A Design Space Study of Density Matrix Parameterizations for Diffusion-Based Quantum State Tomography
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 \emph{geometric conditioning alone does not predict end-to-end performance}: at 3-qubit scale, Hermitian direct (κ= 2.0×) performs worse than Cholesky (κ= 27×) at all shot levels---a 13.5× 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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BibTeX
@misc{chang2026design,
title = {{A Design Space Study of Density Matrix Parameterizations for Diffusion-Based Quantum State Tomography}},
author = {Shuangu Chang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.09625},
eprint = {2608.09625},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.09625},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Design Space Study of Density Matrix Parameterizations for Diffusion-Based Quantum State Tomography
AU - Shuangu Chang
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2608.09625
UR - https://arxiv.org/abs/2608.09625
AB - 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 \emph{geometric conditioning alone does not predict end-to-end performance}: at 3-qubit scale, Hermitian direct (κ= 2.0×) performs worse than Cholesky (κ= 27×) at all shot levels---a 13.5× 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.
ER -