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 0 for each input magic state, the number of input magic states consumed on average to yield an output state with infidelity O (^{2^r}) approaches 2^r+1, which comes close to saturating the conjectured bound in another investigation Bravyi and Haah, Phys. Rev. A 86, 052329 (2012). We show numerically that there exist multilevel protocols such that the average number of magic states consumed to distill from error rate ₈₍=0. 01 to ₎ₔₓ in the range 10^-5--10^-40 is about 14log₁₀ (1/₎ₔₓ) -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 provide insight into the limitations of nearly resource-optimal quantum error correction.