  We study the quantum Ising model on the Sierpi\'{n}ski triangle, whose
Hausdorff dimension is $\log 3/ \log 2 \approx 1.585$, and demonstrate that it
undergoes second-order phase transition with scaling relations satisfied
precisely. We also study the quantum $3$-state Potts model on the
Sierpi\'{n}ski triangle and find first-order phase transition, which is
consistent with a prediction from $\epsilon$-expansion that the transition
becomes first-order for $D > 1.3$. We then compute critical exponents of the
Ising model on higher-dimensional Sierpi\'{n}ski pyramids with various
Hausdorff dimension via Monte-Carlo simulations and real-space RG analysis for
$D\in[1,3]$. We find that only the correlation length exponent $\nu$
interpolates the values of integer-dimensional models. This implies that,
contrary to a generally held belief, the universality class of quantum phase
transition may not be uniquely determined by symmetry and spatial dimension of
the system. This work initiates studies on quantum critical phenomena on graphs
and networks which may be of significant importance in the context of quantum
networks and communication.
