  I study the properties of the quantum critical point of the transverse-field
quantum Ising model on various fractal lattices such as the Sierpi\'nski
carpet, Sierpi\'nski gasket, and Sierpi\'nski tetrahedron. Using a
continuous-time quantum Monte Carlo simulation method and the finite-size
scaling analysis, I identify the quantum critical point and investigate its
scaling properties. Among others, I calculate the dynamic critical exponent and
find that it is greater than one for all three structures. The fact that it
deviates from one is a direct consequence of the fractal structures not being
integer-dimensional regular lattices. Other critical exponents are also
calculated. The exponents are different from those of the classical critical
point, and satisfy the quantum scaling relation, thus confirming that I have
indeed found the quantum critical point. I find that the Sierpi\'nski
tetrahedron, of which the dimension is exactly two, belongs to a different
universality class than that of the two-dimensional square lattice. I conclude
that the critical exponents depend on more details of the structure than just
the dimension and the symmetry.
