  The transverse-field Ising model on the Sierpi\'nski fractal, which is
characterized by the fractal dimension $\log_2^{~} 3 \approx 1.585$, is studied
by a tensor-network method, the Higher-Order Tensor Renormalization Group. We
analyze the ground-state energy and the spontaneous magnetization in the
thermodynamic limit. The system exhibits the second-order phase transition at
the critical transverse field $h_{\rm c}^{~} = 1.865$. The critical exponents
$\beta \approx 0.198$ and $\delta \approx 8.7$ are obtained. Complementary to
the tensor-network method, we make use of the real-space renormalization group
and improved mean-field approximations for comparison.
