  In this paper, we establish Schr\"{o}dinger maximal estimates associated with
the finite type phases \begin{equation*}
\phi(\xi_1,\xi_2):=\xi^m_1+\xi^m_2,\;(\xi_1,\xi_2)\in [0,1]^2, \end{equation*}
where $m \geq 4$ is an even number. Following [12], we prove an $L^2$ fractal
restriction estimate associated with the surfaces \begin{equation*}
F^2_m:=\{(\xi_1,\xi_2,\phi(\xi_1,\xi_2)):\;(\xi_1,\xi_2)\in [0,1]^2\}
\end{equation*} as the main result, which also gives results on the average
Fourier decay of fractal measures associated with these surfaces. The key
ingredients of the proof include the rescaling technique from [16],
Bourgain-Demeter's $\ell^2$ decoupling inequality, the reduction of dimension
arguments from [17] and induction on scales.
