ALT Schematic showing the algorithm for constructing gapped boundaries of a topological generalized Pauli stabilizer code. The diagram flows from top left to top right, then bottom left to bottom right. It begins with a grey plane with repeated purple code on it. It is truncated–depicted by cutting the plane and code with scissors. Numerous operations follow, depicted by a series of arrows and images of computers, introducing boundary gauge operators (purple and green symbols) and then the identification of boundary string operators and boundary anyons (a string of orange lines marked X, bounded by the letters a and a-1 and a red line with multiple z’s marked along it, bounded by the letters b and b-1). The bottom row shows topological spin and braiding statistics (rearranging the red and orange lines), then boundary anyon condensation to create short boundary string operators and topological order completion to construct the gapped boundary Hamiltonian, with a spark of energy violation.

ALT Colorful circular pattern with the text "APS Open Science APCs Waived" in the center.

ALT Curved golden light trails against a dark background, creating a dynamic, flowing effect resembling motion or speed.