Understand final Hamiltonian in Quantum Annealing

Question

Currently, I’m studying quantum annealing in which the system evolves according to the Hamiltonian:

$$ H(t) = H_F + \Gamma(t) H_D,\tag{1} $$

where $H_F$ is a final (classical) Hamiltonian, $H_D$ is a transverse field Hamiltonian and $\Gamma$ is a transverse field coefficient.

My question is about how to imagine what it is like when the transverse field term is not included. Then, we are back to the system evolving according to the classical Hamiltonian alone: $H(t) = H_F.$

For this Hamiltonian how is the potential energy landscape defined? In the case of simulated annealing, in order to define the cost landscape we first need two elements which are (1) the neighborhood rule used to determine neighbors of the states (2) the objective function. Although for (2) we have the energy of eigenstates, what is the rule for determining the connection between eigenstates of the Hamiltonian?

Answer 1

"For this Hamiltonian how is the potential energy landscape defined?"

Most experts in quantum annealing do not use the word "potential energy". The similar term "energy landscape" is typically defined as the eigenvalues of the Hamiltonian (the $H_F$ has a time-independent energy landscape, and the $H(t)$ has a time-dependent one).

"what is the rule for determining the connection between eigenstates of the Hamiltonian?"

We don't typically claim that there is a "rule" determining the connection between the eigenstates of a Hamiltonian, but the off-diagonals of the Hamiltonian are sometimes referred to with various terms such as "tunneling amplitudes" or "couplers" or "coupling constants" (though they can be time-dependent) and they do give some indication as to how closely two states in a particular basis are connected.