[2608.09625] A Design House Examine of Density Matrix Parameterizations for Diffusion-Primarily based Quantum State Tomography

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Summary:Diffusion-based quantum state tomography (QST) has proven promising outcomes, however all present strategies implicitly undertake a single parameterization (usually Cholesky) with out systematic analysis. We current the primary design house research of density matrix parameterizations for diffusion QST, introducing a geometrical framework primarily based on the Jacobian Gram matrix $mathbf{J}^topmathbf{J}$. Our calibration of seven parameterizations at 2- and 3-qubit scales, validated by end-to-end coaching, reveals that emph{geometric conditioning alone doesn’t predict end-to-end efficiency}: at 3-qubit scale, Hermitian direct ($kappa = 2.0times$) performs worse than Cholesky ($kappa = 27times$) in any respect shot levels—a $13.5times$ isotropy benefit that interprets right into a constancy emph{drawback} of as much as $+0.51$. The two-qubit rating (Hermitian $>$ Bloch) reverses at 3 qubits (Bloch 0.907 vs. Hermitian 0.394). We offer a geometrical clarification: unbounded parameterizations endure projection-induced info loss as a result of the PSD constraint {couples} diagonal and off-diagonal coordinates in methods the unconstrained mannequin can’t respect, whereas the Bloch illustration locations the maximally combined state on the middle of the legitimate area, minimizing projection loss.