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Noise Thresholds for Higher Dimensional Systems using the Discrete Wigner Function

Wim van Dam, University of California, Santa Barbara

(Session 12 : Sunday from 11:30am-12:00pm)

Abstract. This work analyzes the non-stabilizer states and non-stabilizer operations that must be present in any quantum circuit if it is to perform better-than-classical quantum computation. That such a non-stabilizer resource is necessary for universal quantum computation is a consequence of the Gottesman-Knill theorem. In particular, we find states and operations that are maximally non-stabilizer in the sense that they require the highest amount of depolarizing noise to make them become stabilizer states and operations respectively. In doing so we make novel use of a theoretical construction known as the discrete Wigner function (DWF). We find d-level (qudit) states whose negativity (in terms of the DWF quasiprobabilites) is maximal, answering a conjecture of Wootters. We find non-Clifford gates, acting on d-level systems, which require very high amounts (rapidly approaching 100 percent as dimension, d, increases) of depolarizing noise to become decomposable in terms of Clifford gates. In previous literature the convex hull of Clifford gates was called the Clifford polytope. This work describes the qudit version of the Clifford polytope, and bounding inequalities that describe this object are derived using a simple argument. Our results have implications for the question of qudit magic state distillation.