Tailor Made Embeddings for Quantum Machine Learning

Abstract: 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.

• The paper introduces a variational autoencoder framework that learns task-specific quantum embeddings recoverable from polynomially many measurements, addressing the exponential tomography cost of amplitude encoding.
• The learned embedding raises MNIST 3-vs-5 validation accuracy from 54.0% with naive amplitude encoding to 85.0%, or 98.5% with a lightweight post-measurement softmax, approaching the 99.7% classical baseline.
• The framework compresses ImageNet into a 13-qubit representation and supports hardware-compatible ansatz encoders, but circuit depth, simulation cost, state preparation, and limited hardware training remain key obstacles.

This paper by Lamarre and Šafránek introduces a variational autoencoder (VAE) framework that learns task-specific quantum embeddings of classical data, addressing a structural weakness of current quantum machine learning (QML): the incompatibility of standard data-encoding schemes with the information-theoretic constraints on measurement. The central claim is that an embedding should not only compress data into a quantum representation but also guarantee that the data can be recovered from a polynomial number of measurements — a property the authors show improves downstream classification substantially.

Motivation and problem statement

The authors ground the work in a concrete limitation of amplitude embeddings. Because classical information is distributed across exponentially many amplitudes, Holevo's bound implies that no polynomial number of measurements can recover the encoded data; full reconstruction requires quantum state tomography, which scales exponentially in the qubit count. The authors argue that this bottleneck may explain the weak empirical performance of amplitude-embedding-based classifiers observed in the large-scale survey by Bowles et al., and that it places any quantum agent at a fundamental disadvantage relative to a classical agent with direct data access.

The paper's proposed remedy is an autoencoder pipeline in which the latent space is constrained to be recoverable: the encoder produces either a quantum state description (amplitudes) or circuit parameters, and a learned neural decoder reconstructs the original image from measurement outcomes alone. The framework is presented as a QML analogue of the historical role autoencoders played in classical deep learning — providing principled embeddings, weight initialization via unsupervised pretraining, and structured latent spaces.

A comparison of standard embeddings motivates the design space: basis encoding uses $O(N \cdot m)$ qubits with trivial recoverability; angle encoding uses $O(N)$ qubits with polynomial recoverability; data re-uploading reduces qubits to $O(k)$ at $O(Nd)$ runtime; amplitude encoding uses $O(\log N)$ qubits but requires exponential tomographic cost for recovery. The autoencoder framework replaces the exponential recovery cost of amplitude embeddings with polynomial measurements plus a learned decoder.

Framework

The architecture is a VAE whose bottleneck is projected into a quantum-compatible format via lightweight projection layers, making it agnostic to the downstream encoding. Two encoder modes are defined:

• Amplitude encoder: the network outputs complex amplitudes $\psi$, normalized via a softmax-like transformation, defining a quantum state. Training avoids quantum simulation entirely, since amplitude encoding reduces to classical linear algebra.
• Ansatz encoder: the network outputs rotation angles that parameterize a data re-uploading circuit, so the latent space is interpreted as the set of circuits realizable within the ansatz. This avoids exponential classical state descriptions and is hardware-executable, but requires explicit circuit simulation during training.

The decoder consumes measurement data — computational-basis probabilities (estimable with polynomially many samples via shadow tomography) or single-qubit expectation values — and reconstructs the input image. For MNIST, a custom VAE with MSE, KL, and a cycle-consistency (state fidelity) loss is used; for CIFAR and ImageNet, the AutoencoderKL architecture from Latent Diffusion Models is adopted with LPIPS perceptual loss and adversarial regularization, with KL weighted at $10^{-8}$.

A deliberate design choice is the strict decoupling of training: the autoencoder is trained first, its weights frozen, and only then is the quantum classifier trained on the frozen latent representation. This isolates the contribution of the quantum circuit from the classical representation learning — a methodological point the authors emphasize in response to concerns that hybrid pipelines can absorb most computation classically.

Reconstruction results

The headline scalability result is the encoding of ImageNet ($256\times256\times3$, roughly 200,000 features) into a 13-qubit latent representation with visually satisfactory reconstruction after only 5 epochs of training (several weeks of computation). The authors claim this is the first demonstration of quantum embedding and reconstruction on ImageNet, which they argue was previously intractable: amplitude embedding requires $O(N)$ state-preparation depth and angle/re-uploading embeddings require $O(N)$ qubits. Reconstruction quality on the ImageNet-to-CIFAR-10 transfer is strong numerically (PSNR 35.21 dB, SSIM 0.969, LPIPS 0.0078, FID 0.61), but the authors themselves flag that this result should be interpreted with caution: the CIFAR images were upsampled to $O(N)$0 prior to encoding, which substantially reduces reconstruction difficulty and explains the near-zero FID.

Native-resolution results are considerably weaker. CIFAR-10 reconstruction with the amplitude encoder yields PSNR of only 14.06 dB and SSIM of 0.287, which the authors characterize as noisy — consistent with a model prioritizing compression over pixel fidelity. The ansatz encoder (10 qubits, 10 re-uploading layers) produces sharp MNIST reconstructions but noticeably blurry CIFAR-10 reconstructions despite using more qubits than the amplitude encoder (7 qubits for CIFAR-10), leading the authors to conclude that expressivity is limited by circuit depth rather than qubit count.

Classification benchmarks

On the MNIST 3-vs-5 task, the results form a clear progression. A circuit-centric variational quantum classifier with naive amplitude embedding fails to converge entirely and is stopped at 255 epochs. Replacing the naive embedding with the learned autoencoder embedding lifts validation accuracy from 54.0% to 85.0% — the paper's claim of a "more than 30 percentage point" improvement over naive amplitude embedding. Adding a two-neuron post-measurement softmax (a dressed-circuit offloading strategy following Mari et al.) raises accuracy to 98.5%, within 1.2 percentage points of a classical MLP (99.7%) trained on the same representations.

The most important ablation concerns attribution. Feeding the encoder's expectation values directly into the same two-neuron classifier, bypassing the quantum circuit entirely, reaches only 66.5% — roughly 30 percentage points below the full pipeline. This supports the claim that the quantum circuit performs a non-trivial transformation rather than serving as a pass-through. The authors are appropriately candid that the softmax layer itself closes most of the remaining gap, and they frame the necessity of classical offloading as an open question about the division of labor between embedding, circuit, and post-processing.

On multiclass MNIST the best model reaches 94% validation accuracy, about 5% below classical state of the art. On CIFAR-10, the circuit-centric classifier with the learned embedding reaches approximately 60%, improving to about 65% with an extra linear layer — a disproportionate gain from minimal classical capacity that again points to circuit depth as the bottleneck. A cross-dataset transfer experiment, reusing the CIFAR-10-trained encoder on CIFAR-100 without retraining, yields 27.5% validation accuracy, within 7.5% of a classical AlexNet baseline, suggesting the embedding generalizes beyond its training distribution.

The ansatz encoder was additionally executed on IBM Yonsei quantum hardware in inference-only mode with frozen parameters, with qualitative reconstructions confirming stability under device noise. The authors concede that full hardware configuration details are unavailable for publication due to a proprietary collaboration.

Robustness of the comparisons

The authors address the well-known critique that reported QML advantages can be artifacts of asymmetric tuning. They perform minimal tuning (learning-rate selection, best-checkpoint retention) and compare against a classical MLP trained on the identical encoding under identical conditions, arguing the 98.5% is therefore conservative. They also note that the naive amplitude baseline they outperform uses no PCA preprocessing, whereas the Bowles et al. baseline relies on PCA — confirming that dimensionality reduction does non-trivial work in that pipeline and that their learned embedding substitutes for it.

Limitations and open questions

Several limitations are conceded explicitly. The amplitude encoder requires a classical description of the full quantum state, scaling exponentially, and cannot be executed on hardware without efficient state-preparation routines; a proposed mitigation via separable products of smaller state descriptions sacrifices long-range entanglement and is left unstudied. The ansatz encoder scales polynomially and is hardware-compatible but is slow to simulate, and its blurry CIFAR-10 reconstructions indicate that scaling will require deeper circuits — with attendant barren plateau risks. The choice of VAE with perceptual loss is justified architecturally rather than by ablation; the authors note that subsequent independent evidence (SSIM correlating with QNN classification performance) supports the choice but that a systematic comparison of AE, VAE, and VQ-VAE variants remains open. Cycle-consistency loss was dropped for the larger datasets due to competing gradient signals, and its reintroduction with tuned weighting is unresolved. Pre-training versus random initialization for the ansatz circuit parameters is planned but not yet evaluated. ImageNet classification was not attempted due to simulation cost, so the framework's end-to-end value at that scale is unverified. The IBM hardware demonstration was inference-only, with no training on hardware and incomplete disclosure of the configuration.

Conclusion

The paper makes a focused contribution: a dataset-agnostic autoencoder framework whose latent spaces are, by construction, recoverable from polynomially many measurements, thereby converting the exponential tomographic cost of amplitude embeddings into a learned decoding problem. The empirical evidence — a 31-point accuracy gain over naive amplitude embedding on MNIST 3-vs-5, a 30-point ablation gap isolating the quantum circuit's contribution, and the first ImageNet-scale quantum embedding at 13 qubits — supports the central thesis that embedding quality is a distinct and previously under-addressed source of QML underperformance. The remaining gap to classical baselines is attributed to circuit capacity and simulation budgets rather than the embedding itself, a diagnosis that is plausible but not conclusively separated from ansatz design choices. The framework's generality across encodings, measurement schemes, and datasets makes it a reusable component for QML experimentation, though its ultimate value on hardware at scale remains contingent on the open problems in circuit depth, trainability, and state preparation the authors identify.