Universal magic state concentration

Abstract

Magic plays a dual role in quantum computation: it promotes stabilizer dynamics from efficient classical simulability to computational universality, but also challenges fault-tolerant architectures, since non-stabilizer operations are harder to protect against noise. Magic state distillation addresses this issue, yet existing protocols remain largely tailored to specific assumptions about the input or noise model and, more fundamentally, no general information-theoretic theory of optimal magic-state conversion currently exists. Here we develop such a characterization for pure qubit states through universal magic state concentration: a fixed stabilizer protocol that converts a few copies of an unknown pure non-stabilizer qubit state into an exact target magic state. Motivated by the impossibility of exact $T$-state concentration, we build six- and eight-copy protocols producing an exact $\mathrm{CCZ}$ state, with six copies being minimal. Their success probabilities are governed by the linearized order-three stabilizer R\'enyi entropy $M^{\mathrm{lin}}_3$. We show that this connection is structural: for up to nine input copies, $M^{\mathrm{lin}}_3$ fully determines the success probability of every universal Clifford-invariant stabilizer protocol. Remarkably, this characterization persists asymptotically: our constructions achieve optimal rates among universal single-output protocols and remain optimal up to logarithmic factors among arbitrary stabilizer protocols. As a corollary, we show that any unknown pure non-stabilizer state suffices for universal quantum computation via probabilistic $\mathrm{CCZ}$-state injection. Together, these results identify the stabilizer R\'enyi entropy as a fundamental operational quantity in magic state distillation.