  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.
