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I can't seem to wrap my head around the suzuki-trotter formula. I have seen This answer but I am still confused of the applicability of the formula. Let me explain:

As I understand it Trotterization lets us use directly the Schrodinger equation $|\psi(t)\rangle = e^{-i H t} |\psi(0)\rangle$ where we approximate $e^{-iHt}=e^{\sum_k H_kt}\approx \bigg(\prod_k e^{-iH_k\Delta t}\bigg)^N$ where $\Delta t=t_{final}/N$. This is formula is approximate to $O(\Delta t^2)$ and we can get further precision if we consider more terms in the expansion with the commutators. In sum, we have a way to get $|\psi(t)\rangle$ from an initial state. But now let's get a familiar physical model to apply this to: the Harmonic oscillator. Then our Hamiltonian is (in 1 dimension):

$$ \hat{H}=\frac{\hat{P}^2}{2m}+\frac{1}{2}m\omega^2\hat{X}^2 $$

if we want to evolve this system we can do so in a position or momentum basis, so that the $\hat X$ operator will just be a diagonal matrix in our discrete space and $\hat P$ will also be diagonal given that we express it as $\hat X$ wrapped around the Fourier transform as $\hat P=\hat{(QFT)}\hspace{1mm}\hat X \hspace{1mm}\hat{(QFT)}^{-1}$. So we clearly have a way to study the system. Another way is to express in terms of the creation and destruction operators $A$ and $A^\dagger$, so that our Hamiltonian can be written as $\hat H=\hat A \hat A^\dagger +\hat I/2$. These operators can be easily be constructed from the position and momentum basis. Normally in a physical system we don't know how to diagonalize a priori our Hamiltonian but in this case we can.

Having said this, here's what I don't get:

  1. Suppose we start with the position basis and our goal is to measure the energy spectrum. In order to do so we could make use of the kickback phase and the phase estimation circuit to know this given that our initial state was already an eigenstate of the Hamiltonian, but how could we know this beforehand? In this case, we could have since its a simple model, but usually we don't, there had to be some state preparation and then evolve it in time, and if that's the case, then this render the algorithm useless.
  2. Another way to view this would be to start with some state that was not an energy eigenstate of the Hamiltonian in the position basis. But how would that be useful?
  3. Then we can look at the energy basis and we can work on that to evolve our system and check its final state, and go again through some phase kickback and estimation to know the eigenvalue, but again, this assumes some state preparation that was already known that it was an eigenstate.

I know I am wrong but I just can't see the applications due to my lack of experience. I also reckon that if my goal was to find the energy spectrum I would better off with the VQE (even though I haven't studied it yet). What are the practical uses of the Trotter formula?

PS: I have looked into these papers for the harmonic oscillator application:

DOI: 10.13140/RG.2.2.18996.81284

DOI: 10.1109/NYSDS.2018.8538959

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