  Measurements allow efficient preparation of interesting quantum many-body
states with long-range entanglement, conditioned on additional transformations
based on measurement outcomes. Here, we demonstrate that the so-called
conformal quantum critical points (CQCP) can be obtained by performing general
single-site measurements in an appropriate basis on the cluster states in
$d\geq2$. The equal-time correlators of the said states are described by
correlation functions of certain $d$-dimensional classical models at finite
temperatures and feature spatial conformal invariance. This establishes an
exact correspondence between the measurement-prepared critical states and
conformal field theories of a range of critical spin models, including familiar
Ising models and gauge theories. Furthermore, by mapping the long-range
entanglement structure of measured quantum states into the correlations of the
corresponding thermal spin model, we rigorously establish the stability
condition of the long-range entanglement in the measurement-prepared quantum
states deviating from the ideal setting. Most importantly, we describe
protocols to decode the resulting quantum phases and transitions without
post-selection, thus transferring the exponential measurement complexity to a
polynomial classical computation. Therefore, our findings suggest a novel
mechanism in which a quantum critical wavefunction emerges, providing new
practical ways to study quantum phases and conformal quantum critical points.
