  Quantum tricriticality of a $J_1$-$J_2$ antiferromagnetic Ising model on a
square lattice is studied using the mean-field (MF) theory, scaling theory, and
the unbiased world-line quantum Monte-Carlo (QMC) method based on the Feynman
path integral formula. The critical exponents of the quantum tricritical point
(QTCP) and the qualitative phase diagram are obtained from the MF analysis. By
performing the unbiased QMC calculations, we provide the numerical evidence for
the existence of the QTCP and numerically determine the location of the QTCP in
the case of $J_1=J_2$. From the systematic finite-size scaling analysis, we
conclude that the QTCP is located at $H_{\rm QTCP}/J_1=3.260(2)$ and
$\Gamma_{\rm QTCP}/J_1=4.10(5)$. We also show that the critical exponents of
the QTCP are identical to those of the MF theory because the QTCP in this model
is in the upper critical dimension. The QMC simulations reveal that
unconventional proximity effects of the ferromagnetic susceptibility appear
close to the antiferromagnetic QTCP, and the proximity effects survive for the
conventional quantum critical point. We suggest that the momentum dependence of
the dynamical and static spin structure factors is useful for identifying the
QTCP in experiments.
