The preparation of long-range entangled states using unitary circuits is limited by Lieb-Robinson bounds, but circuits with projective measurements and feedback ("adaptive circuits") can evade such restrictions. We introduce four classes of local adaptive circuits that enable low-depth preparation of long-range entangled quantum matter characterized by gapped topological orders and conformal field theories (CFTs). The four classes are inspired by distinct physical insights, including tensornetwork constructions, short-range entangled states with non-trivial braiding structures, multiscale entanglement renormalization ansatz (MERA), and parton constructions. A large class of topological orders, including chiral and symmetry-enriched topological orders, can be prepared in constant depth or time, and one-dimensional CFT states and non-abelian topological orders with both solvable and non-solvable groups can be prepared in depth scaling logarithmically with system size. Our work illustrates the practical and conceptual versatility of measurement for state preparation.

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