Suppose we want to perform Quantum Phase Estimation over a Linear Combination of Unitaries Hamiltonian. One of the most efficient ways to do so is to use qubitization:
\begin{equation} Q=(2|0\rangle\langle0|-1)\text{PREP}_H^{\dagger}\text{SEL}_H\text{PREP}_H, \end{equation} which will implement the evolution operator $e^{-i\arccos H/\lambda}= e^{i\arcsin (H/\lambda)} e^{\pi/2}$, e.g. $\arccos$ and $\arcsin$ are equivalent up to a global phase and the direction of the time evolution.
Instead of deciding whether to implement $Q$ or not, we can instead try to control the direction of the time evolution, $e^{\pm i\arcsin H/\lambda}$, which is often easier. For example, if in $\text{SEL}_H$ one implements either $\pm H$, then we will have $e^{i\arcsin (\pm H/\lambda)} = e^{\pm i\arcsin H/\lambda}$, (because $\arcsin$ is an odd function).
Now, to implement either (a block encoding of) $\pm H$ it would seem that the best way is just to apply a $Z$ gate on the control qubit in the Select operation, but then this cannot be right for at least two reasons:
- There is no entangling gate left between the control qubit in QPE and the eigenstate register.
- We seem to be applying $\pm e^{i\arcsin H/\lambda}$ instead of $e^{\pm i\arcsin H/\lambda}$.
However, I still cannot understand why a simple $Z$ gate does not apply the desired phase.
Where is the mistake in the reasoning above?
