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We consider a paradigmatic solvable model of topological order in twodimensions, Kitaev's honeycomb Hamiltonian, and turn it into a measurement-onlydynamics consisting of stochastic measurements of two-qubit bond operators. Wefind an entanglement phase diagram that resembles that of the Hamiltonianproblem in some ways, while being qualitatively different in others. When onetype of bond is dominantly measured, we find area-law entangled phases thatprotect two topological qubits (on a torus) for a time exponential in systemsize. This generalizes the recently-proposed idea of Floquet codes, wherelogical qubits are dynamically generated by a time-periodic measurementschedule, to a stochastic setting. When all types of bonds are measured withcomparable frequency, we find a critical phase with a logarithmic violation ofthe area-law, which sharply distinguishes it from its Hamiltonian counterpart.The critical phase has the same set of topological qubits, as diagnosed by thetripartite mutual information, but protects them only for a time polynomial insystem size. Furthermore, we observe an unusual behavior for the dynamicalpurification of mixed states, characterized at late times by the dynamicalexponent $z = 1/2$ -- a super-ballistic dynamics made possible by measurements.

Topology, criticality, and dynamically generated qubits in a stochastic measurement-only Kitaev model