Efficiently Computable Bounds for Magic State Distillation

Magic-state distillation (or nonstabilizer state manipulation) is a crucial component in the leading approaches to realizing scalable, fault-tolerant, and universal quantum computation. Related to nonstabilizer state manipulation is the resource theory of nonstabilizer states, for which one of the goals is to characterize and quantify nonstabilizerness of a quantum state. In this Letter, we introduce the family of thauma measures to quantify the amount of nonstabilizerness in a quantum state, and we exploit this family of measures to address several open questions in the resource theory of nonstabilizer states. As a first application, we establish the hypothesis testing thauma as an efficiently computable benchmark for the one-shot distillable nonstabilizerness, which in turn leads to a variety of bounds on the rate at which nonstabilizerness 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 previous 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 nonstabilizerness 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 nonstabilizer states and reveals a difference between the resource theory of nonstabilizer states and other resource theories such as entanglement and coherence.

Figure 1

Comparison between $N_{{\theta }_{\max }}({\rho }_{\mathrm{in}}\to H_{+})$ and $N_{\mathcal{M}}({\rho }_{\mathrm{in}}\to H_{+})$ for input ${\rho }_{\mathrm{in}}=(1-p_1-p_2)|H_{+}⟩⟨H_{+}|+p_1$ $|H_{-}⟩⟨H_{-}|+p_2|H_i⟩⟨H_i|$ with $p_2=1/10$.