Local-Time Riemannian Score Matching on the Quantum Pure-State Manifold

Abstract

Score-based diffusion can be defined intrinsically on the manifold of quantum pure states, $\mathbb{CP}^{d-1}$ with the Fubini--Study metric, but no closed-form transition density is available, so the score must be supervised by a local-time teacher taken from the Euclidean limit of the diffusion in normal coordinates. This paper is about what makes that teacher work, and where it stops working. Three training choices turn out not to be incidental: the increment must be divided by the diffusion clock rather than by the elapsed time, since the published expression assumes unit diffusion and a non-unit schedule introduces a time-change mismatch varying by a factor of $400$ across the horizon; the logarithm and exponential maps should be the closed-form Fubini--Study ones, which is the largest single effect we measure; and the global phase must be randomised, because horizontal projection alone does not make a score network descend to the quotient. With these choices the model improves on the published Riemannian local-time baseline in every cell of an eight-benchmark, four-metric comparison over ten seeds, significantly on five of eight after Holm correction, and beats an ambient Euclidean baseline by an order of magnitude everywhere. We then bound what the approximation costs by replacing it with the exact heat kernel of $\mathbb{CP}^{d-1}$, computable up to complex dimension seven, where the local-time teacher loses a factor of $1.2$ to $2.7$.