  We demonstrate the quantum mean estimation algorithm on Euclidean lattice
field theories. This shows a quadratic advantage over Monte Carlo methods which
persists even in presence of a sign problem, and is insensitive to critical
slowing down. The algorithm is used to compute $\pi$ with and without a sign
problem, a toy U(1) gauge theory model, and the Ising model. The effect of
$R_{Z}$-gate synthesis errors on a future fault-tolerant quantum computer is
investigated.
