Novel phases of quantum circuits protected by hidden dynamical symmetries

States evolving in quantum circuits are known to undergo measurement-induced phase transitions from volume-law to area-law entanglement. We argue that many more phases are possible. This statement is made precise in a broad class of circuits, for which we can map the steady state of information dynamics to the ground states of an effective Hamiltonian, possessing a larger symmetry than the symmetry of the physical circuit. The different ground states admitted by the larger effective symmetry, including broken symmetry and SPT phases, correspond to distinct patterns of information flow and scrambling in the circuit. We will illustrate these ideas using two examples. First, we show that the dynamics of spin systems having only Z_2 symmetry, generally map to the ground states of an effective Hamiltonian with D_4 (non-abelian) symmetry. We discuss the physical meaning of the different phases in terms of the information flow and scrambling in the circuit. Second, we show that quadratic fermion circuits, preserving only fermion parity, map to an effective Hamiltonian with U(1) symmetry. We consequently predict a measurement-induced KT transition between a critical phase with log(L) entanglement and a "trivial" area-law phase.