Quantum gate estimation and tomography pipelines routinely combine intrinsically defined 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. We show that omitting the associated chart-volume factor alters the optimization objective itself, modifying its gradient field and stationary-point structure. The gradient discrepancy ∇LG−∇LE=−∇logJexp is nonzero for all v≠0 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; quantitative validation is restricted to single-qubit systems, though the mechanism extends to any regularized optimization on a curved parameter manifold.
Fulton et al. (Sun,) studied this question.