Sequential quantum circuits as maps between gapped phases

Finite-depth quantum circuits preserve the long-range entanglement structure in quantum states and map between states within a gapped phase. To map between states of different gapped phases, we can use sequential quantum circuits, which apply unitary transformations to local patches, strips, or other subregions of a system in a sequential way. The sequential structure of the circuit, on the one hand, preserves entanglement area law and hence the gappedness of the quantum states. On the other hand, the circuit has generically a linear depth, hence, it is capable of changing the long-range correlation and entanglement of quantum states and the phase they belong to. In this paper, we systematically discuss the definition, basic properties, and prototypical examples of sequential quantum circuits that map product states to Greenberger-Horne-Zeilinger states, symmetry-protected topological states, intrinsic topological states, and fracton states. We discuss the physical interpretation of the power of the circuits through connection to condensation, Kramers-Wannier duality, and the notion of foliation for fracton phases.