# help understanding gate to hamiltonian and representation

So I have this question: Given an operator, find some Hamiltonian implementing this operator/gate. I have realized that this is a swap gate and I know the matrix for it. I also know that $$U = \text{exp}(-iHt)$$-- in this case, $$U = G$$. I also know that $$\text{exp}(iA) = I + A + \frac{A^2}{2}! + \frac{A^3}{3}! + \cdots$$ I also know that a $$CNOT=∣0⟩⟨0∣⊗I+∣1⟩⟨1∣⊗X$$ and that $$G$$ operator is composed

My questions/confusions are:

I know $$H$$ will be some combination of tensor product of pauli matrices--is there a direct way to do this/ calculate it or is this just a case by case basis--observing things like that $$XX$$ swaps the states $$∣01⟩$$ and $$∣10⟩$$ and $$ZZ$$ leaves $$∣00⟩$$ and $$∣11⟩$$ unchanged but adds a negative phase to $$∣01⟩$$ and $$∣10⟩$$... etc.? I guess related to this is, given some matrix--is there a specific process to follow to translate it from a matrix representation to a tensor-product of pauli matrices representation? I need help with this question.[question in image: Consider operator G acting on Hilbert space x Hilbert_space as follows: $$∣\phi⟩ ∣\psi⟩ \to ∣\psi⟩ ∣\phi⟩$$ for all $$\psi$$, $$\phi$$ in $$H$$. Find some Hamiltonian implementing the $$G$$ gate if $$G$$ is linear and unitary. 1 .

There are many ways that you could go about doing this. (You might also want to take a look at what you"know" about exponentiating a matrix, because you've exponentiated $$A$$ not $$iA$$.)
One method, whose usage is a bit limited, but works very well in this case, is to think about any special cases of matrix exponentiation you might know about. There's a common identity for $$e^{iAt}$$ where $$A^2=I$$ which you might be able to reverse engineer to find something useful...
More generally, you might find the exponential easier to think about in terms of the spectral decomposition: $$e^{iA}=\sum_ne^{i\lambda_n}|\lambda_n\rangle\langle\lambda_n|$$ where $$A=\sum_n\lambda_n|\lambda_n\rangle\langle\lambda_n|$$. So, given the unitary, you can figure out some properties of the eigenvectors and eigenvalues of $$A$$.