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

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 a design space study of density matrix parameterizations for diffusion QST, introducing a geometric framework based on the Jacobian Gram matrix $\mathbf{J}^\top\mathbf{J}$ that quantifies two competing criteria: isometric conditioning and physical constraint satisfaction. Our calibration of seven parameterizations at 2- and 3-qubit scales reveals three key findings. First, isometry and constraints are orthogonal criteria---no single parameterization optimizes both. Second, theoretical elegance does not predict geometric quality: the exponential map exhibits a spectral range of $149\times$ at 2 qubits and $75\,658\times$ at 3 qubits, while simpler parameterizations remain well-conditioned. Third, geometric conditioning alone does not fully predict end-to-end performance: without classifier-free guidance (CFG), Hermitian direct outperforms Bloch despite worse local isometry; under CFG, the ranking reverses due to amplified boundary effects. End-to-end training confirms that better-conditioned parameterizations converge faster and achieve higher fidelity in the absence of CFG. We provide actionable selection guidelines to guide future QST method design.