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 $\mathcal{O}(\log_2 d)$ 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.
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Amplitude encoding uses $O(\log d)$ qubits for $d$ classical features but maps inputs related by nonzero real scaling to the same physical state. This paper examines the resulting architecture-independent restriction on amplitude-encoded quantum classifiers and whether fixed preprocessing can remove it.
The analysis characterizes preprocessing collisions through proportional feature vectors and shows that nonlinearity alone need not eliminate them. It proposes $\Phi(\mathbf{x})=(\mathbf{x},\mathbf{x}^{\circ2},\cos\mathbf{x})$. The linear and quadratic components rule out proportional images of distinct nonzero inputs, while the cosine component makes the encoding defined at zero. The resulting physical-state map is injective on $\mathbb{R}^d$ and requires $\lceil\log_2(3d)\rceil$ qubits, without additional trainable parameters.
Experiments compare this encoding with conventional amplitude encoding, angle encoding, and a classical neural network using six-fold cross-validation and noiseless state-vector simulation. Each quantum model has 32 trainable rotation parameters. Balanced accuracy improves over conventional amplitude encoding from 0.6991 to 0.8834 on Breast Cancer, 0.7600 to 0.9320 on Wine, 0.2751 to 0.3100 on Digits, and 0.5996 to 0.7717 on a synthetic nonlinear task. Out-of-fold accuracy differences are significant after Holm correction on all four datasets; Digits ROC-AUC and AUPR nevertheless decrease.
The paper notes that angle-encoding comparisons are constrained by PCA reduction to at most eight features. Results exclude hardware noise and finite-shot uncertainty, do not demonstrate quantum advantage, and remain below the classical reference across the reported balanced accuracies.
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Error found: The abstract’s 21–44% improvement range across tasks and both encoding baselines is not supported by the reported balanced accuracies, whose relative gains range from 12.7% to 44.6%.
The abstract describes classification accuracy improvements, but the quantitative results table supplies balanced accuracy rather than the ordinary accuracy values needed to verify that claim.
The displayed general preprocessing objectives do not exclude proportional feature vectors, fail literally at the zero input, and omit the negative-scaling equivalence of physical amplitude-encoded states.
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
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.
Amplitude encoding uses O(logd) qubits for d classical features but maps inputs related by nonzero real scaling to the same physical state. This paper examines the resulting architecture-independent restriction on amplitude-encoded quantum classifiers and whether fixed preprocessing can remove it.
The analysis characterizes preprocessing collisions through proportional feature vectors and shows that nonlinearity alone need not eliminate them. It proposes Φ(x)=(x,x∘2,cosx). The linear and quadratic components rule out proportional images of distinct nonzero inputs, while the cosine component makes the encoding defined at zero. The resulting physical-state map is injective on Rd and requires ⌈log2(3d)⌉ qubits, without additional trainable parameters.
Experiments compare this encoding with conventional amplitude encoding, angle encoding, and a classical neural network using six-fold cross-validation and noiseless state-vector simulation. Each quantum model has 32 trainable rotation parameters. Balanced accuracy improves over conventional amplitude encoding from 0.6991 to 0.8834 on Breast Cancer, 0.7600 to 0.9320 on Wine, 0.2751 to 0.3100 on Digits, and 0.5996 to 0.7717 on a synthetic nonlinear task. Out-of-fold accuracy differences are significant after Holm correction on all four datasets; Digits ROC-AUC and AUPR nevertheless decrease.
The paper notes that angle-encoding comparisons are constrained by PCA reduction to at most eight features. Results exclude hardware noise and finite-shot uncertainty, do not demonstrate quantum advantage, and remain below the classical reference across the reported balanced accuracies.
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.
With the rapid development of quantum machine learning, quantum neural networks (QNNs) have become a research hotspot. However, the quantum gates used to implement feature mapping in this model are all linear transformations, which directly affects the mapping ability of the model. Therefore, how to enhance the mapping capability of QNN is an important issue that has not yet been effectively addressed. This paper proposes a repetitive amplitude encoding method that encodes the probability amplitudes of multiple qubit blocks by repeatedly using the same set of classical data, effectively improving the mapping capability of QNN. Taking the MNIST dataset as an example, the experimental results comparing the repetitive amplitude encoding method with several existing encoding methods show that, firstly, when the number of classes is fixed, the repetitive amplitude encoding is superior to other methods. Secondly, when the number of hidden layers in QNN is fixed, as the number of classes increases, the performance of repetitive amplitude encoding not only consistently outperforms other methods, but this advantage becomes increasingly apparent. Finally, the repetitive amplitude encoding-based QNN was applied to reservoir lithology identification in the field of oil and gas exploration, IRIS and WINe classification datasets. By comparing with classical neural networks, the proposed method was validated for its adaptability to different classification problems and superior classification performance compared to classical neural networks.
Quantum machine learning is often motivated by the idea that quantum systems can expose useful high-dimensional structure that is difficult to access with classical models. We isolate one central component of this claim: the fixed data-encoding map. Amplitude, angle, and basis encoding are evaluated as deterministic feature maps for classical supervised learning under matched output dimensionality and strong classical controls. The benchmark compares these encodings against raw linear models, random Fourier features, polynomial features, PCA, RBF SVMs, and shallow neural networks across diverse classical datasets. Rather than treating performance as a single endpoint, we analyze the geometry of each representation through effective rank, condition number, centered kernel alignment, predictive performance, and practical overhead. The resulting picture is mechanistic: amplitude encoding can remove magnitude information through unit-sphere normalization, angle encoding can become geometrically redundant with raw linear features, and basis encoding can impose a binary Hamming geometry that is poorly aligned with smooth decision structure. These findings do not argue against quantum computation, however, they show that fixed quantum-inspired encoding geometry alone is not a reliable source of machine-learning advantage on classical data.
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.
Efficiently encoding classical visual data into quantum states is essential for realizing practical quantum neural networks (QNNs). However, existing encoding schemes often discard spatial and semantic information when adapting high-dimensional images to the limited qubits of Noisy Intermediate-Scale Quantum (NISQ) devices. We propose a Fidelity-Preserving Quantum Encoding (FPQE) framework that performs near lossless data compression and quantum encoding. FPQE employs a convolutional encoder-decoder to learn compact multi-channel representations capable of reconstructing the original data with high fidelity, which are then mapped into quantum states through amplitude encoding. Experimental results show that FPQE performs comparably to conventional methods on simple datasets such as MNIST, while achieving clear improvements on more complex ones, outperforming PCA and pruning based encodings by up to 10.2\% accuracy on Cifar-10. The performance gain grows with data complexity, demonstrating FPQE's ability to preserve high-level structural information across diverse visual domains. By maintaining fidelity during classical to quantum transformation, FPQE establishes a scalable and hardware efficient foundation for high-quality quantum representation learning.