Asymptotically optimal unitary estimation in $\mathrm{S}\mathrm{U}(3)$ by the analysis of graph Laplacian

Unitary estimation is the task to estimate an unknown unitary operator $U\in \mathrm{S}\mathrm{U}(d)$ with $n$ queries to the corresponding unitary operation, and its accuracy is evaluated by an estimation fidelity. We show that the optimal asymptotic fidelity of $3$-dimensional unitary estimation is given by $F_{\mathrm{e}\mathrm{s}\mathrm{t}}(n,d=3)=1-\frac{56{\pi }^2}{9n^2}+O(n^{-3})$ by the analysis of the graph Laplacian based on the finite element method. We also show the lower bound on the fidelity of $d$-dimensional unitary estimation for an arbitrary $d$ given by $F_{\mathrm{e}\mathrm{s}\mathrm{t}}(n,d)\ge 1-\frac{(d+1)(d-1)(3d-2)(3d-1)}{6n^2}+O(n^{-3})$ achieving the best known lower bound and tight scaling with respect to $n$ and $d$. This lower bound is derived based on the unitary estimation protocol shown in [J. Kahn, Phys. Rev. A 75, 022326, 2007].