Geometric Latent Space Tomography with Metric-Preserving Autoencoders

TL;DR

The paper tackles the exponential scaling of quantum state tomography by introducing geometric latent space tomography, a hybrid quantum-classical autoencoder whose latent space is trained to preserve the quantum Bures geometry. A metric-preserving loss enforces proportionality between latent Euclidean distances and Bures geodesic distances, enabling geometry-aware operations such as fast fidelity estimation and state discrimination with polynomial computational scaling. On 2-qubit mixed states (purity 0.85–0.95), the approach achieves mean fidelity about 0.942 while preserving roughly 78% of the quantum metric structure, and reveals an intrinsic latent manifold of dimension around 6.3 with measurable curvature. The latent space supports interpretable error manifolds and geometry-guided quantum machine learning, offering a practical pathway for NISQ devices and guiding future extensions via shadows, tensor networks, or symmetry constraints to scale to larger systems. Overall, the work demonstrates that enforcing geometric fidelity in neural quantum tomography yields compact, interpretable representations that retain essential quantum distances and enable efficient quantum information processing tasks.

Abstract

Quantum state tomography faces exponential scaling with system size, while recent neural network approaches achieve polynomial scaling at the cost of losing the geometric structure of quantum state space. We introduce geometric latent space tomography, combining classical neural encoders with parameterized quantum circuit decoders trained via a metric-preservation loss that enforces proportionality between latent Euclidean distances and quantum Bures geodesics. On two-qubit mixed states with purity 0.85--0.95 representing NISQ-era decoherence, we achieve high-fidelity reconstruction (mean fidelity $F = 0.942 \pm 0.03$) with an interpretable 20-dimensional latent structure. Critically, latent geodesics exhibit strong linear correlation with Bures distances (Pearson $r = 0.88$, $R^2 = 0.78$), preserving 78\% of quantum metric structure. Geometric analysis reveals intrinsic manifold dimension 6.35 versus 20 ambient dimensions and measurable local curvature ($κ= 0.011 \pm 0.006$), confirming non-trivial Riemannian geometry with $O(d^2)$ computational advantage over $O(4^n)$ density matrix operations. Unlike prior neural tomography, our geometry-aware latent space enables direct state discrimination, fidelity estimation from Euclidean distances, and interpretable error manifolds for quantum error mitigation without repeated full tomography, providing critical capabilities for NISQ devices with limited coherence times.

Figures (4)

• Figure 1: Autoencoder architecture. Classical encoder compresses 15D Pauli measurements to 20D latent space. Quantum circuit decoder reconstructs the density matrix. Dual training objective: reconstruction fidelity ($\mathcal{L}_{\text{recon}}$) and metric preservation ($\mathcal{L}_{\text{metric}}$).
• Figure 2: Training dynamics over 226 epochs with early stopping. (A) Reconstruction and metric preservation losses. (B) Validation fidelity. (C) Combined loss with metric weight $\lambda=0.06$.
• Figure 3: Latent space preserves quantum geometry. (A) Principal components of latent representations and theoretical Bures-fidelity curve. (B) Latent versus Bures distances for 500 state pairs with linear fit (red, $R^2 = 0.776$) and identity line (dashed).
• Figure 4: Geometric correspondence. Density projections of the quantum state manifold (left, MDS on Bures distances) and learned latent space (right, first two principal components) show preserved geometric structure with matching cluster topology.