Clifford Circuits on Qudits of Mixed Dimension

Associated to every finite abelian group G, there is a generalization of qubit Clifford group associated to G. We demonstrate that every G-Clifford gate can be decomposed into some distinguished elementary types: Fourier transforms, quadratic phase gates, and automorphism gates. Combined with earlier results on classical simulation of normalizer circuits, this implies a generalized Gottesman-Knill-type result. We additionally provide circuits for a universal quantum computing scheme based on local two-qudit G-Clifford gates and magic states.