Shortest surface walk across a unit cube

An ant at one vertex of a unit cube wants to reach the diagonally opposite vertex, walking only on the cube's surface. If $d$ is the length of the shortest such route, what is $d^2$?

$5$. Unfold two adjacent faces into a $1\times2$ rectangle; the opposite vertex becomes its far corner, and the straight line has length $\sqrt{1^2+2^2}=\sqrt5$.