A Design Space Study of Density Matrix Parameterizations for Diffusion-Based Quantum State Tomography
quant-ph, math-ph, math.MP
Submitted: 2026-08-10
Updated: 2026-09-19
Code: https://github.com/csjnb/DM-QST-EDM
License: http://creativecommons.org/licenses/by/4.0/
The gist: Diffusion-based quantum state tomography (QST) has shown promising results, but all existing methods implicitly adopt a single parameterization (typically Cholesky) without systematic evaluation.
Terminology
Abstract
Diffusion-based quantum state tomography (QST) has shown promising results, but all existing methods implicitly adopt a single parameterization (typically Cholesky) without systematic evaluation. We present the first design space study of density matrix parameterizations for diffusion QST, introducing a geometric framework based on the Jacobian Gram matrix J J. Our calibration of seven parameterizations at 2- and 3-qubit scales, validated by end-to-end training, reveals that geometric conditioning alone does not predict end-to-end performance: at 3-qubit scale, Hermitian direct (κ= 2.0 times) performs worse than Cholesky (κ= 27 times) at all shot levels---a 13.5 times isotropy advantage that translates into a fidelity disadvantage of up to +0.51. The 2-qubit ranking (Hermitian > Bloch) reverses at 3 qubits (Bloch 0.907 vs. Hermitian 0.394). We provide a geometric explanation: unbounded parameterizations suffer projection-induced information loss because the PSD constraint couples diagonal and off-diagonal coordinates in ways the unconstrained model cannot respect, whereas the Bloch representation places the maximally mixed state at the center of the valid region, minimizing projection loss.
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