  There has been much recent progress on exotic surface critical behavior, yet
the classical-quantum correspondence of special and extraordinary-log
criticality remains largely unclear. Employing worm Monte Carlo simulations, we
explore the surface criticality at an emergent superfluid-Mott insulator
critical point in the Villain representation, which is believed to connect
classical and quantum O(2) critical systems. We observe a special transition
with the thermal and magnetic renormalization exponents $y_t=0.58(1)$ and
$y_h=1.690(1)$ respectively, which are close to recent estimates from models
with discrete spin variables. The existence of extraordinary-log universality
is evidenced by the critical exponent $\hat{q}=0.58(2)$ from two-point
correlation and the renormalization-group parameter $\alpha=0.28(1)$ from
superfluid stiffness, which obey the scaling relation of extraordinary-log
critical theory and recover the logarithmic finite-size scaling of critical
superfluid stiffness in open-edge quantum Bose-Hubbard model. Our results
bridge recent observations of surface critical behavior in the classical
statistical mechanical models [Parisen Toldin, Phys. Rev. Lett. 126, 135701
(2021); Hu $et$ $al.$, $ibid.$ 127, 120603 (2021); Parisen Toldin $et$ $al.$,
$ibid.$ 128, 215701 (2022)] and the open-edge quantum Bose-Hubbard model [Sun
$et$ $al.$, Phys. Rev. B 106, 224502 (2022)].
