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arXiv:2512.15801

Geometric Latent Space Tomography with Metric-Preserving Autoencoders

S. M. Yousuf Iqbal Tomal, Abdullah Al Shafin

quant-phSubmitted Dec 16, 2025• Announced Dec 18, 2025• 0 citations
arXivHTMLPDF

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.

Comments
9 pages, 4 figures, 5 tables

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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±0.03F = 0.942 \pm 0.03F=0.942±0.03) with an interpretable 20-dimensional latent structure. Critically, latent geodesics exhibit strong linear correlation with Bures distances (Pearson r=0.88r = 0.88r=0.88, R2=0.78R^2 = 0.78R2=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±0.006κ= 0.011 \pm 0.006κ=0.011±0.006), confirming non-trivial Riemannian geometry with O(d2)O(d^2)O(d2) computational advantage over O(4n)O(4^n)O(4n) 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.
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