Fourier series weight in quantum machine learning

TL;DR

This work investigates the role of Fourier series in quantum machine learning by modeling data with Hamiltonian-encoded feature maps and variational circuits, framing the output as a partial Fourier series with frequencies determined by the encoding Hamiltonians. It demonstrates that the quantum model can perform trigonometric interpolation and function-approximation, and extends to binary and multiclass classification, supported by Hamiltonian-simulation techniques and QSP-inspired perspectives. Through Pennylane-based experiments, the paper provides evidence that Fourier weights can be extracted from quantum circuits and used to approximate periodic functions, characterize interpolation quality, and achieve competitive classification accuracy (around ~92%). The study also discusses limitations related to hardware noise, barren plateaus, and high-frequency behavior, while outlining future work on hyperparameter optimization, coefficient estimation, and applying these insights to real-world problems in finance and mobility.

Abstract

In this work, we aim to confirm the impact of the Fourier series on the quantum machine learning model. We will propose models, tests, and demonstrations to achieve this objective. We designed a quantum machine learning leveraged on the Hamiltonian encoding. With a subtle change, we performed the trigonometric interpolation, binary and multiclass classifier, and a quantum signal processing application. We also proposed a block diagram of determining approximately the Fourier coefficient based on quantum machine learning. We performed and tested all the proposed models using the Pennylane framework.

Figures (14)

• Figure 1: This figure is the standard quantum machine learning model in the literature with only one embedding block and an Ansatz as a parameterized quantum circuit with $m$ parameters. This model has a limitation: in the worst case, when the input data is not very well coded, for many parameters $m$ we add to our parametric function/circuit, we cannot find the best model that generalizes the input data. There have been various design proposals for how a quantum circuit should be lloyd2020quantumsuzuki2020analysisschuld2019quantum.
• Figure 2: Scenario (a) will be a univariate function. This function will be a sine, a cosine, a logarithm, and a rectangular pulse. Scenario (b) is a multivariate function that can be an arithmetic operation on $m$ univariate functions. Scenario (c) is where the input data is regrouped in a classical dataset. Said dataset is the result of a study on classical machine learning models. However, the data can be from any analysis, even if quantum processes after being observed.
• Figure 3: This is the model we use to analyze the weight of the Fourier series in quantum machine learning. Depending on some scaling parameters $\beta$ and the input data $\vec{x}$, the Feature Map will be responsible for coding our data. We rely heavily on the fact that the Feature Map ($F(\vec{x},\beta)$) must be variational, that the data's loading is repeated in all the layers, and that the variational circuit ($V(\vec{\theta})$) searches the best function within the space of functions that defines the capacity of the variational circuit ($U(\vec{x},\beta,\vec{\theta})= F(\vec{x},\beta)V(\vec{\theta)}$). With this configuration, we get closer to a quantum neural network that interpolates like a trigonometric function or classifies depending on the problem.
• Figure 4: This figure shows a quantum circuit of one qubit with only one repetition (Layer). This circuit implements a trigonometric interpolation (Fourier series), where the repeat frequency of the feature map, thus of the data, defines the angular frequency of the Fourier series: $\sum_{\omega} c_{\omega} e^{i \omega x}$. Our Feature Map, which implements our data encoding strategy, determines the frequencies $\omega$, and our variational quantum circuit determines the coefficients $c_{\omega}$. In blue, the function that implements the feature map, with $\beta$, the scaling hyperparameter, with $x$ the input data. Said data is encoded in the $RY$ and $RZ$ parameterized gates. In green is the variational circuit that forms the proposed parameterized function, with $\theta$, the parameter, one parameter per qubit. Without repeating the layer and reloading the data $x$, this circuit will only learn a $sine$ or $cosine$ function.
• Figure 5: This figure shows a quantum circuit of three qubits with only one repetition (Layer). This circuit implements a trigonometric interpolation (Fourier series), where the repeat frequency of the feature map, thus of the data, defines the angular frequency of the Fourier series: $\sum_{\omega} c_{\omega} e^{i \omega x}$. Our Feature Map, which implements our data encoding strategy, determines the frequencies $\omega$, and our variational quantum circuit determines the coefficients $c_{\omega}$. Each qubit represents a variable of our multivariable function. The function implements the feature map in blue, with $\beta$, the scaling hyperparameter, with $x$ the input data. Said data is encoded in the $RY$ and $RZ$ parameterized gates. In green is the variational circuit that forms the proposed parameterized function, with $\theta$, our parameter, one parameter per qubit. We have three. The $CNOT$ gates help us interleave the data so we can only read in the first qubit. This allows us to reduce the effect of the barren plateau in the case of a classifier.