  The interplay of unitary evolution and projective measurements is a modern
interest in the study of many-body entanglement. On the one hand, the
competition between these two processes leads to the recently-discovered
measurement-induced phase transition (MIPT). On the other, measurement-based
quantum computation (MBQC) is a well-known computational model studying how
measurements simulate unitary evolution utilizing the entanglement of special
resources such as the 2D cluster state. The entanglement properties enabling
MBQC may be attributed to symmetry-protected topological (SPT) orders,
particularly subsystem symmetric (SSPT) orders. It was recently found that the
1D cluster state may be associated with an SPT phase in random circuits
respecting a global $Z_2 \times Z_2$ symmetry, and furthermore that all phase
transitions in this scenario belong to the same universality class. As
resources with greater computational power feature greater symmetry, it is
fruitful to investigate further any relationship between levels of symmetry in
MIPTs and MBQC. In this paper we investigate MIPTs on a torus with three levels
of symmetry-respecting unitary evolution interspersed by measurements. Although
we find two area-law phases and one volume-law phase with distinct entanglement
structures for each ensemble, the phase transition from the volume-law phase to
the area-law phase associated with the 2D cluster state has variable
correlation length exponent $\nu$. Whereas $\nu\approx 0.90$ for unconstrained
Clifford unitaries and $\nu\approx 0.83$ for globally-symmetric Cliffords,
subsystem-symmetric Cliffords feature a much smaller value $\nu\approx 0.38$.
It is speculated that the hierarchy of distinct transitions seen in these
random monitored quantum circuit models might have consequences for
computational universality in MBQC.
