Thank you for your answer. I had noticed the relationship between the $q_i$ and the diagonal of $L$, implying the equalities of the sums. Unfortunately I do believe too much information is lost to have a sharper relation. Your insight was nonetheless helpful.

@JosiahPark the link above doesn't work anymore. Here is one that does: arxiv.org/pdf/1510.05517.pdf If in the future this one doesn't work, for reference the title is "Sparse Hanson-Wright inequalities for subgaussian quadratic forms" by Shuheng Zhou.