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Measurement-based quantum computation describes a scheme where entanglementof resource states is utilized to simulate arbitrary quantum gates via localmeasurements. Recent works suggest that symmetry-protected topologicallynon-trivial, short-ranged entanged states are promising candidates for such aresource. Miller and Miyake [NPJ Quantum Information 2, 16036 (2016)] recentlyconstructed a particular $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$symmetry-protected topological state on the Union-Jack lattice and establishedits quantum computational universality. However, they suggested that the sameconstruction on the triangular lattice might not lead to a universal resource.Instead of qubits, we generalize the construction to qudits and show that theresulting $(d-1)$ qudit nontrivial $\mathbb{Z}_d \times \mathbb{Z}_d \times\mathbb{Z}_d$ symmetry-protected topological states are universal on thetriangular lattice, for $d$ being a prime number greater than 2. The sameconstruction also holds for other 3-colorable lattices, including theUnion-Jack lattice.

Universal quantum computing using $(\mathbb{Z}_d)^3$ symmetry-protected topologically ordered states