Multilevel distillation of magic states for quantum computing

We develop a procedure for distilling magic states used in universal quantum computing that requires substantially fewer initial resources than prior schemes. Our distillation circuit is based on a family of concatenated quantum codes that possess a transversal Hadamard operation, enabling each of these codes to distill the eigenstate of the Hadamard operator. A crucial result of this design is that low-fidelity magic states can be consumed to purify other high-fidelity magic states to even higher fidelity, which we call "multilevel distillation." When distilling in the asymptotic regime of infidelity $ε \rightarrow 0$ for each input magic state, the number of input magic states consumed on average to yield an output state with infidelity $O(ε^2^r)$ approaches $2^r+1$, which comes close to saturating the conjectured bound in [Phys. Rev. A 86, 052329]. We show numerically that there exist multilevel protocols such that the average number of magic states consumed to distill from error rate $ε_\mathrmin = 0.01$ to $ε_\mathrmout$ in the range $10^-5$ to $10^-40$ is about $14\log_10(1/ε_\mathrmout) - 40$; the efficiency of multilevel distillation dominates all other reported protocols when distilling Hadamard magic states from initial infidelity 0.01 to any final infidelity below $10^-7$. These methods are an important advance for magic-state distillation circuits in high-performance quantum computing, and they provide insight into the limitations of nearly resource-optimal quantum error correction.