Reference-Measure Geometry in Quantum Parameter Estimation: When Coordinate Surrogates Optimize the Wrong Objective

Quantumgateestimationandtomographypipelinesroutinelycombineintrinsicallydefined likelihoods with priors or regularization terms specified in local Euclidean coordinates. This practice implicitly replaces the Haar reference measure on SU(2) with Lebesgue measure, specifying a different statistical model rather than a reparametrization of the intended one. Weshowthat omitting the associated chart-volume factor alters the optimization objective itself, modifying its gradient field and stationary-point structure. The mismatch persists arbitrarily close to the identity, so that flat-coordinate surrogate objectives can converge to points that are non-stationary for the corresponding Haar-consistent objective even in regimes where local Gaussian approximations are assumed valid. We prove a formal non-equivalence proposition and validate a leading-order Fisher-information correction analytically and numerically. Large-scale multi-start optimization experiments (N = 11,900 runs) demonstrate that the discrepancy is regime-dependent and most pronounced under moderate-to-strong regularization or limited data. The fix requires a single-line modification to any gradient-based optimizer. These results identify reference-measure selection as an explicit modeling decision with direct consequences for optimization and inference in gate-set tomography, randomized benchmarking, and Bayesian gate estimation on curved parameter manifolds.