  Quantum Bruhat graph is a weighted directed graph on a finite Weyl group
first defined by Brenti-Fomin-Postnikov. It encodes quantum Monk's rule and can
be utilized to study the $3$-point Gromov-Witten invariants of the flag
variety. In this paper, we provide an explicit formula for the minimal weights
between any pair of permutations on the quantum Bruhat graph, and consequently
obtain an Ehresmann-like characterization for the tilted Bruhat order.
Moreover, for any ordered pair of permutations $u$ and $v$, we define the
tilted Richardson variety $T_{u,v}$, with a stratification that gives a
geometric meaning to intervals in the tilted Bruhat order. We provide a few
equivalent definitions to this new family of varieties that include Richardson
varieties, and establish some fundamental geometric properties including their
dimensions and closure relations.
