# How to determine appropriate time evolution for phase estimation algorithm?

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!!

• See Appendix F in "Elucidating Reaction Mechanisms on Quantum Computers", 2016. arxiv.org/abs/1605.03590 Feb 28, 2022 at 9:10

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.
• Thanks so much for the answer! If we set $tE_{max}=2\pi$, are we assuming we know the maximum energy value? If so, can I let $t = 2\pi/E_{max}$ to run the time-evolution?
• Yes, exactly. The challenge is to find out $E_{\max}$ or at least a reasonable upper bound on it. This can often be inferred from some properties of your Hamiltonian. Mar 1, 2022 at 7:41