# Simulate Hamiltonians with Pauli operations (controlled time evolution)

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:)

• 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) Nov 12 '20 at 0:23
• @C. Kang Thanks for the comment! I still prefer the exponentiation approach:)
– ZR-
Nov 12 '20 at 0:32

So we know that $$e^{i t H_2}$$ has the following circuit:
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:
• 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. Nov 12 '20 at 18:14