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Let's say I am given a Hamiltonian $H$, whose ground state is efficiently preparable. We know that $||H|| \leq 1$. Let that ground state be $|\psi_{0}\rangle$, with energy $E_{0}$. We also know that the unitary $U$ such that \begin{equation} U |\psi_{0}\rangle |00\cdots0\rangle = |\psi_{0}\rangle |E_{0}\rangle, \end{equation} is efficiently preparable.

Consider the promise problem. For the yes instances, \begin{equation} E_{0} \geq c, \end{equation} and for the no instances, \begin{equation} E_{0} \leq s, \end{equation} where $c - s \geq \frac{1}{O\big(2^{poly(n)}\big)}$. Would this promise problem be in $BQP$?

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As you've written it, yes. The implication of what you've written is that $|E_0\rangle$ is prepared exactly. Since it has been done efficiently, we know that you must have used only a number of qubits that's polynomial in $n$. Thus, I can measure those poly($n$) qubits and exactly determine the value of $E_0$. Once I have an exact classical value, it's no problem to compare it to $c$ and $s$, again each expressed using poly($n$) bits.

However, I feel that this problem is not generally a good reflection of what one would have to do in practice. But I'm rather rusty on this sort of thing, and wouldn't want to commit myself further!

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