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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César Leonardo Clemente López
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2Can you show the circuit (or a portion of it?) you can reconstruct it but may depend on the circuits complexity – C. Kang Feb 15 '21 at 07:14
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5You 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. – DaftWullie Feb 15 '21 at 07:55
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@kang it is basically a combination of grovers and QPE – César Leonardo Clemente López Feb 15 '21 at 14:30
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@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? – Mark Spinelli Feb 16 '21 at 01:47
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1@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. – DaftWullie Feb 16 '21 at 07:35
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What I've realised in writing that previous comment is that I'm implicitly assuming that $H$ should not vary in time. Is that the desired case? – DaftWullie Feb 16 '21 at 07:36