One can use QAOA to find the ground state and its energy by minimizing the expectation value of a target Hamiltonian (say, $H_T$) with respect to the QAOA ansatz and obtaining optimal parameters associated with the ground state of the target Hamiltonian.
Is there a way to obtain the parameters associated with the QAOA ansatz such that it approximates the QAOA ansatz to the first excited state of a given target Hamiltonian ($H_T$)?
One could, of course, add a term proportional to the outer product of the ground state of the target Hamiltonian to $H_T$, and then the ground state of this modified Hamiltonian (call this $H_T'$) would be the first excited state that we want (given the scalar associated with the outer product of the ground state is big enough). Basically, the first excited state of $H_T$ is the ground state of $H_T'$. But the issue is the parameters we obtain here, after classical optimization, are associated with a QAOA ansatz with the target as $H_T'$.
What I want are parameters associated with a QAOA ansatz with $H_T$ as the target and $H_M$ as the mixer (and not for $H_T'$ and $H_M$) such that these parameters approximate the QAOA ansatz to the first excited state of $H_T$.
My aim isn't to find the first excited state itself but the parameters that can approximate the QAOA ansatz (with $H_T$ and $H_M$ as the target and mixer) to the first excited state of $H_T$.
