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Coherent Projector-Overlap QFNNs with Chebyshev Responses: Expressivity, Approximation, and Coherent Depth

Andrej Sum-Shik

Published Sep 14, 2026• 0 citations
Publisher

Abstract

Abstract Variational quantum models combine data encodings, trainable unitaries, and terminal measurements. We introduce a coherent quantum feedforward neural network (QFNN) with an explicit neural-style activation: trainable projector overlaps act as pre-activations, and alternating reflections implement exact Chebyshev responses. For a state |ψ⟩, an orthogonal projector Q, and p = ⟨ψ|Q|ψ⟩, d reflection pairs produce Td(2p − 1). For pure-state inputs, we characterize the complete idealized degree-at-most-D one-layer hypothesis space as ρ ↦→ Tr(Aρ⊗D) with A ∈ Herm(SymD H). This yields its exact dimension, polynomialfidelity kernel, strict degree hierarchy, domain-restricted capacity, interpolation properties, and finite separation from single-copy observable models. We derive Chebyshev approximation bounds, prove Universal Approximation Theorems for scalar and finite-dimensional continuous targets, and obtain norm- and margin-sensitive generalization bounds. The construction extends to fully coherent depth without intermediate measurement. The exact activation law and idealized Universal Approximation Theorem persist at every prescribed finite depth, while a process-feature representation and an exact Chebyshev-composition subarchitecture clarify the expressive and resource roles of recursion. Six-seed simulations on eight-class MNIST, Fashion-MNIST, and KMNIST obtain exact test accuracies of 90.57 ± 0.20%, 74.55 ± 0.51%, and 67.61 ± 0.65%, respectively. Together, these results provide an explicit mathematical framework for neural-style polynomial responses in coherent variational quantum architectures.

DOI
10.21203/rs.3.rs-10870861/v1

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Abstract Variational quantum models combine data encodings, trainable unitaries, and terminal measurements. We introduce a coherent quantum feedforward neural network (QFNN) with an explicit neural-style activation: trainable projector overlaps act as pre-activations, and alternating reflections implement exact Chebyshev responses. For a state |ψ⟩, an orthogonal projector Q, and p = ⟨ψ|Q|ψ⟩, d reflection pairs produce Td(2p − 1). For pure-state inputs, we characterize the complete idealized degree-at-most-D one-layer hypothesis space as ρ ↦→ Tr(Aρ⊗D) with A ∈ Herm(SymD H). This yields its exact dimension, polynomialfidelity kernel, strict degree hierarchy, domain-restricted capacity, interpolation properties, and finite separation from single-copy observable models. We derive Chebyshev approximation bounds, prove Universal Approximation Theorems for scalar and finite-dimensional continuous targets, and obtain norm- and margin-sensitive generalization bounds. The construction extends to fully coherent depth without intermediate measurement. The exact activation law and idealized Universal Approximation Theorem persist at every prescribed finite depth, while a process-feature representation and an exact Chebyshev-composition subarchitecture clarify the expressive and resource roles of recursion. Six-seed simulations on eight-class MNIST, Fashion-MNIST, and KMNIST obtain exact test accuracies of 90.57 ± 0.20%, 74.55 ± 0.51%, and 67.61 ± 0.65%, respectively. Together, these results provide an explicit mathematical framework for neural-style polynomial responses in coherent variational quantum architectures.
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.
This paper proposes and evaluate Hybrid Quantum Convolutional Neural Networks (HQCNNs) that integrate Structured Variational Quantum Circuits (SVQCs) into classical convolutional networks for image classification. Prior quanvolutional architectures often rely on unstructured variational circuits, leading to barren-plateau training landscapes, unpredictable convergence, and poor scaling of expressivity as qubit count grows. Moreover, simplistic data-encoding schemes can either fail to capture spatial correlations or demand impractically deep circuits, while arbitrary entanglement patterns obscure the relationship between circuit design and learning capacity. To address these challenges, we design four HQCNN architectures with progressively increased quantum encoding precision, entanglement topology, and classical post-processing depth. Each HQCNN replaces the convolutional layer with a quanvolutional layer: local image patches are encoded into small qubit registers via an Enhanced Novel Enhanced Quantum Representation (ENEQR), then processed by the SVQCs, and then measured to produce quantum feature maps. The four versions vary in quantum structure: Version 1 uses a simple linear 5-qubit chain (q=2), Version 2 adds wrap-around entanglement (closed-loop topology), Version 3 augments the classical head with a deeper batch-normalized multilayer perceptron, and Version 4 increases encoding fidelity to q=4 (7 qubits per patch). All models end-to-end using the Adam optimizer and cross-entropy loss on the MNIST dataset for 10 epochs using a PennyLane simulator. Version 4 achieves the best test accuracy (96% on MNIST). Fourier spectrum analysis reveals that richer entanglement and higher-fidelity encoding broaden the circuits’ harmonic spectrum, correlating with enhanced representational capacity and improved classification performance. These results confirm that carefully structured quantum layers and enhanced encoding can enrich feature representations and boost accuracy, albeit with increased simulation overhead and training complexity.
Quantum physics-informed neural networks (QPINNs) solve partial differential equations (PDEs) by training parameterized quantum circuits against physics-based residuals, yet the role of the embedding that maps coordinates into quantum states remains insufficiently understood. In this research, we introduce a unified embedding framework that formulates embedding as a functional transformation shaping the feature representation available to the variational circuit, and we propose two embeddings within it. The Linear Quantum Neural Network Trainable Embedding QPINN (LQNN-TE-QPINN) generates data-dependent features with an auxiliary quantum circuit and adds them to the input coordinates, instead of multiplying them as in our previous formulation. The adaptive-frequency QPINN (AdaFreq-QPINN) promotes the frequency-scaling factors of an arccos-Chebyshev encoding to trainable parameters. We evaluate both against direct and fixed analytical frequency encodings and alternative hybrid architectures on one- and two-dimensional Burgers equations. In one dimension, LQNN-TE-QPINN achieves a relative L2L_2L2​ error of 0.0916 with 720 trainable parameters, compared with 0.4501 for direct-encoding QPINN and 0.1233 for a locally adaptive classical physics-informed neural network with approximately eleven times more parameters, while AdaFreq-QPINN attains the lowest error among the analytical embeddings with only four additional parameters. In two dimensions, LQNN-TE-QPINN attains the lowest training objective, although its solution error is not clearly separated from that of a data re-uploading QPINN. Under simulated hardware noise, LQNN-TE-QPINN retains the lowest absolute derivative error while degrading most relative to its noiseless baseline. These results demonstrate that embedding design substantially influences the approximation and optimization behavior of QPINNs.
Abstract The field of artificial neural networks is expected to strongly benefit from recent developments of quantum computers. In particular, quantum machine learning, a class of quantum algorithms which exploit qubits for creating trainable neural networks, will provide more power to solve problems such as pattern recognition, clustering and machine learning in general. The building block of feed-forward neural networks consists of one layer of neurons connected to an output neuron that is activated according to an arbitrary activation function. The corresponding learning algorithm goes under the name of Rosenblatt perceptron. Quantum perceptrons with specific activation functions are known, but a general method to realize arbitrary activation functions on a quantum computer is still lacking. Here, we fill this gap with a quantum algorithm which is capable to approximate any analytic activation functions to any given order of its power series. Unlike previous proposals providing irreversible measurement–based and simplified activation functions, here we show how to approximate any analytic function to any required accuracy without the need to measure the states encoding the information. Thanks to the generality of this construction, any feed-forward neural network may acquire the universal approximation properties according to Hornik’s theorem. Our results recast the science of artificial neural networks in the architecture of gate-model quantum computers.
We introduce QFARE, a hybrid quantum-classical architecture for MNIST digit classification. Our approach employs a classical variational autoencoder (VAE) to compress 28×28 grayscale images into 64-dimensional latent representations. These compressed features are then loaded into an 8-qubit register. We augment this register with 4 additional ancilla qubits, whose partial mid-circuit measurements and subsequent conditional resets induce nonunitary transformations on the main qubits. This effectively provides a nonlinearity mechanism in an otherwise unitary quantum circuit, analogous to activation functions in classical neural networks. We train the resulting model end-to-end using a reinforcement learning algorithm (TD3) that explores the space of circuit parameters without relying on gradients of the quantum circuit, thus avoiding barren plateaus. Our theoretical framework proves that these measurement-based feedforward operations can simulate broad classes of nonunitary channels, guaranteeing universal classification capability when paired with a classical classifier. On a noiseless simulator, our experiments demonstrate that QFARE achieves around 99% accuracy on MNIST with significantly fewer parameters than purely classical neural network baselines. We also investigate the impact of circuit depth, the number of ancilla qubits, and how these aspects scale. This work shows how feedforward-enabled discrete-qubit designs can yield powerful and parameter-efficient classifiers for standard image recognition tasks.
Abstract Quantum machine learning, focusing on quantum neural networks (QNNs), remains a vastly uncharted field of study. Current QNN models primarily employ variational circuits on an ansatz or a quantum feature map, often requiring multiple entanglement layers. This methodology not only increases the computational cost of the circuit beyond what is practical on near-term quantum devices but also misleadingly labels these models as neural networks, given their divergence from the structure of a typical feed-forward neural network (FFNN). Moreover, the circuit depth and qubit needs of these models scale poorly with the number of data features, resulting in an efficiency challenge for real-world machine learning tasks. We introduce a bona fide QNN model, which seamlessly aligns with the versatility of a traditional FFNN in terms of its adaptable intermediate layers and nodes, absent from intermediate measurements such that our entire model is coherent. This model stands out with its reduced circuit depth and number of requisite CNOT gates, achieving a more than 50% reduction in both compared to prevailing QNN models. Furthermore, the qubit count in our model remains unaffected by the data’s feature quantity. We test our proposed model on various benchmarking datasets such as the breast cancer diagnostic (Wisconsin) and credit card fraud detection datasets. Our model achieved an accuracy of 91% on the breast cancer dataset and 85% on the credit card fraud detection dataset, outperforming existing QNN methods by 5–10% while requiring approximately 50% fewer quantum resources. These results showcase the advantageous efficacy of our approach, paving the way for the application of quantum neural networks to relevant real-world machine learning problems.