  By imaginary-time evolution with Hamiltonian, an arbitrary state arrives in
the system's ground state. In this work, we conjecture that this dynamics can
be simulated by measurement-only circuit (MoC), where each projective
measurement is set in a suitable way. Based on terms in the Hamiltonian and
ratios of their parameters (coefficients), we propose a guiding principle for
the choice of the measured operators called stabilizers and also the
probability of projective measurement in the MoC. In order to examine and
verify this conjecture of the parameter ratio and probability ratio
correspondence in a practical way, we study a generalized (1+1)-dimensional
$Z_2$ lattice gauge-Higgs model, whose phase diagram is very rich including
symmetry-protected topological phase, deconfinement phase, etc. We find that
the MoC constructed by the guiding principle reproduces phase diagram very
similar to that of the ground state of the gauge-Higgs Hamiltonian. The present
work indicates that the MoC can be broadly used to produce interesting phases
of matter, which are difficult to be simulated by ordinary Hamiltonian systems
composed of stabilizer-type terms.
