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I can't seem to wrap my head around the Trotter-Suzuki 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 formula is approximate up 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 annihilation 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:

Simulation of Quantum Harmonic Oscillator With Its Introduction to A Fermionic System, and

Quantum Computation and Visualization of Hamiltonians using Discrete Quantum Mechanics and IBM QISKit

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Your question is not so much about the usefulness of the Trotter-Suzuki formula per se, but is rather about the usefulness of Hamiltonian simulation in general (and eigenvalue sampling in particular), of which the Trotter-Suzuki method is but one of several approaches.

In particular the question is more along the lines of "what's the point of doing Hamiltonian simulation to calculate the energy of an eigenstate when we are not even sure that we are in an eigenstate in the first place?" But, quantum mechanics is linear, and expectation is linear as well!

What I mean is that if you know how to prepare a pure state $|\psi\rangle$ that is not necessarily known to be an eigenstate of a Hamiltonian $H$ but is rather in a linear superposition thereof, and if you know how to use the Trotter-Suzuki formula to simulate controlled versions of $U=\exp(-iHt)$, then surely you can use the inverse Quantum Fourier Transform as you propose to calculate, in superposition, the energies of each of the supported eigenstates. Repeatedly preparing your state $|\psi\rangle$ and measuring the ancilla register that stores these energies lets you determine the average energy of $|\psi\rangle$. (The accuracy of your estimate of the energy depends on the number of times you prepare $|\psi\rangle$ and sample the ancilla qubits, the number of terms you extrapolate out in your Trotter-Suzuki formula, and the number of qubits in your ancilla register.)

Certainly it's axiomatic that after measuring the ancilla register then the state $|\psi\rangle$ necessarily 'collapses' to an eigenstate having the measured eigenvalue. But you need not measure this ancilla and instead you can do other eigenvalue surgery on the ancilla qubits while in superposition, such as calculating their inverse with the HHL algorithm and kicking back the phases by uncomputing the Trotter-Suzuki simulation.

Thus, the Trotter-Suzuki formula is useful because it allows for Hamiltonian simulation, which, among other things, is what enables the Quantum Phase Estimation algorithm to estimate the energies of various states and to do other cooler tricks like the HHL linear-systems algorithm.

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You chose too simple example. The power of the formula is in a case that the Hamiltonian is sum of many terms, usually to show the behavior of many particles, that their Hamiltonian is sum of terms, where each term represent the energy of each particle, or the interaction Hamiltonian between particles.

Then, since the formula is converting to multiplication of operators, you can simulate by operating with each operator, sequentially one after another.

Also for little amount of terms, it is useful, because it is not trivial to implement general quantum operator, but it might be easier to act with X and than with Z for example

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