Multilevel distillation of magic states for quantum computing
Cody Jones · Physical Review A · 2013
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 $\ensuremath{\epsilon}\ensuremath{\rightarrow}0$ for each input magic state, the number of input magic states consumed on average to yield an output state with infidelity $O({\ensuremath{\epsilon}}^{{2}^{r}})$ 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 ${\ensuremath{\epsilon}}_{\mathrm{in}}=0.01$ to ${\ensuremath{\epsilon}}_{\mathrm{out}}$ in the range ${10}^{\ensuremath{-}5}$--${10}^{\ensuremath{-}40}$ is about $14{\mathrm{log}}_{10}(1/{\ensuremath{\epsilon}}_{\mathrm{out}})\ensuremath{-}40$; 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 ${10}^{\ensuremath{-}7}$. 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.