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Learning Disentangled Representations with Quantum Variational Autoencoders
Gaoyuan Wang, Jerry Tan, Mark Gerstein
TL;DR
It is unclear whether QVAEs can learn disentangled, interpretable factors, in part because a qubit-level latent dimension is not straightforward to identify in an exponentially large Hilbert space. The paper combines theoretical analysis with controlled experiments and reports factorized, semantically interpretable representations, with individual qubits serving as meaningful latent factors.
Problem
Whether QVAEs can learn disentangled, interpretable factors—and how these factors are encoded in quantum latent space—remains unclear because individual quantum latent dimensions are nontrivial to define.
Source: §1 · p. 2Method
The paper analyzes quantum latent dimensions theoretically and studies their relation to known factors through controlled experiments on synthetic problems and MNIST variants.
Source: §1 · p. 3Results
The learned quantum representation decomposes into independent subsystems encoding meaningful factors in disentangled form, and individual qubits can serve as meaningful latent dimensions.
Source: §1 · p. 3, §5 · p. 13Takeaways & Limitations
The findings provide a basis for interpreting quantum latent representations in quantum machine learning and scientific data analysis.
Source: §1 · p. 3In the spectral-lines dataset, the effect of β, and thus regularization, is reported as rather small.
Source: §4 · p. 11
Abstract
from arXiv · showhide
Variational autoencoders are powerful representation learning models that map complex data into low-dimensional latent spaces, enabling the discovery of interpretable and disentangled factors. Such representations can facilitate the interpretation and controllable generation of data describing complex scientific systems. Understanding how these factors are organized and encoded in latent space is therefore important for developing reliable representation learning models. Recently, quantum variational autoencoders (QVAEs) have been proposed as quantum representation models, demonstrating informative latent representations and improved latent-space occupancy through quantum regularization. However, it remains unclear whether and how QVAEs can learn disentangled and interpretable latent factors. A key challenge in investigating quantum latent factors is that a small number of qubits spans an exponentially large Hilbert space, making the notion of an individual quantum latent dimension nontrivial. Here, we investigate what constitutes an individual quantum latent dimension and whether it can encode a distinct factor. We develop theoretical insights into quantum latent dimensions and support them with empirical studies on representative synthetic problems, including MNIST variants. Across three datasets, we demonstrate that QVAEs can discover factorized and semantically interpretable latent representations, with individual qubits functioning as meaningful latent factors. These results establish a foundation for understanding quantum latent spaces and their potential for structured and interpretable representation learning.
1 Introduction
The paper asks whether QVAEs can learn disentangled, interpretable factors despite the difficulty of identifying individual latent dimensions in an exponentially large Hilbert space. It combines theoretical analysis with controlled experiments to examine how quantum latent degrees of freedom relate to known data factors.
1 Introduction
- Representation learning maps high-dimensional scientific data into lower-dimensional latent representations intended to capture underlying factors and support interpretation.Such methods have been applied to physical systems, molecular and protein modeling, and biomedical data analysis.Source: §1 · pp. 1–2
- QVAEs use a maximally mixed-state prior, but whether they learn disentangled, interpretable factors—and how those factors are encoded—remains unresolved.The Hilbert space grows exponentially with qubit count, and its basis states do not naturally correspond to qubits or meaningful factors.Source: §1 · p. 2
- The study combines theoretical analysis with controlled experiments on synthetic problems and MNIST variants with known factors.It reports that individual qubits can function as meaningful latent dimensions and that ζ-QVAE discovers factorized, semantically interpretable representations in an unsupervised setting.Source: §1 · p. 3
2 Quantum Representation Learning
This section develops intuitions about how quantum autoencoders compress data and how regularization may encourage disentanglement aligned with computational qubits. It treats reduced states of selected qubits as candidate latent dimensions and assesses their information about known factors.
2.1 Quantum Autoencoders Without Regularization
- At the bottleneck, the autoencoder preserves k latent qubits and discards m = d − k trash qubits, optimizing which information belongs in each subspace.The preserved subspace uses zero states on trash qubits, which are traced out and later zero-padded for reconstruction.Source: §2.1 · p. 3
- The autoencoder can discard low-variance shared features or redundant correlated dimensions with minimal information loss.Discarding insignificant eigenbases loses instance-specific information; the account describes sacrificing low-significance dimensions as reducing reconstruction error.Source: §2.1 · pp. 3–4
- The ensemble density matrix’s eigenvalues indicate average occupation, so large-eigenvalue directions are more frequently occupied by ensemble states.Eigenvectors with small eigenvalues have small average overlap with the ensemble states.Source: §2.1 · p. 4
- A quantum autoencoder compresses an ensemble by preserving instance-specific information while discarding redundant or insignificant information.Without regularization, a QVAE is effectively a quantum autoencoder trained on reconstruction loss alone.Source: §2.1 · pp. 3–4
- Quantum autoencoders resemble Schumacher compression, but unlike it, they apply the same unitary independently to each instance rather than jointly across a sequence.The ensemble von Neumann entropy is a loose lower bound on QAE compression, with equality only in a degenerate case.Source: §2.1 · p. 4
- Unlike PCA, which discards low-variance directions, a quantum autoencoder’s ensemble eigenvalues measure occupation and top-k truncation alone may retain redundant dimensions.Its end-to-end training leaves room for regularization to guide the encoder toward a latent structure.Source: §2.1 · p. 4
2.2 Impact of Regularization
- QVAE regularization with a maximally mixed-state prior penalizes correlations between preserved dimensions, trading reconstruction quality for disentanglement.The account relates this capacity constraint to β-VAE’s pressure to prioritize influential factors that reduce reconstruction error.Source: §2.2 · p. 5
2.3 Disentangled Factors in Quantum Representations
- The paper treats each reduced state of selected qubits as a candidate latent dimension and suggests tuning qubit count against expected factors and reconstruction quality.It motivates qubit-level alignment by a training preference: a factor localized on one qubit gives focused gradient feedback, whereas a factor spread across qubits involves more gates and parameters. Factor information is assessed with regressors on reduced states; lower error or higher accuracy indicates more information, with MSE losses normalized by random-state baselines.Source: §2.3 · pp. 5–6
3 Learning Disentanglement Factors in MNIST Images
The MNIST studies test whether individual qubits encode discrete class labels and a continuous rotation factor. With regularization, the target-factor information is concentrated in the associated qubit, supporting qubits as meaningful latent dimensions.
3 Learning Disentanglement Factors in MNIST Images
- The studies use balanced sets of 1,000 images: digits 0–4 with three latent qubits, and digits 3 and 7 with random rotations and two latent qubits.The 8 × 8 images are encoded onto six input qubits; single-qubit latent states are obtained by tracing out the other qubits.Source: §3 · p. 6, §3.1 · p. 6, §3.2 · p. 8
3.1 Results on Raw MNIST Digit Images
- At β = 0.2, qubit 0 has a smooth, structured class representation with limited class overlap; without regularization, it is less structured and classes overlap more.Qubits 1 and 2 show little class correlation, while qubit 0 carries most class information.Source: §3.1 · pp. 6–7, Figure 1 · p. 7
3 Learning Disentanglement Factors in MNIST Images
- Across discrete-class and rotation tasks, regularization concentrates target-factor information in qubit 0 while the remaining qubits carry little target-factor information.The reported class accuracy is highest for qubit 0 at β = 0.1 and 0.2; rotation-angle MSE for qubit 0 dips at β = 0.2.Source: §3.1 · pp. 7–8, Figure 2 · p. 8, §3.2 · p. 8
3.2 Results on MNIST Digits with Random Rotations
- At β = 0.2, qubit 0 varies smoothly with rotation angle, whereas qubit 1 shows little alignment; without regularization, small and large rotations overlap in qubit 0.The angle-to-state relationship is nonlinear, and qubit 0's rotation-prediction MSE dips at β = 0.2 while qubit 1 remains near the random-state baseline.Source: §3.2 · p. 8, Figure 3 · p. 9, Figure 4 · p. 10
4 Learning Disentanglement Factors in Spectral Lines Dataset
On 64-dimensional Gaussian spectral lines, ζ-QVAE assigns position primarily to one qubit and width to the other two. The position association strengthens with regularization, while width remains informative and the latent distribution becomes more structured at the optimal β.
4 Learning Disentanglement Factors in Spectral Lines Dataset
- The study uses 64-dimensional Gaussian spectra controlled by peak position and width, extracting latent subsystems with partial traces and averaging normalized predictor MSE over nine runs.The two-qubit bottleneck was insufficient for accurate width encoding, motivating the three-qubit analysis.Source: §4 · p. 10
- Across most β values and multiple runs, one qubit encodes position and two encode width, consistent with width varying nonuniformly while position shifts the spectrum globally.The position factor is inferred much less accurately from the width qubits, and width much less accurately from the position qubit.Source: §4 · pp. 10–11, Figure 5 · p. 11
- Position MSE falls from 0.1217 at β = 0 to 0.0766 at β = 0.2 (about 37%), while aligned width MSE does not improve significantly.Aligned qubits remain informative for width across β, while position becomes less associated with the remaining subsystem at β = 0.2.Source: §4 · p. 11
- Although β has a small effect on averaged normalized MSE, the latent-state distribution improves significantly in structure at the optimal β.The β = 0.2 distribution shows improved coverage and organization compared with β = 0.Source: §4 · p. 11, Figure 6 · p. 12
- For qubit 0, position and width MSEs are 0.0123 and 0.1548 at β = 0, and 0.0097 and 0.1704 at β = 0.2, respectively.The latent-state distributions also show qubit 0 aligned with position and weakly associated with width.Source: §4 · pp. 11–12, Figure 6 · p. 12
5 Discussion
The discussion reports that an unsupervised QVAE with a maximally mixed prior decomposes its learned representation into independent subsystems encoding meaningful, disentangled factors. It frames this as a step toward interpretable quantum machine learning and proposes further comparisons in established application domains.
5 Discussion
- With a maximally mixed prior, the fully unsupervised quantum representation decomposes into independent subsystems, each encoding a meaningful factor in disentangled form.Source: §5 · p. 13
- These representations reside in Hilbert space and obey quantum probability, distinguishing them fundamentally from classical representations.Source: §5 · p. 13
- Future work may test quantum embeddings in computer vision, protein–protein docking, and natural language processing to characterize differences from classical representation learning.Source: §5 · p. 13
6 Method
The method adapts ζ-QVAE to classical inputs using amplitude encoding, a trash-qubit bottleneck, and inverse-unitary reconstruction, with reconstruction and latent-prior losses balanced by β. It evaluates MNIST variants and synthetic spectra under a specified training setup.
6.1 QVAE Architecture
- Amplitude embedding maps classical inputs to quantum states; an encoder compresses them by tracing out NT trash qubits, and the inverse decoder reconstructs from a zero-state extension.The decoder uses the inverse encoder transformation.Source: §6.1 · p. 15
- The objective weights negative-fidelity reconstruction against Jensen–Shannon regularization toward a maximally mixed latent prior, with β setting their relative strength.Source: §6.1 · p. 15
6.2 Dataset
- MNIST experiments use 200 images each of digits 0–4 and 500 each of digits 3 and 7, with the latter assigned uniform 0°–180° rotations after alignment.Images are resized to 8×8; the 3s and 7s are aligned to a common baseline orientation and intensity before rotation.Source: §6.2 · pp. 15–16
- The 64-dimensional synthetic Gaussian spectra vary peak position μ∈[10, 50] and width σ∈[1, 8], with these factors used as ground-truth labels.Source: §6.2 · p. 16
6.3 Training Details
- COBYLA training runs for at most 60 epochs, stops after 20 epochs without training-score improvement, and uses rhobeg 1.0 with three RealAmplitudes layers.Source: §6.3 · pp. 16–17