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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?

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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.

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    $\begingroup$ I don't know much about this stuff. I have a question. So, are finding the ground state of Ising Hamiltonian and QUBO problems mathematically equivalent? Are they the same class of problems? $\endgroup$
    – FDGod
    Apr 7 at 9:01
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    $\begingroup$ @FDGod: Yes, they are both NP-hard problems. One can be translated to another with simple linear transformation. See for example here: arxiv.org/abs/1302.5843 $\endgroup$ Apr 7 at 9:11
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    $\begingroup$ My first question is whether it is possible to get the same solution of the Ising model using VQE instead of QAOA if we have a quantum computer that can perform arbitrary state preparation. If QAOA is subset of VQE, I think it is possible and I don't know why people are just only focusing on QAOA for solving the Ising model instead of using VQE. $\endgroup$
    – William
    Apr 8 at 14:50
  • $\begingroup$ @William: Yes, the results should be same in case of QAOA and VQE. You can see this in result section of my first linked paper (pg. 16). In my case, I even found out that VQE needs less iterations than QAOA. However, I solved only a toy problem (six variables), so I would not draw strong conclusions from that. Concerning your second question, it is often better to use an algorithm specialized for your particular problem. Moreover, there are not only QAOA and VQE for solving Ising model. Please, have a look at quantum annealers (provided by company D-Wave). $\endgroup$ Apr 8 at 15:19
  • $\begingroup$ @William: This may help: quantumcomputing.stackexchange.com/questions/8566/… $\endgroup$ Apr 8 at 15:22

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