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.

• The paper introduces a continuous-variable quantum Fourier layer that maps Cooley–Tukey FFT butterflies to phase rotations and 50:50 beam splitters, enabling a 2D Fourier transform with O(m log m + n log n) gates.
• The paper encodes an m×n matrix into the cross-covariance of a bipartite Gaussian state through its singular value decomposition, avoiding noncommuting two-mode-squeezing operations while preserving the matrix structure.
• The paper demonstrates machine-precision spectral filtering and heat-equation integration in simulation, but the Gaussian design remains classically simulable and requires costly classical SVD preprocessing and hardware validation.

Overview and motivation

Fourier representations underpin operator learning for partial differential equations, and this paper develops a continuous-variable (CV) quantum realization of the two-dimensional discrete Fourier transform (DFT) within Gaussian photonic circuits. The central observation is a structural isomorphism between the Cooley–Tukey (CT) butterfly network of the fast Fourier transform and the native gate set of CV quantum computing: each radix-2 butterfly, which maps a pair $(a,b)$ to $(a+\omega_n^k b,\, a-\omega_n^k b)$ with twiddle factor $\omega_n^k = e^{-2\pi i k/n}$, decomposes exactly into a phase rotation $R(\varphi_k+\pi)$ followed by a 50:50 beam splitter $\mathrm{BS}(\pi/4,0)$. Because the beam splitter is unitary, the $1/\sqrt{2}$ normalization per stage arises automatically. The authors claim this is the first CV quantum Fourier layer (CV–QFL) that explicitly integrates CV circuits with the CT framework.

The motivation stems from a recognized gap in quantum machine learning (QML): existing CV encodings embed classical data as vectors or local pointwise features, which prevents the representation of a full discretized field as a globally structured object at the quantum-state level. Such structured representation is a prerequisite for multidimensional spectral processing in operator-learning settings such as Fourier Neural Operators.

Bipartite Gaussian encoding

The encoding scheme embeds a matrix $D \in \mathbb{R}^{m \times n}$ exactly into the cross-covariance block of a bipartite Gaussian state over $m+n$ bosonic modes. A naive entry-wise encoding via two-mode squeezing (TMS) gates, assigning $$r_{ij} = \tfrac12 \operatorname{arcsinh}(\lambda D_{ij})$$, is shown to be physically unrealizable because TMS gates acting on shared modes do not commute, so later operations distort previously established correlations.

The authors resolve this by working in the singular value decomposition $D = U\Sigma V^\top$. In the rotated basis, each singular value corresponds to an independent mode pair, so squeezing parameters $(a+\omega_n^k b,\, a-\omega_n^k b)$0 can be assigned without inter-register interference. Applying the Clements-decomposed interferometers $(a+\omega_n^k b,\, a-\omega_n^k b)$1 and $(a+\omega_n^k b,\, a-\omega_n^k b)$2 to the two registers then reconstructs the full matrix in the $(a+\omega_n^k b,\, a-\omega_n^k b)$3-quadrature cross-block: $(a+\omega_n^k b,\, a-\omega_n^k b)$4, with $(a+\omega_n^k b,\, a-\omega_n^k b)$5 and vanishing $(a+\omega_n^k b,\, a-\omega_n^k b)$6 cross-correlations. For real data matrices, $(a+\omega_n^k b,\, a-\omega_n^k b)$7 and $(a+\omega_n^k b,\, a-\omega_n^k b)$8 are orthogonal, so the symplectic transformation reduces exactly to the SVD reconstruction. The encoding circuit has gate count $(a+\omega_n^k b,\, a-\omega_n^k b)$9 and depth $\omega_n^k = e^{-2\pi i k/n}$0, dominated by the interferometric stages. The inter-register entanglement spectrum is fixed entirely by the singular values of $\omega_n^k = e^{-2\pi i k/n}$1, since local symplectic transformations preserve entanglement.

The CV–QFL and its complexity

The Fourier layer applies independent CT QFT circuits to the two registers. Because the two-dimensional DFT factorizes as $\omega_n^k = e^{-2\pi i k/n}$2, the transform acts on the bipartite state as a tensor-product structure, and the result is read directly from the covariance matrix:

$\omega_n^k = e^{-2\pi i k/n}$3

with the real part carrying cosine components and the imaginary part sine components, both accessible via homodyne detection. Notably, unlike the one-dimensional case, the $\omega_n^k = e^{-2\pi i k/n}$4 block contributes a nontrivial term because both registers are transformed; the authors handle this explicitly in the derivation.

The claimed complexity advantage is that the bipartite CV implementation requires $\omega_n^k = e^{-2\pi i k/n}$5 gates with depth $\omega_n^k = e^{-2\pi i k/n}$6, versus $\omega_n^k = e^{-2\pi i k/n}$7 for a classical two-dimensional FFT on a flattened grid. This is a genuine structural reduction in gate count, but it should be interpreted carefully: the classical FFT operates on $\omega_n^k = e^{-2\pi i k/n}$8 data points, while the CV circuit operates on $\omega_n^k = e^{-2\pi i k/n}$9 modes whose cross-correlations encode $R(\varphi_k+\pi)$0 numbers. The comparison is between circuit resources for the transform itself, not an end-to-end computational speedup.

Results: filtering and heat equation

Both experiments are simulated on the Gaussian backend of Strawberry Fields, which propagates only first and second moments and is therefore itself efficiently classically simulable.

Spectral low-pass filtering. A $R(\varphi_k+\pi)$1 signal with exactly sparse spectrum (8 nonzero bins) is corrupted by white Gaussian noise at input SNR of $R(\varphi_k+\pi)$2 dB. Filtering is implemented optically via the LossChannel, whose phase-insensitivity rescales each complex Fourier coefficient as $R(\varphi_k+\pi)$3 without altering phase, so binary transmissivities realize a separable rectangular mask. The CV–QFL achieves an SNR improvement of $R(\varphi_k+\pi)$4 dB versus $R(\varphi_k+\pi)$5 dB for a classical circular-mask baseline; the gap is attributed to the larger rectangular mask retaining 10.8% versus 7.7% of bins. Against a classical reference using the same rectangular mask, the maximum pointwise error is $R(\varphi_k+\pi)$6, i.e., machine precision. This confirms exactness of the photonic spectral processing, though the comparison is against simulation, not hardware.

Heat equation. On a $R(\varphi_k+\pi)$7 grid with $R(\varphi_k+\pi)$8, the exact spectral propagator $R(\varphi_k+\pi)$9 is realized by programming LossChannel transmissivities $\mathrm{BS}(\pi/4,0)$0, exploiting the same $\mathrm{BS}(\pi/4,0)$1 scaling. The only architectural difference from the filter is the transmissivity profile. Maximum absolute error against the classical pseudospectral solver remains at $\mathrm{BS}(\pi/4,0)$2 level across all four time steps up to $\mathrm{BS}(\pi/4,0)$3. These results establish that Gaussian photonic circuits can implement exact Fourier-domain PDE integration, at least for diagonal-in-Fourier-space propagators.

Limitations and open questions

The paper is candid about several restrictions. First, the architecture is entirely Gaussian and hence efficiently simulable classically; the authors explicitly disclaim any computational quantum advantage, noting that non-Gaussian resources would be needed for expressivity beyond classical simulation, at substantially greater experimental complexity. Second, the encoding is intrinsically two-dimensional, and extension to three-dimensional or spatio-temporal fields requires new register structures. Third, the SVD must be computed classically in advance at cost $\mathrm{BS}(\pi/4,0)$4, a significant preprocessing bottleneck for large inputs; the authors acknowledge that the encoding's value lies in structured optical representation rather than in data-loading speedup for arbitrary classical matrices. Fourth, the spectral processing is separable mode-by-mode, and richer inter-register couplings remain unexplored. Finally, all validation is numerical simulation; no photonic hardware results are reported, so loss, noise, and squeezing-budget constraints of real devices are untested. The paper also leaves open whether the framework can be integrated into a trainable quantum neural operator for many-body Hamiltonian dynamics, where the authors argue classical simulation becomes intractable and quantum advantage is most plausible.

Conclusion

This paper establishes a precise correspondence between the CT butterfly network and Gaussian photonic gates, and builds on it a bipartite Gaussian encoding that embeds a two-dimensional matrix exactly into the cross-covariance block of a multimode state. The resulting CV–QFL performs an exact two-dimensional QFT with gate complexity $\mathrm{BS}(\pi/4,0)$5 and reproduces both spectral filtering and Fourier-domain heat-equation integration to machine precision in simulation. Its principal value is architectural rather than computational: it supplies a native spectral primitive for CV photonics, most naturally suited to inputs that are already optical, and provides a foundation for CV quantum neural operators. Whether the framework can transcend the Gaussian, classically simulable regime and demonstrate advantage on intrinsically quantum problems remains the central open question.