A Design Space Study of Density Matrix Parameterizations for Diffusion-Based Quantum State Tomography

• URL: http://arxiv.org/abs/2608.09625v2
• Date: Thu, 13 Aug 2026 08:48:25 GMT
• Title: A Design Space Study of Density Matrix Parameterizations for Diffusion-Based Quantum State Tomography
• Abstract summary: We present the first design space study of density matrix parameterizations for diffusion QST.<n>Our calibration of seven parameterizations at 2- and 3-qubit scales, validated by end-to-end training, reveals that emphgeometric conditioning alone does not predict end-to-end performance.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\mathbf{J}^\top\mathbf{J}$. Our calibration of seven parameterizations at 2- and 3-qubit scales, validated by end-to-end training, reveals that \emph{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 \emph{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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