Efficiently computable bounds for magic state distillation

Magic-state distillation (or non-stabilizer state manipulation) is a crucial component in the leading approaches to realizing scalable, fault-tolerant, and universal quantum computation. Related to non-stabilizer state manipulation is the resource theory of non-stabilizer states, for which one of the goals is to characterize and quantify non-stabilizerness of a quantum state. In this paper, we introduce the family of thauma measures to quantify the amount of non-stabilizerness in a quantum state, and we exploit this family of measures to address several open questions in the resource theory of non-stabilizer states. As a first application, we establish the hypothesis testing thauma as an efficiently computable benchmark for the one-shot distillable non-stabilizerness, which in turn leads to a variety of bounds on the rate at which non-stabilizerness can be distilled, as well as on the overhead of magic-state distillation. We then prove that the max-thauma can be used as an efficiently computable tool in benchmarking the efficiency of magic-state distillation and that it can outperform pervious approaches based on mana. Finally, we use the min-thauma to bound a quantity known in the literature as the regularized relative entropy of magic. As a consequence of this bound, we find that two classes of states with maximal mana, a previously established non-stabilizerness measure, cannot be interconverted in the asymptotic regime at a rate equal to one. This result resolves a basic question in the resource theory of non-stabilizer states and reveals a difference between the resource theory of non-stabilizer states and other resource theories such as entanglement and coherence.