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I'm studying about the hybrid quantum algorithms and I have a question about the VQE and QAOA.

It seems like QAOA is a part of VQE that use a certain ansatz for solving combinatorial optimization problem by finding a highest computational basis that can be found from the lowest energy of a given Ising Hamiltonian.

Let's suppose that I have a quantum computer that can prepare aribitrary quantum states.

If we use the VQE to solve the combinatorial optimization problem with this computer, can't we get the same solution as can be obtained from the QAOA?

If so, is there a reason why people are only focusing on the QAOA for the combinatorial optimization problem solving?

William
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VQE is more general algorithm allowing to look for ground state of a general Hamiltonian. On the other hand, QAOA is intended only for finding ground state of Ising Hamiltonian. So, we can say that QAOA is "subset" of VQE

The reason why people often focus on QAOA instead of VQE is that Ising Hamiltonian is related to quadratic unconstrained binary optimization (QUBO) having wide application in business (for example: traveling salesman problem, quadratic assignment problem, portfolio optimization, many logistic problems etc.). Despite that QAOA was originally intended for QUBO, it can be adapted also for binary optimization of higher degree (see here).

For a brief discussion on the difference between these two techniques, see this paper, pg. 9. A description of QAOA and its application for QUBO problems is discussed here, pg. 20. In the papers you can also find several references which can help you in deeper understanding of both algorithms. Disclaimer: I am author of these papers.

Martin Vesely
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