  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 un-measured 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.
