Noise Thresholds for Higher Dimensional Systems using the Discrete Wigner Function

TL;DR

This paper determines the noise thresholds at which non-stabilizer states and gates in higher-dimensional quantum systems become classically simulable, using the discrete Wigner function to identify optimal robustness and critical noise rates.

Contribution

It introduces a method to compute noise thresholds for non-stabilizer resources in qudit systems using the discrete Wigner function, deriving optimal robustness bounds.

Findings

Critical noise rate approaches 100% as dimension increases.

Facets of the qudit Clifford polytope are derived.

Robust states are proven to be optimal.

Abstract

For a quantum computer acting on d-dimensional systems, we analyze the computational power of circuits wherein stabilizer operations are perfect and we allow access to imperfect non-stabilizer states or operations. If the noise rate affecting the non-stabilizer resource is sufficiently high, then these states and operations can become simulable in the sense of the Gottesman-Knill theorem, 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, and consequently the most robust non-stabilizer states and non-Clifford gates. In doing so, we make use of the discrete Wigner function and derive facets of the so-called qudit Clifford polytope i.e. the inequalities defining the convex hull of all qudit Clifford gates. Our results for robust states are provably optimal. For robust gates we find a critical noise rate that, as dimension increases, rapidly approaches the the theoretical optimum of 100%. Some connections with the question of qudit magic state distillation are discussed.