I would like to calculate an observable's expectation value of a state, the ground state, or time evolution of a finite system with $N$ spins under an Hamiltonian $H$.

For the sake of discussion assume $N=16$ so we can use IBM QC.

How to translate a given Hamiltonian into Quantum logic gates in order to simulate the system evolution or its statistics.

If it makes life easier assume a local hamiltonian or any lattice based model such as an Ising model Hamiltonian:

$$

H(\sigma) = - \sum_{\langle i~j\rangle} J_{ij} \sigma_i \sigma_j -\mu \sum_{j} h_j\sigma_j

$$

As a side note, I was intrigued by this question which mentions Andre Lucas paper:

Ising formulations of many NP problems and thought that it would be nice to know how to translate an hamiltonian to a QC circuit.