I want to calculate $$y^T \mbox{diag}(A^T B A) \,y$$ where

• $y$ is a $n \times 1$ vector.
• $A$ is a $m \times n$ matrix where $n \gg m$.
• $B$ is a $m \times m$ symmetric positive definite matrix; the Cholesky decomposition $B = LL^T$ is precomputed if it is needed.

Is it possible to calculate the above expression at a cost of $O(m n)$ flops?