If I have a designed circuit to solve a particular problem. Is there a systematic way how to generate the Hamiltonian from it?

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    $\begingroup$ Can you show the circuit (or a portion of it?) you can reconstruct it but may depend on the circuits complexity $\endgroup$ – C. Kang Feb 15 at 7:14
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    $\begingroup$ You mean you're specifying a unitary $U$ that you want to implement as a circuit, and you would like to convert that into a Hamiltonian $H$ such that $U=e^{iHt}$? If so, the simple answer is: no. $\endgroup$ – DaftWullie Feb 15 at 7:55
  • $\begingroup$ @kang it is basically a combination of grovers and QPE $\endgroup$ – César Leonardo Clemente López Feb 15 at 14:30
  • $\begingroup$ @DaftWullie, we know we can go from $H$ to something very close to $U$ for nice enough hamiltonians (e.g. $k$-sparse with oracle access), but in what sense is this not reversible? Is it too under-specified? $\endgroup$ – Mark S Feb 16 at 1:47
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    $\begingroup$ @MarkS For example, even if we first computed $U$ (which would entirely miss the point of computing!), of course we could evaluate $H=-i\ln U$. However, there is a lot of freedom here to modify the $H$, because you can add arbitrary multiples of $2\pi$ to each eigenvalue, and there's even more freedom if the eigenvalues of $U$ are not unique. How you control those choices in order to impose some sort of reasonable properties on the Hamiltonian structure is a horrific question. $\endgroup$ – DaftWullie Feb 16 at 7:35

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