Variational quantum algorithms (VQAs) and their applications in the field of quantum machine learning through parametrized quantum circuits (PQCs) are thought to be one major way of leveraging noisy intermediate-scale quantum computing devices. However, differences in the performance of certain VQA architectures are often unclear since established best practices, as well as detailed studies, are missing. In this paper, we build upon the work by Schuld [] and Vidal [] by comparing how well popular ansatz for PQCs learn different one-dimensional truncated Fourier series. We also examine dissipative quantum neural networks (dQNN) as introduced by Beer [] and propose a data reupload structure for dQNNs to increase their capability for this regression task. By comparing the results for different PQC architectures, we can provide guidelines for designing efficient PQCs.
Variational quantum algorithms (VQAs) and their applications in the field of quantum machine learning through parametrized quantum circuits (PQCs) are thought to be one major way of leveraging noisy intermediate-scale quantum computing devices. However, differences in the performance of certain VQA architectures are often unclear since established best practices, as well as detailed studies, are missing. In this paper, we build upon the work by Schuld [] and Vidal [] by comparing how well popular ansatz for PQCs learn different one-dimensional truncated Fourier series. We also examine dissipative quantum neural networks (dQNN) as introduced by Beer [] and propose a data reupload structure for dQNNs to increase their capability for this regression task. By comparing the results for different PQC architectures, we can provide guidelines for designing efficient PQCs.
Abstract In this work, we explore the application of 1D quantum convolutional layers for time series forecasting, framing the problem within the context of machine learning and quantum circuit expressivity. By encoding multiple time steps as features into a quantum circuit, the forecasting task becomes intrinsically multidimensional. We leverage the recent theoretical results that describe Variational Quantum Circuits (VQCs) as multidimensional Fourier series, to design a more expressive 1D quantum convolutional neural network in which an enhanced performance is obtained by reuploading the data. We aim to relate the number of accessible coefficients in each architecture design with the obtained metrics. Our analysis incorporates key machine learning concepts such as expressivity and the presence of barren plateaus, evaluating their impact on generalization and training efficiency. Notably, we show that even with a limited number of trainable parameters, quantum circuits can approximate high-degree Fourier functions, underscoring their expressive capacity and computational efficiency. Our results demonstrate that more expressive ansatz, characterized by richer Fourier spectra, consistently yield improved predictive performance as the number of qubits increases. This study bridges theoretical insights from quantum physics with practical advances in machine learning model design for time series forecasting.
Leveraging the principles of quantum mechanics—specifically superposition and parallelism— quantum computation holds the potential to surpass the computational power limitations of classical computers. However, due to constraints such as the limited number of available qubits and the presence of decoherence noise in quantum hardware, designing efficient quantum algorithms capable of running on Noisy Intermediate-Scale Quantum (NISQ) devices has become a pivotal research focus. The Variational Quantum Algorithm (VQA) represents a hybrid quantum-classical computational paradigm that is executable on NISQ devices and can potentially demonstrate quantum advantage. In VQAs, the loss function is evaluated on a quantum computer, optimized by a classical computer, and the updated parameters are subsequently fed back to the quantum circuit for the next iteration. The Quantum Convolutional Neural Network (QCNN) is a prominent implementation within the VQA framework. It performs convolutional and pooling operations via Parameterized Quantum Circuits (PQCs) , endowing it with potentially superior feature extraction capabilities compared to its classical convolutional neural network counterparts. This paper begins with the foundational theories of quantum computing, provides a detailed exposition of the VQA workflow and QCNN architecture, and summarizes VQAs based on PQCs. It then reviews recent research progress on QCNNs from a practical application standpoint and concludes by outlining current challenges and future research directions.
Abstract In this work, we apply 1D quantum convolution in the task of time series forecasting. By encoding multiple points into the quantum circuit to predict subsequent data, each point becomes a feature and the problem becomes a multidimensional one. Taking as basis previous theoretical works which demonstrated that Variational Quantum Circuits (VQCs) can be expressed as multidimensional Fourier series, the capabilities of different architectures and ansatz are explored. This analysis considers the concepts of circuit expressivity and the presence of barren plateaus. Analyzing the problem within the framework of the Fourier series enabled the incorporation of data reuploading in the architecture, resulting in enhanced performance. Rather than a strict requirement for the number of free parameters to exceed the degrees of freedom of the Fourier series, our findings suggest that even a limited number of parameters can produce Fourier functions of higher degrees. This highlights the remarkable expressive power of quantum circuits. This observation is also significant in reducing training times. The ansatz with greater expressivity and number of non-zero Fourier coefficients consistently delivers favorable results across different scenarios, with performance metrics improving as the number of qubits increases.
In this study, we apply 1D quantum convolution to address the task of time series forecasting. By encoding multiple points into the quantum circuit to predict subsequent data, each point becomes a feature, transforming the problem into a multidimensional one. Building on theoretical foundations from prior research, which demonstrated that Variational Quantum Circuits (VQCs) can be expressed as multidimensional Fourier series, we explore the capabilities of different architectures and ansatz. This analysis considers the concepts of circuit expressibility and the presence of barren plateaus. Analyzing the problem within the framework of the Fourier series enabled the design of an architecture that incorporates data reuploading, resulting in enhanced performance. Rather than a strict requirement for the number of free parameters to exceed the degrees of freedom of the Fourier series, our findings suggest that even a limited number of parameters can produce Fourier functions of higher degrees. This highlights the remarkable expressive power of quantum circuits. This observation is also significant in reducing training times. The ansatz with greater expressibility and number of non-zero Fourier coefficients consistently delivers favorable results across different scenarios, with performance metrics improving as the number of qubits increases.
Parameterized quantum circuits (PQC) have emerged as a quantum analogue of deep neural networks and can be trained for discriminative or generative tasks and can be trained with gradient-based optimization on near-term quantum devices [1], [2], [3]. In the current era of quantum computing, known as the noisy intermediate scale quantum (NISQ) era [4], these devices contain a moderate number of qubits (< 100), and algorithmic performance is strongly impacted by hardware noise. Additionally, the training of PQCs are hybrid algorithms, in which the computational workflow is split between quantum and classical computing platforms.
Quantum neural networks (QNNs), typically implemented via parameterized quantum circuits (PQCs), offer a potential route to quantum advantage in learning. We review recent progress through the Fourier structure of PQCs: the encoding Hamiltonian fixes the accessible frequency set, while training estimates the corresponding Fourier coefficients. This perspective explains how QNNs can mitigate the spectral bias that limits classical deep neural networks and motivates task-specialized designs. In particular, quantum ordinary differential equation (QODE) solver, quantum convolutional neural networks (QCNNs) and quantum reinforcement learning (QRL) exploit frequency pinching to better capture high-frequency image features and structured value landscapes. We further discuss trainability on noisy intermediate-scale quantum (NISQ) hardware, emphasizing Lie-algebraic and Fourier views of barren plateaus and practical mitigation strategies. Together, the Fourier and Lie perspectives yield concrete design principles and motivate quantum-native architectures built around spectral bias avoidance rather than classical imitation. Quantum neural networks (QNNs), implemented via parameterized quantum circuits (PQCs), promise a path to quantum advantage in learning by addressing the limitations of classical deep neural networks. Here, the authors explore the Fourier structure of PQCs to mitigate spectral bias, enhancing applications like quantum ODE solvers and quantum reinforcement learning, with implications for designing quantum-native architectures.