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arXiv:2610.05297

Do Neural Networks Learn Structure-Preserving Maps? A Case Study in Latent-to-Hilbert Embeddings

Muhammad Adnan Shahzad

cs.LGquant-phSubmitted Oct 4, 2026• Announced Oct 6, 2026• 0 citations
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Abstract

We ask whether a neural network can learn a structure-preserving map from a compressed latent space to a Hilbert-space representation. Using an 8-dimensional autoencoder bottleneck on MNIST and $n$-qubit product-state targets from PCA-based angle encoding, we report four findings. Although the target angles are generated by a nonlinear sigmoid transformation of the latent projections, the resulting mapping is well approximated by a linear function over the observed latent distribution: linear regression from $z$ to the true target angles achieves $R^{2} = 0.98$, while regression to the MLP's recovered angles achieves $R^{2} = 0.91$. The learned map's primary direction is strongly aligned with the target-induced direction, with cosine similarity $0.989$, while remaining nearly orthogonal to the input's principal direction, with cosine similarity $0.002$. The map is genuinely rank-4: removing any singular direction degrades inner-product preservation by $2.5$--$5.9\times$ despite a singular-value spectrum with two dominant and two small values. The learned subspace does not coincide with the PCA basis used to construct the target, and different random seeds recover the same primary direction but diverge in higher ranks. Finally, kernel ridge regression with an RBF kernel outperforms a tuned MLP (IP error $0.0144$ vs.\ $0.0197$), suggesting that for approximately linear structure-preserving mappings, classical kernel methods may be a simpler and more effective alternative.

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Neural Networks for Programming Quantum Annealers

Samuel Bosch, Bobak T. Kiani, Rui Liang Yang +2 more

Quantum machine learning has the potential to enable advances in artificial intelligence, such as solving problems intractable on classical computers. Some fundamental ideas behind quantum machine learning are similar to kernel methods in classical machine learning. Both process information by mapping it into high-dimensional vector spaces without explicitly calculating their numerical values. We explore a setup for performing classification on labeled classical datasets, consisting of a classical neural network connected to a quantum annealer. The neural network programs the quantum annealer's controls and thereby maps the annealer's initial states into new states in the Hilbert space. The neural network's parameters are optimized to maximize the distance of states corresponding to inputs from different classes and minimize the distance between quantum states corresponding to the same class. Recent literature showed that at least some of the "learning" is due to the quantum annealer, connecting a small linear network to a quantum annealer and using it to learn small and linearly inseparable datasets. In this study, we consider a similar but not quite the same case, where a classical fully-fledged neural network is connected with a small quantum annealer. In such a setting, the fully-fledged classical neural-network already has built-in nonlinearity and learning power, and can already handle the classification problem alone, we want to see whether an additional quantum layer could boost its performance. We simulate this system to learn several common datasets, including those for image and sound recognition. We conclude that adding a small quantum annealer does not provide a significant benefit over just using a regular (nonlinear) classical neural network.

arXiv:2308.06807•Aug 13, 2023•1 citation
We ask whether a neural network can learn a structure-preserving map from a compressed latent space to a Hilbert-space representation. Using an 8-dimensional autoencoder bottleneck on MNIST and nnn-qubit product-state targets from PCA-based angle encoding, we report four findings. Although the target angles are generated by a nonlinear sigmoid transformation of the latent projections, the resulting mapping is well approximated by a linear function over the observed latent distribution: linear regression from zzz to the true target angles achieves R2=0.98R^{2} = 0.98R2=0.98, while regression to the MLP's recovered angles achieves R2=0.91R^{2} = 0.91R2=0.91. The learned map's primary direction is strongly aligned with the target-induced direction, with cosine similarity 0.9890.9890.989, while remaining nearly orthogonal to the input's principal direction, with cosine similarity 0.0020.0020.002. The map is genuinely rank-4: removing any singular direction degrades inner-product preservation by 2.52.52.5--5.9×5.9\times5.9× despite a singular-value spectrum with two dominant and two small values. The learned subspace does not coincide with the PCA basis used to construct the target, and different random seeds recover the same primary direction but diverge in higher ranks. Finally, kernel ridge regression with an RBF kernel outperforms a tuned MLP (IP error 0.01440.01440.0144 vs.\ 0.01970.01970.0197), suggesting that for approximately linear structure-preserving mappings, classical kernel methods may be a simpler and more effective alternative.
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.
We implement a hybrid quantum-classical model for image classification that compresses MNIST digit images into a low-dimensional feature space and then maps these features onto a 5-qubit quantum state. First, an autoencoder compresses each 28×2828\times2828×28 image (784 pixels) into a 64-dimensional latent vector, preserving salient features of the digit with minimal reconstruction error. We further reduce the latent representation to 5 principal components using Principal Component Analysis (PCA), to match the 5 available qubits. These 5 features are encoded as rotation angles in a quantum circuit with 5 qubits. The quantum feature map applies single-qubit rotations (RyR_yRy​ gates) proportional to the feature values, followed by a Hadamard gate and a cascade of entangling CNOT gates to produce a non-product entangled state. Measuring the 5-qubit state yields a 32-dimensional probability distribution over basis outcomes, which serves as a quantum-enhanced feature vector for classification. A classical neural network with a softmax output is then trained on these 32-dimensional quantum feature vectors to predict the digit class. We evaluate the hybrid model on the MNIST dataset and compare it to a purely classical baseline that uses the 64-dimensional autoencoder latent features for classification. The results show that the hybrid model can successfully classify digits, demonstrating the feasibility of integrating quantum computing in the classification pipeline, although its accuracy (about 75\% on test data) currently falls below the classical baseline (about 98\% on the same compressed data).
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.
Autoencoders transformed classical machine learning by solving the curse of dimensionality, enabling principled weight initialization and learning compact, structured representations. In this work, we extend this paradigm to quantum machine learning by introducing a variational autoencoder framework that learns task-specific quantum embeddings of classical data. We demonstrate that high-dimensional datasets, including ImageNet, can be compressed into a 13-qubit quantum representation while remaining reconstructable through a learned decoder. On MNIST (3 vs 5), our approach achieves 98.5% validation accuracy using a circuit-centric quantum classifier, within 1.2 percentage points of a classical neural network baseline (99.7%) and more than 30 percentage points above a naive amplitude-embedding approach. Unlike amplitude embeddings, which require full quantum state tomography for recovery, or angle embeddings, which generally rely on circuit inversion under restrictive assumptions, the proposed framework reconstructs the original data from only a polynomial number of measurements. The framework was further validated on IBM quantum hardware, confirming that the learned embeddings remain stable and reconstructable under real device noise.
Neural network quantum states (NQS) for representing spin-12\frac{1}{2}21​ systems are typically built from Multi-Layer Perceptrons (MLPs) with binary visible variables, in which the input spins are first combined through linear affine maps before more expressive nonlinear transformations are applied. The standard way for representing multi-categorical systems, namely spin-1 and spin-2, or the q-state quantum Potts systems with more than two local degrees of freedom, is a unary or one-hot encoded MLP architecture. Following a few similar studies, I observe that while the one-hot construction is the mathematically faithful encoding for systems with more than two local states such as spin-SSS models with S>1/2S > 1/2S>1/2 or qqq-state Potts models, its parameter count grows with the number of local states, and the resulting optimization landscape can hinder convergence. I find numerically that allowing log⁡Ψ\log ΨlogΨ to be a nonlinear function of the raw multi-valued spin variable is a natural categorical generalization: it preserves the labelling freedom of the local basis and reproduces the one-hot model with strictly fewer parameters, often with improved trainability. I first benchmark this idea on shallow Restricted Boltzmann Machines (RBMs), equipping them with several distinct nonlinear connections, for spin-1, 2, and 3 Heisenberg chains. I then turn to Kolmogorov-Arnold Networks (KANs), where each edge carries a learnable univariate nonlinearity, and show that they provide a strictly more expressive realization of the same principle. Finally, I demonstrate that this framework captures the critical behaviour of the quantum Potts Hamiltonian, recovering its phase transition.
Quantum machine learning has the potential to enable advances in artificial intelligence, such as solving problems intractable on classical computers. Some fundamental ideas behind quantum machine learning are similar to kernel methods in classical machine learning. Both process information by mapping it into high-dimensional vector spaces without explicitly calculating their numerical values. We explore a setup for performing classification on labeled classical datasets, consisting of a classical neural network connected to a quantum annealer. The neural network programs the quantum annealer's controls and thereby maps the annealer's initial states into new states in the Hilbert space. The neural network's parameters are optimized to maximize the distance of states corresponding to inputs from different classes and minimize the distance between quantum states corresponding to the same class. Recent literature showed that at least some of the "learning" is due to the quantum annealer, connecting a small linear network to a quantum annealer and using it to learn small and linearly inseparable datasets. In this study, we consider a similar but not quite the same case, where a classical fully-fledged neural network is connected with a small quantum annealer. In such a setting, the fully-fledged classical neural-network already has built-in nonlinearity and learning power, and can already handle the classification problem alone, we want to see whether an additional quantum layer could boost its performance. We simulate this system to learn several common datasets, including those for image and sound recognition. We conclude that adding a small quantum annealer does not provide a significant benefit over just using a regular (nonlinear) classical neural network.