Well, matrix H seems to be degenerate in a fundamental way. Not only is its determinant 0, but its rank is 2 (for both N=10 and N=100). Having a 10*10 matrix of rank 2 is usually a computationally bad thing, let alone a 100*100 matrix of rank 2. So adding just a little bit of a well-behaved matrix to it seems to help computationally. Without knowing the exact algorithm behind qp it is hard to be more specific.

I am surprised though how little it took to improve things. And I am not at all 100% sure it is the correct approach. Let's wait for the answers.