Determining the J matrix of the ISING Hamiltonian

Question

I am reading from the paper of Utkarsh et al., Solving Vehicle Routing Problem Using Quantum Approximate Optimization Algorithm (arXiv).

And I couldn't quite understand the notation to which the authors describe the formulation of the J matrix in the ISING Hamiltonian.

$$ H_{\text{ISING}} = -\sum_i \sum_{j < i} J_{ij} s_i s_j + \sum_i h_i s_i + d $$

where $J_{ij}$ is defined as $$ J_{ij} = -\frac{Q_{ij}}{4} \quad \forall i < j, \quad J_{ii} = 0 \quad \forall i $$

where the $Q$ Matrix is the matrix present in the QUBO formulation.

I do not understand whether the $J$ matrix is an upper triangular matrix with all the rest of the elements 0 or is it symmetric or skewed-symmetric?

My confusion arises from the fact that elements $\forall i > j$ are not specified.

Answer 1

Remember that $s_is_j=s_js_i$, so effectively you need $J$ to be symmetric with 0 diagonal. I suspect your confusion comes from a slightly weird choice of inequality (typo?). Just change it to $$ J_{ij}=-\frac{Q_{ij}}{4}\qquad\forall i>j $$ and that sorts everything out.