Circuit construction for Hamiltonian simulation

I would like to know how to design a quantum circuit that given a Hermitian matrix $$\hat{H}$$ and time $$t$$, maps $$|\psi\rangle$$ to $$e^{i\hat{H}t} |\psi\rangle$$. Thank you for your answer.

• There are many advanced algorithms for compiling a Hamiltonian into a series of gates; are you looking for a summary of some of the simpler approaches, or a list of resources to start you off? – ahelwer Feb 27 at 16:44
• in terms of what elementary gates? If you do not restrict the gate set the answer is trivially the circuit with the single gate $U=e^{i H t}$ – glS Feb 28 at 20:27