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I had a question last week regarding the simulation of Hamiltonians composed of the sum of Pauli products: How can I simulate Hamiltonians composed of Pauli matrices? I'm having a follow-up question: still for those two Hamiltonians: $$ H_{1} = X_1+ Y_2 + Z_1\otimes Z_2 \\ H_{2} = X_1\otimes Y_2 + Z_1\otimes Z_2 $$ How can I perform the 'controlled version' of them? The thing really confused me is the 'tensor product term': for both $H_1$ and $H_2$, the two qubits are coupled, but if I want to do the controlled time-evolution simulation, should I couple the whole thing with the third qubit? If so, how to do that?

Thanks:)

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    $\begingroup$ The last post you used implicitly used exponentiation - do you want to use that approach or VQEs? (what's the end goal here might be a better question) $\endgroup$
    – C. Kang
    Commented Nov 12, 2020 at 0:23
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    $\begingroup$ @C. Kang Thanks for the comment! I still prefer the exponentiation approach:) $\endgroup$
    – ZR-
    Commented Nov 12, 2020 at 0:32

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So we know that $e^{i t H_2}$ has the following circuit:

enter image description here

From this answer along with page 13 from this paper we can try to build the controlled-version of $e^{i t H_2}$ as follow:

enter image description here

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  • $\begingroup$ Thank you so much for the answer! Is there still a global phase gate needed on the control qubit? $\endgroup$
    – ZR-
    Commented Nov 12, 2020 at 14:34
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    $\begingroup$ The global phase gate is not needed if we don't have $e^{i I t}$ from my understanding. So unless our Hamiltonian is something like $H = XY + ZZ + II$ then you won't be needing it. $\endgroup$
    – KAJ226
    Commented Nov 12, 2020 at 18:14
  • $\begingroup$ Got it. Thanks!! $\endgroup$
    – ZR-
    Commented Nov 12, 2020 at 20:30

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