Measurement-induced phase transitions in the toric code

We study how distinct phases of matter can be generated by performing random single-qubit measurements on a subsystem of toric code. 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 un-measured boundary between phases with different orders and entanglement scaling. The results follow from a parton construction of the toric code, using which random Pauli measurements map to a classical loop model in 2 dimensions. The short and long loop phases in the classical model and their transitions completely determine the entanglement and quantum order in the measured toric code. We also comment on the usefulness of such an approach to produce and manipulate phases of matter, and realise measurement-induced phase transitions, in current quantum simulators.