This article presents an argument for why quantum computers could unlock new methods for machine learning. We argue that spectral methods, in particular those that learn, regularise, or otherwise manipulate the Fourier spectrum of a machine learning model, are often natural for quantum computers. For example, if a generative machine learning model is represented by a quantum state, the Quantum Fourier Transform allows us to manipulate the Fourier spectrum of the state using the entire toolbox of quantum routines, an operation that is usually prohibitive for classical models. At the same time, spectral methods are surprisingly fundamental to machine learning: A spectral bias has recently been hypothesised to be the core principle behind the success of deep learning; support vector machines have been known for decades to regularise in Fourier space, and convolutional neural nets build filters in the Fourier space of images. Could, then, quantum computing open fundamentally different, much more direct and resource-efficient ways to design the spectral properties of a model? We discuss this potential in detail here, hoping to stimulate a direction in quantum machine learning research that puts the question of ``why quantum?'' first.
Figures & tables
Figure 1: A common simplicity bias of machine learning models is their smoothness, which is linked to a decay of the model function’s Fourier spectrum. But designing such a decay in Fourier space is usually computationally expensive. Can quantum computers help here?
Figure 2: A sketch of the idea of smoothing an empirical distribution of training samples in Fourier space. The empirical distribution is sparse, with support only on the training data. Therefore, its Fourier spectrum is dense and has support on high-order frequencies which can be seen as a consequence of finite data effects. By applying a low-pass filter in Fourier space we impose smoothness on the resulting distribution.
Figure 3: An example of smoothing an empirical distribution in Fourier space to solve the generative learning problem. Starting from an empirical distribution pX of four binary data samples (top left), we compute the (Walsh) Fourier spectrum (bottom left). We then apply a filter of the form p^θ(k)=(1−2θ)∣k∣p^X(k) , where θ∈[0,1] controls the spectral decay rate. Moving back into direct space, we get a model that generalised from the training data by making the empirical distribution smoother.
Figure 4: An example of smoothing empirical distributions with quantum computers, using the same filter as in Figure 3 . Fourier filtering is performed on the amplitudes of the quantum state. We encode the empirical distribution (top left) into a superposition ∣ψX⟩ (mid left), to which we apply the (Walsh) Fourier transform (bottom left). We then apply a non-unitary transformation (with the help of ancillas) to implement the filter ψ^θ(k)∝(1−2θ)∣k∣ψ^X(k) (bottom right). Moving back into direct space (mid right), we get a quantum model with a new distribution of amplitudes, whose measurement distribution constitutes the generative model. Comparing to Figure 3 , it is clear that manipulating the Fourier coefficients of the amplitudes of a quantum model leads to a different generative model, but with a similar structure: high probabilities are amplified and low ones suppressed.
Figure 5: Illustration of the relation between a group and a homogeneous space, which are spaces where every point can be reached by acting with a group element. Group elements that map to the same point live in cosets of the stabiliser subgroup (whose elements map x0 to itself). We can associate the homogeneous space with the quotient space G/H .
Figure 6: Fourier basis functions χk(x) for the cyclic group Zd with imaginary part in blue and real part in red.
Figure 7: Fourier basis functions χk(x) for the boolean group Z2n with imaginary part in blue and real part in red.
Figure 8: Examples for basis functions m^(λ)i,j of the “marginal” subspaces, which are indicator functions that are 1 for certain permutation patterns. Shown are basis functions from the three marginal subspaces corresponding to frequencies indexed by the integer partitions (3,1) , (2,2) and (2,1,1) . It is visible that the highest order (bottom) has support over fewer permutations, than the lowest order (top). The marginal subspaces lend interpretation to Fourier coefficients, which define different subspaces that are “pure” versions of those spanned by the marginal basis.
A central design principle in modern machine learning and artificial intelligence is to align a model's inductive bias with the structure of its input data. For matrix-valued inputs, relevant matrix-level relationships can be characterised through spectral values and spectral subspaces; however, common coordinate-wise rotation-gate data-encoding unitaries used in most quantum machine learning models do not explicitly construct such a matrix-level representation. We introduce Quantum Spectral Models (QSMs), in which we construct the generator of the data-encoding unitary directly from each input matrix. We study three QSM variants based on symmetric, global block, and non-overlapping patch-local block Hamiltonians. Their outputs admit truncated Fourier representations in which input-dependent spectral gaps supply candidate phase carriers, while spectral subspaces help determine their coefficients. We evaluate the QSMs and comparison quantum models on two matrix representations of Pendigits and two controlled synthetic tasks defined by spectral statistics. At the largest evaluated circuit depth, QSM variants lead the tested quantum models in mean test accuracy across all four benchmarks. The patch-local QSM leads on Pendigits, whereas the global block-Hamiltonian QSM leads on the controlled spectral tasks. Ablations show a task-dependent reversal: subspace-preserving controls perform better on Pendigits, whereas spectral-value-only controls lead among the tested ablations on the synthetic tasks. Together, these results shed new light on quantum machine-learning model design by showing how input-conditioned spectral representations can provide an analysable inductive bias, while offering a broader perspective on structure-aware model design in machine learning and artificial intelligence.
Peiyong Wang, Udaya Parampalli, Casey R. Myers
CSIRO Technology · Research Way, Clayton VIC 3168, Australia · School of Computing and Information Systems +6
Quantum machine learning models based on parameterized circuits can be viewed as Fourier series approximators. However, they often struggle to learn functions with multiple frequency components, particularly high-frequency or non-dominant ones; a phenomenon we term the quantum Fourier parameterization bias. Inspired by recent advances in classical Fourier neural operators (FNOs), we adapt the multi-stage residual learning idea to the quantum domain, iteratively training additional quantum modules on the residuals of previous stages. We evaluate our method on a synthetic benchmark composed of spatially localized frequency components with diverse envelope shapes (Gaussian, Lorentzian, triangular). Systematic experiments show that the number of qubits, the encoding scheme, and residual learning are all crucial for resolving multiple frequencies; residual learning alone can improve test MSE significantly over a single-stage baseline trained for the same total number of epochs. Our work provides a practical framework for enhancing the spectral expressivity of quantum models and offers new insights into their frequency-learning behavior.
Ammar Daskin
Department of Computer Engineering, Istanbul Medeniyet University, Istanbul, Turkiye, 34000
Quantum kernel methods are promising for near-term quantum ma- chine learning, yet their behavior under data corruption remains insuf- ficiently understood. We analyze how quantum feature constructions degrade under controlled additive noise. We introduce Spectral Phase Encoding (SPE), a hybrid construc- tion combining a discrete Fourier transform (DFT) front-end with a diagonal phase-only embedding aligned with the geometry of diagonal quantum maps. Within a unified framework, we compare QK-DFT against alternative quantum variants (QK-PCA, QK-RP) and classi- cal SVM baselines under identical clean-data hyperparameter selection, quantifying robustness via dataset fixed-effects regression with wild cluster bootstrap inference across heterogeneous real-world datasets. Across the quantum family, DFT-based preprocessing yields the smallest degradation rate as noise increases, with statistically sup- ported slope differences relative to PCA and RP. Compared to classical baselines, QK-DFT shows degradation comparable to linear SVM and more stable than RBF SVM under matched tuning. Hardware exper- iments confirm that SPE remains executable and numerically stable for overlap estimation. These results indicate that robustness in quan- tum kernels depends critically on structure-aligned preprocessing and its interaction with diagonal embeddings, supporting a robustness-first perspective for NISQ-era quantum machine learning.
Pablo Herrero Gómez, Antonio Jimeno Morenilla, David Muñoz-Hernández +1
Department of Computer Science Technology and Computation, University of Alicante, Alicante, Spain