  In quantum many-body spin systems, the interplay between the entangling
effect of multi-qubit Pauli measurements and the disentangling effect of
single-qubit Pauli measurements may give rise to two competing effects. By
introducing a randomized measurement pattern with such bases, a phase
transition can be induced by altering the ratio between them. In this work, we
numerically investigate a measurement-based model associated with the $(2+1)$d
$\mathbb{Z}_2$ Fradkin-Shenker Hamiltonian model, encompassing the deconfining,
confining, and Higgs phases. We determine the phase diagram in our
measurement-only model by employing entanglement measures. For the bulk
topological order, we use the topological entanglement entropy. We also use the
mutual information between separated boundary regions to diagnose the boundary
phase transition associated with the Higgs or the bulk SPT phase. We observe
the structural similarity between our phase diagram and the one in the standard
quantum Hamiltonian formulation of the Fradkin-Shenker model with the open
rough boundary. First, a deconfining phase is detected by nonzero and constant
topological entanglement entropy. Second, we find a (boundary) phase transition
curve separating the Higgs=SPT phase from the rest. In certain limits, the
topological phase transitions reside at the critical point of the formation of
giant homological cycles in the bulk 3d spacetime lattice, as well as the bond
percolation threshold of the boundary 2d spacetime lattice when it is
effectively decoupled from the bulk. Additionally, there are analogous
mixed-phase properties at a certain region of the phase diagram, emerging from
how we terminate the measurement-based procedure. Our findings pave an
alternative pathway to study the physics of Higgs=SPT phases on quantum devices
in the near future.
