I have a question about how to prepare a state $|\psi\rangle$ for quantum phase estimation (QPE). My question is about whether the state prepared in QPE has to be the exact eigenstate of the operator or whether it is sufficient for applications to use another initial state that is easy to prepare.
My present understanding of phase estimation laid out below.
Phase estimation allows you to estimate the eigenvalue of a unitary operator applied to an eigenstate $|\psi \rangle $ by estimating the value of $\theta$ in the eigenvalue equation below.
\begin{equation} U |\psi \rangle = e^{2 \pi i \theta} |\psi \rangle \end{equation}
The algorithm works by preparing a state $|\psi \rangle $ and then applying some controlled $U$ operations between a state preparation register and a register of "evaluation qubits". The inverse Quantum Fourier transform is then applied before the evaluation register is measured. The phase $\theta$ can be known with greater precision by increasing the number of evaluation qubits.
A pytket circuit diagram of a trivial instance of QPE is shown below where I estimate the phase applied by a single qubit rotation gate. This is similar to a pedagogical example given in the qiskit textbook.
Here the $U$ is defined as follows
\begin{equation} U(\varphi) := \begin{pmatrix} 1 & 0 \\ 0 & e^{i \varphi} \end{pmatrix}. \end{equation}
In this case its super easy to just use the Pauli $X$ gate to prepare the $|1 \rangle$ eigenstate of $U$.
$$ \begin{equation} U(\varphi)|1\rangle = e^{i \varphi}|1\rangle \, . \end{equation} $$
My question is as follows... For more interesting applications of QPE does the state prepared have to be the eigenstate of the operator $U$ ? I don't know of a reason to expect that preparing the eigenstate $|\psi \rangle$ of some operator $U$ is an easy thing to do in general (especially without knowing the eigenvalues of $U$ to begin with).
I have heard that phase estimation is a "projective" algorithm and so projects into the eigenspace of $U$ even if the initial state isn't exactly $|\psi \rangle$. I wasn't sure exactly how to interpret this. Is it sufficient for the initial state to be "close" to $|\psi \rangle $ in some sense?
I would appreciate if someone could provide more detail on how this works or point me in the direction of some useful references. These can include how QPE is used in practice in chemistry etc.