Summary: The exact quantum entropy of BPS black holes can be evaluated using localization in supergravity. An important ingredient in this program, that has been lacking so far, is the one-loop effect arising from the quadratic fluctuations of the exact deformation (the \(\mathcal{QV}\) operator). We compute the fluctuation determinant for vector multiplets and hyper multiplets around \(Q\)-invariant off-shell configurations in four-dimensional \( \mathcal{N}=2 \) supergravity with \(\mathrm{AdS}_2\times \mathrm S^2\) boundary conditions, using the Atiyah-Bott fixed-point index theorem and a subsequent zeta function regularization. Our results extend the large-charge on-shell entropy computations in the literature to a regime of finite charges. Based on our results, we present an exact formula for the quantum entropy of BPS black holes in \( \mathcal{N}=2 \) supergravity. We explain cancellations concerning \( \frac{1}{8} \)-BPS black holes in \( \mathcal{N}=8 \) supergravity that were observed in [A. Dabholkar et al., “Localization & exact holography”, Preprint, arXiv:1111.1161]. We also make comments about the interpretation of a logarithmic term in the topological string partition function in the low energy supergravity theory.