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For a quantum computer acting on d-dimensional systems, we analyze thecomputational power of circuits wherein stabilizer operations are perfect andwe allow access to imperfect non-stabilizer states or operations. If the noiserate affecting the non-stabilizer resource is sufficiently high, then thesestates and operations can become simulable in the sense of the Gottesman-Knilltheorem, reducing the overall power of the circuit to no better than classical.In this paper we find the depolarizing noise rate at which this happens, andconsequently the most robust non-stabilizer states and non-Clifford gates. Indoing so, we make use of the discrete Wigner function and derive facets of theso-called qudit Clifford polytope i.e. the inequalities defining the convexhull of all qudit Clifford gates. Our results for robust states are provablyoptimal. For robust gates we find a critical noise rate that, as dimensionincreases, rapidly approaches the the theoretical optimum of 100%. Someconnections with the question of qudit magic state distillation are discussed.

Noise Thresholds for Higher Dimensional Systems using the Discrete Wigner Function