Change QAOA observable to problem Hamiltonian

[Misty-W]

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 for binary encoding the problem Hamiltonian is given by:

$$H_{i,j} = diag(vec(D^T))$$

where, for a 10-qubit circuit as in the UCC benchmarks,

$$H_{i,j} = diag (1, −1, −1, −1, −1, 1, −1, −1, −1, −1, 1, −1, −1, −1, −1, 1, −1, −1, −1, 1)$$

$H_{ij}$ is then input into Qiskit's built-in Operator to facilitate computing the observable.

[natestemen]

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.

[Misty-W]

Can you help me understand how this observable is chosen?

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.
Although the authors use MAXCUT to illustrate the binary encoding and "one-hot" encoding methods, the authors mention that they applied both methods to solving a Barabasi-Albert weighted graph and report the results.

[natestemen]

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.

[natestemen]

The variable obs_str still needs to be set in order to be recorded in the results.

[natestemen]

This observable only represents an operator acting on 4 qubits, but the circuit we are running is 10 qubits. Do we just assume the operator acts as identity on the other 6 qubits?