Say that we have some (integer) weights $w_{1,1},w_{1,2},...,,w_{m,m}$ and a target sum $W$.
Suppose that we want to find whether there are $a_1,...,a_m \in \{0,1\}$ such that
$$\sum_{i = 1}^{m} \sum_{j = i}^m a_i a_j w_{i,j} = W.$$
I read through the Wikipedia article on the quadratic knapsack problem. It mentions (but does not completely spell out why) that pseudo polynomial dynamic programming algorithms can only function as a heuristic for the QKP.
Has it been shown that there is no pseudo polynomial time algorithm for this 0-1 quadratic subset sum problem under reasonable assumptions?