  We develop solvable models of large-$N$ hybrid quantum circuits on qubits and
fermions with long-range power-law interactions and continuous local
monitoring, which provide analytical access to the entanglement phase diagram
and error-correcting properties of many-body entangled non-equilibrium states
generated by such dynamics. In one dimension, the long-range coupling is
irrelevant for $\alpha>3/2$, where $\alpha$ is the power-law exponent, and the
models exhibit a conventional measurement-induced phase transition between
volume- and area-law entangled phases. For $1/2<\alpha<3/2$ the long-range
coupling becomes relevant, leading to a nontrivial dynamical exponent at the
measurement-induced phase transition. More interestingly, for $\alpha<1$ the
entanglement pattern receives a sub-volume correction for both area-law and
volume-law phases, indicating that the phase realizes a quantum error
correcting code whose code distance scales as $L^{2-2\alpha}$. While the
entanglement phase diagram is the same for both the interacting qubit and
fermionic hybrid Brownian circuits, we find that long-range free-fermionic
circuits exhibit a distinct phase diagram with two different fractal entangled
phases.
