Mean estimation when you have the source code; or, quantum Monte Carlo methods

Suppose $\boldsymbol{y}$ is a real random variable, and one is given access to ``the code'' that generates it (for example, a randomized or quantum circuit whose output is $\boldsymbol{y}$). We give a quantum procedure that runs the code $O(n)$ times and returns an estimate $\widehat{\boldsymbolμ}$ for $μ= \mathrm{E}[\boldsymbol{y}]$ that with high probability satisfies $|\widehat{\boldsymbolμ} - μ| \leq σ/n$, where $σ= \mathrm{stddev}[\boldsymbol{y}]$. This dependence on $n$ is optimal for quantum algorithms. One may compare with classical algorithms, which can only achieve the quadratically worse $|\widehat{\boldsymbolμ} - μ| \leq σ/\sqrt{n}$. Our method improves upon previous works, which either made additional assumptions about $\boldsymbol{y}$, and/or assumed the algorithm knew an a priori bound on $σ$, and/or used additional logarithmic factors beyond $O(n)$. The central subroutine for our result is essentially Grover's algorithm but with complex phases.ally Grover's algorithm but with complex phases.