Measurement-induced phase transitions in the toric code

We show how distinct phases of matter can be generated by performing random single-qubit measurements on a subsystem of toric code. Using a parton construction, such measurements map to random Gaussian tensor networks, and in particular, random Pauli measurements map to a classical loop model in which watermelon correlators precisely determine measurement-induced entanglement. Measuring all but a 1d boundary of qubits realizes hybrid circuits involving unitary gates and projective measurements in $1+1$ dimensions. We find that varying the probabilities of different Pauli measurements can drive transitions in the unmeasured boundary between phases with different orders and entanglement scaling, corresponding to short- and long-loop phases in the classical model. Furthermore, by utilizing single-site boundary unitaries conditioned on the bulk measurement outcomes, we generate mixed-state ordered phases and transitions that can be experimentally diagnosed via linear observables. This demonstrates how parton constructions provide a natural framework for measurement-based quantum computing setups to produce and manipulate phases of matter.