Is there a standard way to translate an Hamiltonian into QC circuit?

Question

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.

Comments

An analogous question for quantum optical systems is answered in my paper 

U. Leonhardt and A. Neumaier, Explicit effective Hamiltonians for general linear quantum-optical networks, J. Optics B: Quantum Semiclass. Opt. 6 (2004), L1-L4.quant-ph/0306123

Maybe you can translate it to your setting.

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I think you want the Solovay-Kitaev algorithm, see https://arxiv.org/pdf/quant-ph/0505030.pdf