BuresTomFlow: Bures-Geometric Flow Matching for Posterior Quantum Tomography

Finite-shot quantum tomography often leaves a posterior over many density matrices compatible with the same measurement record, but fast reconstruction methods collapse this uncertainty to a single state. We frame this setting as amortized posterior sampling and introduce BuresTomFlow, a measurement-conditioned Flow Matching sampler for full-rank density matrices. The method transports base density matrices along Bures-Wasserstein paths on the positive matrix cone, projects the path back to the trace-one state space, and trains the velocity field with a Bures-aligned tangent loss. Thus the sampler is trained in the same fidelity-based geometry used to judge posterior quality. We evaluate BuresTomFlow in dense four-qubit simulations with count, shadow, and mixed measurement records. Against Cholesky, log-density, and generic Riemannian Flow Matching controls at comparable learned-sampler runtime, BuresTomFlow improves Bures distance, credible-ball coverage, and held-out observable prediction. In the main comparison across multiple seeds, mean Bures distance decreases from $0.618$ for the Cholesky baseline to $0.367$ for BuresTomFlow, and mean coverage error decreases from $0.203$ to $0.100$. These results show that the value of Bures geometry is not merely enforcing physical states, but producing calibrated posterior uncertainty when finite-record tomography cannot be reduced to one reconstruction.