As quantum technology advances, mid-circuit measurement capabilities have emerged across various platforms, essential for the long-term goal of fault-tolerant quantum computation. Inspired by this background and the work [1], we have proposed and studied a concept we term measurement-based quantum simulation (MBQS), where dynamical quantum simulation of lattice gauge theory is driven by single-qubit measurements on a family of symmetry-protected topologically ordered states (SPT states). We provide a versatile framework for constructing and analyzing bulk SPT lattice models starting from general boundary Calderbank-Shor-Steane (CSS) codes — a natural generalization of lattice gauge theories — based on tensor products of chain complexes. Motivated by the MBQS construction, we study in detail a wave function overlap between the SPT state and the product state that is related to the projective measurements. The overlap equals statistical partition functions of various spin models related to CSS codes, much in the spirit of strange correlators [2], with examples including those for gauge theories and the Xu-Moore model. We introduce a ‘sibling SPT state’ whose overlap construction leads to a Kramers-Wannier-Wegner dual of the statistical partition function. Finally, we demonstrate that a related overlap construction gives a spacetime blueprint for a quantum circuit that consists solely of randomized non-commuting many-body projective measurements (i.e., a measurement-only circuit), which also shows that entanglement transitions in the circuit can be viewed as percolation problems. Our numerical results match expected percolation theories in key limits. This talk is a compilation of results in Refs. [3, 4, 5].