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Quantum circuit dynamics with local projective measurements can realize arich spectrum of entangled states of quantum matter. Motivated by the physicsof the Kitaev quantum spin liquid [1], we study quantum circuit dynamics in(2+1)-dimensions involving local projective measurements, in which themonitored trajectories realize (i) a phase with topological quantum order or(ii) a "critical" phase with a logarithmic violation of area-law-scaling of theentanglement entropy along with long range tripartite entanglement. A Majoranaparton description of these dynamics, which provides an out-of-equilibriumgeneralization of the parton description of the Kitaev honeycomb model, permitsan analytic understanding of the universal properties of these two phases,including the entanglement properties of the steady-state, the dynamics of thesystem on the approach to equilibrium, and the phase transition between thesestates. In the topologically-ordered phase, two logical qubits can be encodedin an initial state and protected for a time which scales exponentially in thelinear dimension of the system, while no robust encoding of quantum informationpersists in the critical phase. Extensive numerical simulations of thesemonitored dynamics confirm our analytic predictions.

Monitored Quantum Dynamics and the Kitaev Spin Liquid