Quantum neural networks (QNNs) based on parametrized quantum circuits are promising candidates for machine learning applications, yet many architectures lack clear connections to classical models, potentially limiting their ability to leverage established classical neural network techniques. We examine QNNs built from SWAP test circuits and discuss their equivalence to classical two-layer feedforward networks with quadratic activations under amplitude encoding. Evaluation on real-world and synthetic datasets shows that while this architecture learns many practical binary classification tasks, it has fundamental expressivity limitations: polynomial activation functions do not satisfy the universal approximation theorem, and we show analytically that the architecture cannot learn the parity check function beyond two dimensions, regardless of network size. To address this, we introduce generalized SWAP test circuits with multiple Fredkin gates sharing an ancilla, implementing product layers with polynomial activations of arbitrary even degree. This modification enables successful learning of parity check functions in arbitrary dimensions as well as binary n-spiral tasks, and we provide numerical evidence that the expressivity enhancement extends to alternative encoding schemes such as angle (Z) and ZZ feature maps. We validate the practical feasibility of our proposed architecture by implementing a classically pretrained instance on the IBM Torino quantum processor, achieving 84% classification accuracy on the three-dimensional parity check despite hardware noise. Our work establishes a framework for analyzing and enhancing QNN expressivity through correspondence with classical architectures, and demonstrates that SWAP test-based QNNs possess broad representational capacity relevant to both classical and potentially quantum learning tasks.
Quantum neural networks (QNNs) based on parametrized quantum circuits are promising candidates for machine learning applications, yet many architectures lack clear connections to classical models, potentially limiting their ability to leverage established classical neural network techniques. We examine QNNs built from SWAP test circuits and discuss their equivalence to classical two-layer feedforward networks with quadratic activations under amplitude encoding. Evaluation on real-world and synthetic datasets shows that while this architecture learns many practical binary classification tasks, it has fundamental expressivity limitations: polynomial activation functions do not satisfy the universal approximation theorem, and we show analytically that the architecture cannot learn the parity check function beyond two dimensions, regardless of network size. To address this, we introduce generalized SWAP test circuits with multiple Fredkin gates sharing an ancilla, implementing product layers with polynomial activations of arbitrary even degree. This modification enables successful learning of parity check functions in arbitrary dimensions as well as binary n-spiral tasks, and we provide numerical evidence that the expressivity enhancement extends to alternative encoding schemes such as angle (Z) and ZZ feature maps. We validate the practical feasibility of our proposed architecture by implementing a classically pretrained instance on the IBM Torino quantum processor, achieving 84% classification accuracy on the three-dimensional parity check despite hardware noise. Our work establishes a framework for analyzing and enhancing QNN expressivity through correspondence with classical architectures, and demonstrates that SWAP test-based QNNs possess broad representational capacity relevant to both classical and potentially quantum learning tasks.
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.
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 neural networks (QNN) have been proposed as a promising architecture for quantum machine learning. There exist a number of different quantum circuit designs being branded as QNNs, however no clear candidate has presented itself as more suitable than the others. Rather, the search for a ``quantum perceptron" -- the fundamental building block of a QNN -- is still underway. One candidate is quantum perceptrons designed to emulate the nonlinear activation functions of classical perceptrons. Such sigmoid quantum perceptrons (SQPs) inherit the universal approximation property that guarantees that classical neural networks can approximate any function. However, this does not guarantee that QNNs built from SQPs will have any quantum advantage over their classical counterparts. Here we critically investigate both the capabilities and performance of SQP networks by computing their effective dimension and effective capacity, as well as examining their performance on real learning problems. The results are compared to those obtained for other candidate networks which lack activation functions. It is found that simpler, and apparently easier-to-implement parametric quantum circuits actually perform better than SQPs. This indicates that the universal approximation theorem, which a cornerstone of the theory of classical neural networks, is not a relevant criterion for QNNs.
Quantum machine learning models based on parametrized quantum circuits, also called quantum neural networks (QNNs), are considered to be among the most promising candidates for applications on near-term quantum devices. Here we explore the expressivity and inductive bias of QNNs by exploiting an exact mapping from QNNs with inputs x to classical perceptrons acting on x⊗x (generalised to complex inputs). The simplicity of the perceptron architecture allows us to provide clear examples of the shortcomings of current QNN models, and the many barriers they face to becoming useful general-purpose learning algorithms. For example, a QNN with amplitude encoding cannot express the Boolean parity function for n≥3, which is but one of an exponential number of data structures that such a QNN is unable to express. Mapping a QNN to a classical perceptron simplifies training, allowing us to systematically study the inductive biases of other, more expressive embeddings on Boolean data. Several popular embeddings primarily produce an inductive bias towards functions with low class balance, reducing their generalisation performance compared to deep neural network architectures which exhibit much richer inductive biases. We explore two alternate strategies that move beyond standard QNNs. In the first, we use a QNN to help generate a classical DNN-inspired kernel. In the second we draw an analogy to the hierarchical structure of deep neural networks and construct a layered non-linear QNN that is provably fully expressive on Boolean data, while also exhibiting a richer inductive bias than simple QNNs. Finally, we discuss characteristics of the QNN literature that may obscure how hard it is to achieve quantum advantage over deep learning algorithms on classical data.
Quantum machine learning (QML) is promising for potential speedups and improvements in conventional machine learning (ML) tasks (e.g., classification/regression). The search for ideal QML models is an active research field. This includes identification of efficient classical-to-quantum data encoding scheme, construction of parametric quantum circuits (PQC) with optimal expressivity and entanglement capability, and efficient output decoding scheme to minimize the required number of measurements, to name a few. However, most of the empirical/numerical studies lack a clear path towards scalability. Any potential benefit observed in a simulated environment may diminish in practical applications due to the limitations of noisy quantum hardware (e.g., under decoherence, gate-errors, and crosstalk). We present a scalable quantum-classical hybrid deep neural network (DeepQMLP) architecture inspired by classical deep neural network architectures. In DeepQMLP, stacked shallow Quantum Neural Network (QNN) models mimic the hidden layers of a classical feed-forward multi-layer perceptron network. Each QNN layer produces a new and potentially rich representation of the input data for the next layer. This new representation can be tuned by the parameters of the circuit. Shallow QNN models experience less decoherence, gate errors, etc. which make them (and the network) more resilient to quantum noise. We present numerical studies on a variety of classification problems to show the trainability of DeepQMLP. We also show that DeepQMLP performs reasonably well on unseen data and exhibits greater resilience to noise over QNN models that use a deep quantum circuit. DeepQMLP provided up to 25.3% lower loss and 7.92% higher accuracy during inference under noise than QMLP.
Quantum Neural Networks (QNNs) offer a framework for learning functions from classical data, but their expressive power depends critically on how inputs are encoded into quantum states. Angle encoding can require resources that scale with the number of features, whereas amplitude encoding represents d features using only O(log2d) qubits. This compactness comes at a representational cost: normalization makes conventional amplitude encoding invariant to nonzero real rescaling of the input, up to an unobservable global phase. We show that this invariance induces an exact, architecture-independent restriction on the resulting hypothesis class. No downstream quantum circuit can distinguish inputs differing by nonzero real scaling. We develop a general condition under which deterministic preprocessing can remove this ambiguity and show that nonlinearity alone is insufficient. Based on this analysis, we introduce a fixed nonlinear feature map that provably distinguishes all real inputs under subsequent amplitude encoding, without adding trainable parameters and while retaining logarithmic qubit scaling. We evaluate the resulting QNN on three canonical classification benchmarks and a nonlinear synthetic dataset against conventional amplitude and angle encoding, as well as a classical neural network. Across these tasks, the proposed encoding consistently improves classification accuracy over both quantum encoding baselines by 21$-$44 %. Our results establish normalization-induced scale invariance as a concrete representational bottleneck of amplitude encoding and demonstrate that it can be removed while preserving a compact quantum representation, marking a step toward quantum models capable of capturing arbitrary relationships in classical data.