  In this paper, we construct 2-dimensional bipartite cluster states and
perform single-qubit measurements on the bulk qubits. We explore the
entanglement scaling of the unmeasured 1-dimensional boundary state and show
that under certain conditions, the boundary state can undergo a volume-law to
an area-law entanglement transition driven by variations in the measurement
angle. We bridge this boundary state entanglement transition and the
measurement-induced phase transition in the non-unitary 1+1-dimensional circuit
via the transfer matrix method. We also explore the application of this
entanglement transition on the computational complexity problems. Specifically,
we establish a relation between the boundary state entanglement transition and
the sampling complexity of the bipartite $2$d cluster state, which is directly
related to the computational complexity of the corresponding Ising partition
function with complex parameters. By examining the boundary state entanglement
scaling, we numerically identify the parameter regime for which the $2$d
quantum state can be efficiently sampled, which indicates that the Ising
partition function can be evaluated efficiently in such a region.
