>>11827920I see. It seems your problem is not really an algorithmic one IMO, but a more analytic one. At least, that's the more interesting aspect of it to me, trying to obtain some nontrivial bounds based on the parametrization of qutrits.
As for the algorithm itself (I wouldn't call it a quantum algorithm really, since everything with product states can be done efficiently classically), if you just want to run some heuristics, seems straightforward.
>generate initial state on qutrits, >compute >do something like gradient ascent or Nelder-Mead, constrained to the 's being unitaryAs you probably already know, the main issue is how you decide the initial state, which radically affects the convergence rate (and accuracy, i.e. not getting stuck in local minima). It's going to depend entirely on the specific instance of the Hamiltonian, so I doubt there's a "good" universal ansatz. However, this does seem basically like Hartree-Fock, albeit in an entirely different setting (qubits/trits instead of fermions, generic 2-local Hamiltonians instead of the electronic Hamiltonian). Of course, quantum algorithms like VQE and QAOA have already been trying to figure out good initial conditions too (QAOA is really just VQE but with a specific structure to the ansatz and applied to diagonal Hamiltonians only).
I think an interesting thing to observe is that the variational approach will give you a lower bound to the eigenvalue, while analytic methods typically provide upper bounds. Might be interesting to see how these two estimates compare, and you could even compare against exact diagonalization for a couple of small values of .
