|\mathbf{k}| \leq 10$ and a separable rectangular mask with $|k_x| \leq 10$ and $|k_y| \leq 10$. The signal frequency components lie within both masks.
• Figure 5: Spectral low-pass filtering on a $64\times64$ image. Left to right: clean signal; noisy input ($\mathrm{SNR}=-3.0\,\mathrm{dB}$); classical output (circular mask, $+10.7\,\mathrm{dB}$); CV--QFL output (rectangular separable mask, $+9.4\,\mathrm{dB}$). The photonic circuit implements the rectangular spectral mask to machine-precision accuracy.