This paper presents an efficient approach to quantifying image registration uncertainty based on a low-dimensional representation of geometric deformations. In contrast to previous methods, we develop a Bayesian diffeomorphic registration framework in a bandlimited space, rather than a high-dimensional image space. We show that a dense posterior distribution on deformation fields can be fully characterized by much fewer parameters, which dramatically reduces the computational complexity of model inferences. To further avoid heavy computation loads introduced by random sampling algorithms, we approximate a marginal posterior by using Laplace's method at the optimal solution of log-posterior distribution. Experimental results on both 2D synthetic data and real 3D brain magnetic resonance imaging (MRI) scans demonstrate that our method is significantly faster than the state-of-the-art diffeomorphic registration uncertainty quantification algorithms, while producing comparable results.