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Suppose I run the Qiskit's QAOA algorithm.

qaoa = QAOA(operator=qubit_operator, p=p, optimizer=optimizer)
result = qaoa.run(quantum_instance)

There is a built-in method that gets me the most likely binary solution for this problem. sample_most_likely(result['eigenstate']) But how do I get the whole probability distribution of binary solutions? Ultimately, I would like to calculate the expected value for my problem, not the value based solely on the most likely string.

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You should be able to use: result['eigenstate']

That should spit out something like (if you are using 4 qubits):

{'0001': 27,
 '0010': 20,
 '0011': 1821,
 '0100': 25,
 '0101': 591,
 '0110': 46,
 '0111': 5347,
 '1001': 18,
 '1010': 2,
 '1011': 226,
 '1101': 11,
 '1111': 58}
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