Universal magic state concentration

Abstract

Magic plays a dual role in quantum computation: it promotes stabilizer dynamics from efficient classical simulability to universality, but it presents a central challenge for fault tolerance, since non-stabilizer operations are harder to protect against noise. Magic state distillation addresses this issue; however, existing protocols typically assume prior structure in the input, such as proximity to the target or a specified noise model. Here we introduce 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 obstruction to exact $T$-state concentration, we show that $\mathrm{CCZ}$ states behave fundamentally differently. Six input copies are necessary and sufficient to distill one exact $\mathrm{CCZ}$ state, with an optimal success probability determined by the linearized order-three stabilizer Rényi entropy $M^{\mathrm{lin}}_3$. Beyond this, we show that $M^{\mathrm{lin}}_3$ governs the optimal state dependence of any protocol up to nine input copies, and we showcase an eight-copy protocol with improved success probability. Furthermore, block repetition of our protocols yields asymptotic distillation rates that achieve optimal scaling up to logarithmic factors. As a corollary, any unknown pure qubit magic state suffices for universal quantum computation via exact $\mathrm{CCZ}$ injection. Together, these results identify the stabilizer Rényi entropy as a fundamental operational quantity in magic state distillation.