How to determine appropriate time evolution for phase estimation algorithm?

Question

In phase estimation algorithms, we have $U|\psi\rangle = e^{2\pi i\theta}|\psi\rangle$, where $|\psi\rangle$ is an eigenvector and $ e^{2\pi i\theta}$ is the corresponding eigenvalue. Since $U$ represents the time-evolution of the quantum system, suppose the original Hamiltonian has an eigenvalue $E_1$, then after certain time-evolution, the value becomes $e^{-iE_1t}$, I wonder to simulate the time-evolution in the algorithm, how can we determine the time $t$?

Should we let $-iE_1t = 2\pi i\theta$ to solve for $t$? Also, given that $\theta\in[0,1]$, to make phase estimation less confusing, can we scale the Hamiltonian being simulated to make its eigenvalues within $[0,2\pi]$?

Thanks!!

Answer 1

In general, you're doing phase estimation because you don't know the eigenvalue $E_1$. So, while you might ideally like to choose a specific value of $\theta$ and rearrange for $t$, because you don't know $E_1$, you cannot.

Instead, you need to try and determine some basic properties of your Hamiltonian $H$. For example, if you can bound all the energies $E$ to be between $0\leq E\leq E_\max$, then you can choose $tE_{\max}=2\pi$ which will guarantee that whatever the corresponding value of $\theta$ is, it certainly falls in the range 0 to $2\pi$.

Note that if your lower bound is non-zero, you can always change $H\rightarrow H-E_{\min}I$ and it'll work just the same.

Yes, you could certainly choose to rescale $H$ to incorporate the factor of $2\pi$ if you wanted to.