  It has recently been discovered that random quantum circuits provide an
avenue to realize rich entanglement phase diagrams, which are hidden to
standard expectation values of operators. Here we study (2+1)D random circuits
with random Clifford unitary gates and measurements designed to stabilize
trivial area law and topologically ordered phases. With competing single qubit
Pauli-Z and toric code stabilizer measurements, in addition to random Clifford
unitaries, we find a phase diagram involving a tricritical point that maps to
(2+1)D percolation, a possibly stable critical phase, topologically ordered,
trivial, and volume law phases, and lines of critical points separating them.
With Pauli-Y single qubit measurements instead, we find an anisotropic
self-dual tricritical point, with dynamical exponent $z \approx 1.46$,
exhibiting logarithmic violation of the area law and an anomalous exponent for
the topological entanglement entropy, which thus appears distinct from any
known percolation fixed point. The phase diagram also hosts a
measurement-induced volume law entangled phase in the absence of unitary
dynamics.
