the quantum bruhat graph, initially introduced by brenti, fomin, and postnikov, is a weighted directed graph defined on finite weyl groups. it serves as a valuable tool for exploring the quantum cohomology ring of the flag variety. in this presentation, we present a combinatorial formula for the minimal weights between any pair of permutations within the quantum bruhat graph. furthermore, for an ordered pair of permutations $u$ and $v$, we introduce the tilted richardson variety $t_{u,v}$ demonstrating its equivalence to the two-pointed curve neighborhood of opposite schubert varieties $x_u$ and $x^v$ in the minimal degree $d$. we establish a deodhar-like decomposition for tilted richardson varieties, leveraging it to prove several results. this is joint work with shiliang gao and yibo gao.