Fig. 2.

The Toffoli cost of estimating the projectile kinetic energy with traditional Monte Carlo sampling and the mean estimation algorithm from Kothari and O’Donnell (81) (KO). Both techniques have SE that linearly depends on the square root of the variance of the observable. The number of samples required for fixed SE in Standard Monte Carlo mean estimation scales as $\mathcal{O}(1/{ϵ}^2)$ while the KO algorithm scales as $\mathcal{O}(1/ϵ)$ but with larger constant factors originating from code (circuit) for the random variable and quantum phase estimation on the Grover like iterate used in the algorithm. More details of the algorithm’s main subroutines are provided in SI Appendix, section VI.