  For a nonsingular projective variety $X$, the virtual Tevelev degree in
Gromov-Witten theory is defined as the virtual degree of the morphism from
$M_{g,n}(X,d)$ to the product $M_{g,n} \times X^n$. After proving a simple
formula for the virtual Tevelev degree in the (small) quantum cohomology ring
of $X$ using the quantum Euler class, we provide several exact calculations for
flag varieties and complete intersections. In the cominuscule case (including
Grassmannians, Lagrangian Grassmannians, and maximal orthogonal Grassmannians),
the virtual Tevelev degrees are calculated in terms of the eigenvalues of an
associated self-adjoint linear endomorphism of the quantum cohomology ring. For
complete intersections of low degree (compared to dimension), we prove a
product formula. The calculation for complete intersections involves the
primitive cohomology. Virtual Tevelev degrees are better behaved than arbitrary
Gromov-Witten invariants, and, by recent results of Lian and Pandharipande, are
much more likely to be enumerative.
