Can we use Hadamard test to estimate phases?

Question

There have been some questions discussing the Hadamard test and quantum phase estimation (QPE), but I did not find the answer to the following question. Suppose we are given $|\psi\rangle$, which is an eigenstate of $U$ such that $U|\psi\rangle = \exp(i\theta)|\psi\rangle$, and we are asked to estimate the phase $\theta$. Surely we can use QPE to estimate it, but can't we do the same with Hadamard test? In particular, my question consists of two parts

1. Can we use Hadamard test to measure the real and imaginary part of $\langle\psi |U|\psi\rangle$ separately to find $\theta$? If the answer is yes, what is the advantage of QPE?

2. How many measurements do I need in order to make sure the error of estimation is below $\epsilon$?

Answer 1

So QPE using $\mathcal{O}(1/\epsilon)$ queries to $U$ outputs an estimate of the eigenphase $\theta$ given a corresponding eigenvector with additive error and $\Omega(1)$ probability.

The method using separate Hadamard Tests, that you mentioned, works. However, it requires $\mathcal{O}(1/\epsilon^2)$ samples to just get $\epsilon$-estimates of the real and imaginary parts of the eigenvalue. These samples can be viewed as queries to $U$. This complexity is based off of the central limit theorem/law of large numbers or Chernoff bounds as @Mark S mentioned. Thus, the method you mentioned is at least quadratically slower, in query complexity, than QPE.

If the goal is to estimate $\theta$, you need to compute the $\arctan$ of the estimates of the imaginary and real parts. This makes the error for $\theta$ no longer additive and would need to be taken into account to determine the number of samples required for the desired error on $\theta$.

On another note, there are versions of QPE that make use of single Hadamard Tests and don't suffer from this problem: Iterative Quantum Phase Estimation, which performs multiple single qubit QPEs, or the semi-classical QFT method that uses a non-unitary variant of the inverse QFT.