Change QAOA observable to problem Hamiltonian #260
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Can you help me understand how this observable is chosen?
Starting from the circuit we are running: qaoa_barabasi_albert which comes from benchpress. I don't know much about this circuit, but I assume it is trying to solve some problem related to the Barabási-Albert model. Naively, I wouldn't expect this QAOA circuit to have the same Hamiltonian as a QAOA circuit solving a MAXCUT problem. My knowledge of QAOA is quite limited, so I may be missing something.
It's unfortunate that there isn't more documentation on this circuit in benchpress.
Since I was unable to find the problem Hamiltonian or any other observable used in Benchpress, I looked in the literature for QAOA approaches to solving Barabasi-Albert weighted graphs and found the paper I linked in this PR. |
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Ah okay, thanks for the explanation, that helps me understand much more (and more than the reading I was doing). I wonder if we should reach out to the benchpress team to get some info on how the qaoa circuits are generated and ask what Hamiltonian they are using.
I have at one blocking comment, and one question.
| @@ -235,8 +235,37 @@ def simulate_expvals( | |||
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| else: | |||
| density_matrix = simulate_density_matrix(compiled_circuit) | |||
| obs_str = "Z" * compiled.num_qubits | |||
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The variable obs_str still needs to be set in order to be recorded in the results.
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Good catch. I wonder why it didn't raise an error when I ran it. Let me check again...
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Both issues are fixed-
- I set
obs_strto"H_p"(problem Hamiltonian) because the sum of Pauli strings is way too long to display. - The missing
obs_strdidn't throw an error because the 'if' statement should have readif circuit_name == "qaoa_barabasi_albert"notif circuit_name == "qaoa"
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Using H_p doesn't seem ideal, but a good compromise. We should record the observable string somewhere for easy reference.
Good find with that bug!
Use `qiskit.quantum_info.SparsePauliOp` to construct problem Hamiltonian for QAOA observable
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Nice! I was initially a little worried about using SparsePauliOp instead of an instance of Operator (since that what density_matrix.expectation_value expects), but it turns out that they are both derived from LinearOp, so all checks out.
How does this impact results?
Fixes #255.
A natural choice of observable for the QAOA circuits is the problem Hamiltonian, instead of the default "ZZZZ..." observable.
In https://arxiv.org/pdf/2009.01095, we see that the problem Hamiltonian for a weighted graph with binary encoding is:
where the matrix$H_{i,j}$ is a diagonal matrix modeling the interaction between vertices $i$ and $j$ , and $w_{i,j}$ is the weight of the edge between vertices i and j as well as the resulting unitary evolution.
For$k = 2$ , which corresponds to one qubit per vertex of the graph, the local Hamiltonian is
which can be written