Multilevel distillation of magic states for quantum computing

We develop a procedure for distilling magic states used in universal quantum computing which requires substantially fewer resources than prior schemes. Our distillation circuit is based on a family of concatenated quantum codes with a transversal Hadamard operation which can distill the eigenstate of the Hadamard operator. A crucial result of this design is that low-fidelity magic states can be consumed to purify high-fidelity magic states to even higher fidelity, which we call ``multilevel distillation.'' We show numerically that there exist multilevel protocols such that the average number of magic states consumed to distill from error rate $\epsilon_{\mathrm{in}} = 0.01$ to $\epsilon_{\mathrm{out}}$ in the range $10^{-5}$ to $10^{-40}$ is about $14\log_{10}(1/\epsilon_{\mathrm{out}}) - 40$; 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.