Symmetry-protected topologically ordered phases for measurement-based quantum computation

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Measurement-based quantum computation (MBQC) is a computational scheme to simulate spacetime dynamics on the network of entanglement using local measurements and classical communication. The pursuit of a broad class of useful entanglement encountered a concept of symmetry-protected topologically ordered (SPTO) phases in condensed matter physics. A natural question is "What kinds of SPTO ground states can be used for universal MBQC in a similar fashion to the 2D cluster state?" 2D SPTO states are classified not only by global on-site symmetries but also by subsystem symmetries, which are fine-grained symmetries dependent on the lattice geometry. Recently, all ground states within SPTO cluster phases on the square and hexagonal lattices have been shown to be universal, based on the presence of subsystem symmetries and associated structures of quantum cellular automata. Motivated by this observation, we analyze the computational capability of SPTO cluster phases on all vertex-translative 2D Archimedean lattices. We show that there are four different "fundamental" subsystem symmetries, called here ribbon, cone, fractal, and 1-form symmetries, for cluster phases, and the former three types one-to-one correspond to three classes of Clifford quantum cellular automata. We conclude that nine out of the eleven Archimedean lattices support universal cluster phases protected by one of the former three symmetries, while the remaining lattices with the 1-form symmetry have a different capability related to error correction.