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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.

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    $\begingroup$ 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? $\endgroup$ – ahelwer Feb 27 at 16:44
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    $\begingroup$ in terms of what elementary gates? If you do not restrict the gate set the answer is trivially the circuit with the single gate $U=e^{i H t}$ $\endgroup$ – glS Feb 28 at 20:27

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