Variational quantum algorithms (VQAs) have shown advantages over classical approaches in solving many problems, yet their quantum neural networks (QNNs) models still perform poorly in complex tasks like multi-class classification and high-frequency time-series prediction, raising questions about their feasibility. While existing studies have characterized their performance from several perspectives, but research on their learning performance and visualization remains scarce. This work develops the universal approximation theory of variational quantum algorithms for multivariate function fitting, supported by a verification in visual training analysis on two-input binary datasets. We first visualize how a single-qubit VQA with single-encoding process four types of binary data, revealing inherent limitations, and then formulate the measurement expectation of VQA as a partial Fourier series, establishing its universal approximation capacity and showing that the Fourier spectrum is determined by the data encoding strategy: multiple encoding layers broaden the spectrum and enhance learning performance. Guided by this theory, we adjust the circuit structure and achieve nearly 100% accuracy across all four binary classification tasks.
Variational quantum algorithms (VQAs) have shown advantages over classical approaches in solving many problems, yet their quantum neural networks (QNNs) models still perform poorly in complex tasks like multi-class classification and high-frequency time-series prediction, raising questions about their feasibility. While existing studies have characterized their performance from several perspectives, but research on their learning performance and visualization remains scarce. This work develops the universal approximation theory of variational quantum algorithms for multivariate function fitting, supported by a verification in visual training analysis on two-input binary datasets. We first visualize how a single-qubit VQA with single-encoding process four types of binary data, revealing inherent limitations, and then formulate the measurement expectation of VQA as a partial Fourier series, establishing its universal approximation capacity and showing that the Fourier spectrum is determined by the data encoding strategy: multiple encoding layers broaden the spectrum and enhance learning performance. Guided by this theory, we adjust the circuit structure and achieve nearly 100% accuracy across all four binary classification tasks.
Quantum neural networks (QNNs) have emerged as a leading strategy to establish applications in machine learning, chemistry, and optimization. While the applications of QNN have been widely investigated, its theoretical foundation remains less understood. In this paper, we formulate a theoretical framework for the expressive ability of data re-uploading quantum neural networks that consist of interleaved encoding circuit blocks and trainable circuit blocks. First, we prove that single-qubit quantum neural networks can approximate any univariate function by mapping the model to a partial Fourier series. We in particular establish the exact correlations between the parameters of the trainable gates and the Fourier coefficients, resolving an open problem on the universal approximation property of QNN. Second, we discuss the limitations of single-qubit native QNNs on approximating multivariate functions by analyzing the frequency spectrum and the flexibility of Fourier coefficients. We further demonstrate the expressivity and limitations of single-qubit native QNNs via numerical experiments. We believe these results would improve our understanding of QNNs and provide a helpful guideline for designing powerful QNNs for machine learning tasks.
Quantum machine learning has established as an interdisciplinary field to overcome limitations of classical machine learning and neural networks. This is a field of research which can prove that quantum computers are able to solve problems with complex correlations between inputs that can be hard for classical computers. This suggests that learning models made on quantum computers may be more powerful for applications, potentially faster computation and better generalization on less data. The objective of this paper is to investigate how training of quantum neural network (QNNs) can be done using quantum optimization algorithms for improving the performance and time complexity of QNNs. A classical neural network can be partially quantized to create a hybrid quantum-classical neural network which is used mainly in classification and image recognition. In this paper, a QNN structure is made where a variational parameterized circuit is incorporated as an input layer named as Variational Quantum Neural Network (VQNNs). We encode the cost function of QNNs onto relative phases of a superposition state in the Hilbert space of the network parameters. The parameters are tuned with an iterative quantum approximate optimisation (QAOA) mixer and problem hamiltonians. VQNNs is experimented with MNIST digit recognition (less complex) and crack image classification datasets (more complex) which converges the computation in lesser time than QNN with decent training accuracy.
Quantum Machine Learning (QML) is a newly emerging research area at the intersection of classical machine learning (CML) and quantum computing (QC). Data is becoming voluminous rapidly, so it is challenging for classical computers to train Machine Learning (ML) models over massive datasets. The hope is that quantum physics' intrinsic features, such as entanglement, superposition, and interference, could be exploited as resources for training ML models on big datasets that would otherwise be relatively impossible for classical computers. It is theoretically proven that quantum computers have an exponential-time advantage over their classical counterparts in solving several problems, e.g., complex large dimensional matrix multiplication, factorization problem, unstructured database search, etc. QML models attempt to find a quantum advantage over their classical counterparts. Variational Quantum Classifiers (VQC) are hybrid quantum neural networks to perform the task of classification using QML models. VQC models in the present NISQ (Noisy Intermediate Scale Quantum — 50 to 100 qubits) era can produce comparable and even better results than Classical models. In this article, we examined the performance of a VQC while performing a simple binary classification task. In this article, we use a VQC to evaluate this method's performance empirically. We constructed a VQC to predict the label of fresh input for the typical Iris dataset comprising pairings of target outputs and training inputs. Our quantum classifier can reasonably predict species labels using only four qubits. These levels are fairly good compared to the accuracy levels attained by classical classifiers.
Quantum Machine Learning methods are becoming a key component for various types of tasks making predictions or decisions based on datasets. Recent efforts and researches on quantum computing point out the significance of quantum speedup advantage, especially for learning processes that require enormous amount of computational resources. Advances in both quantum hardware design and hybrid quantum-classical software frameworks accommodate a paradigm shift from classical to quantum. In consideration with this quantum leap notion, we investigate the capability of variational quantum algorithms (VQA) on a real world problem of user localization dealing with the binary classification task. This paper introduces a VQA with four variants that differ in the number of layers related to the variational quantum circuit (VQC) part of the VQA. The samples from a publicly available user localization dataset are first preprocessed through padding, scaling and normalization. Next, they are mapped into three qubit quantum states using amplitude encoding as a data embedding scheme. Unitary transformation of the mapped quantum data in the VQC is followed by a measurement in computational basis to produce predictions for the labels. The error between true and predicted labels is computed in a classical manner and a cost function minimization process is executed with the aid of gradient descent algorithm. The updated training parameters from the optimization stage are fed into the VQC and this process is repeated until the learnable parameters converge. The simulation results demonstrate that the designed VQA for binary classification achieves an accuracy value of 99% in the training phase. Moreover, the ratio of predicted labels to true labels approaches to 93% during the validation of actual user locations based on the signal strength received from the routers that are positioned at different places in a facility. Keywords—quantum machine learning, user localization, variational quantum algorithm, variational quantum circuit, amplitude encoding
Abstract Quantum Neural Networks (QNNs) offer a promising framework for integrating quantum computing principles into machine learning, yet their practical capabilities and limitations remain insufficiently studied. In this work, we systematically investigate the trainability and approximation properties of QNNs by benchmarking diverse circuit architectures and encoding strategies across synthetic and real-world datasets.We analyze several ansätze, including Hamming, binary, exponential, ternary, turnpike and Golomb, by evaluating their ability to learn synthetic data modeled as random finite Fourier series. To assess real-world applicability, we further evaluate QNNs on two time-series classification tasks: a Fischertechnik pneumatic leak detection dataset and the publicly available NASA bearing fault dataset.Our experiments show that while broader frequency spectra can theoretically enhance expressivity, practical trainability is strongly influenced by architectural factors such as qubit count and circuit depth. Notably, we find that QNNs perform best when the frequency spectrum is tailored to the target function’s complexity but remains as compact as possible. Moreover, architectures with identical frequency spectra can differ in trainability, with configurations using more qubits and fewer layers generally performing better, except in the single-layer case.These findings provide guidelines for selecting QNN ansätze and offer new insights into the interplay between expressivity and trainability in quantum machine learning.
Quantum computers can be used for supervised learning by treating parametrized quantum circuits as models that map data inputs to predictions. While a lot of work has been done to investigate the practical implications of this approach, many important theoretical properties of these models remain unknown. Here, we investigate how the strategy with which data are encoded into the model influences the expressive power of parametrized quantum circuits as function approximators. We show that one can naturally write a quantum model as a partial Fourier series in the data, where the accessible frequencies are determined by the nature of the data-encoding gates in the circuit. By repeating simple data-encoding gates multiple times, quantum models can access increasingly rich frequency spectra. We show that there exist quantum models which can realize all possible sets of Fourier coefficients, and therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators.