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We consider dual unitary operators and their multi-leg generalizations thathave appeared at various places in the literature. These objects can be relatedto multi-party quantum states with special entanglement patterns: the sites arearranged in a spatially symmetric pattern and the states have maximalentanglement for all bipartitions that follow from the reflection symmetries ofthe given geometry. We consider those cases where the state itself is invariantwith respect to the geometrical symmetry group. The simplest examples are thosedual unitary operators which are also self dual and reflection invariant, butwe also consider the generalizations in the hexagonal, cubic, and octahedralgeometries. We provide a number of constructions and concrete examples forthese objects for various local dimensions. All of our examples can be used tobuild quantum cellular automata in 1+1 or 2+1 dimensions, with multipleequivalent choices for the ``direction of time''.

Multi-directional unitarity and maximal entanglement in spatially symmetric quantum states