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?



Posted 2020-11-11T23:44:54.610

Reputation: 988

1The 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) – C. Kang – 2020-11-12T00:23:08.593

1@C. Kang Thanks for the comment! I still prefer the exponentiation approach:) – Zhengrong – 2020-11-12T00:32:47.480



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

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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:

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Posted 2020-11-11T23:44:54.610

Reputation: 6 322

Thank you so much for the answer! Is there still a global phase gate needed on the control qubit? – Zhengrong – 2020-11-12T14:34:32.820

1The 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. – KAJ226 – 2020-11-12T18:14:27.183

Got it. Thanks!! – Zhengrong – 2020-11-12T20:30:43.787