In the context of QAOA, I often see the problem Hamiltonian being called an "Ising Hamiltonian", and shortly after, I that the Hamiltonian is a quadratic function of the spin variables.
Is this required? If I tried to optimize a function of boolean variables which is a polynomial of degree 3 using QAOA, this would not fit the description of a quadratic. But I could still make the following substitution: $$x=\frac{1-s}{2}$$
Does QAOA require that the problem Hamiltonian be an Ising Hamiltonian as a quadratic function of the spin variables?
Perhaps related: Why does the problem Hamiltonian of QAOA always consist of $Z$ and $I$ gates? Creating Ising Hamiltonian with Qiskit
EDIT In particular, I'm trying to express the following constraint: either $\sum_{i} x_i > 0$ or $\sum_{j} x_j > 0$ (or both). There might be an easier way to express this but currently my expression is of the following form:
$$(\sum_{i} x_i - k_1) (\sum_{j} x_j - k_2) ... (\sum_{i} x_i - K_1) (\sum_{j} x_j - K_2)$$