  We study a class of (1+1)D symmetric random quantum circuits with two
competing types of measurements in addition to random unitary dynamics. The
circuit exhibits a rich phase diagram involving robust symmetry-protected
topological (SPT), trivial, and volume law entangled phases, where the
transitions are hidden to expectation values of operators and can only be
accessed through the entanglement entropy averaged over quantum trajectories.
In the absence of unitary dynamics, we find a purely measurement-induced
critical point with logarithmic scaling of the entanglement entropy, which we
map exactly to two copies of a classical 2D percolation problem. We perform
numerical simulations that indicate this transition is a tricritical point that
splits into two critical lines in the presence of arbitrarily sparse unitary
dynamics with an intervening volume law entangled phase. Our results show how
measurements alone are sufficient to induce criticality and logarithmic
entanglement scaling, and how arbitrarily sparse unitary dynamics can be
sufficient to stabilize volume law entangled phases in the presence of rapid
yet competing measurements.
