Multilevel distillation of magic states for quantum computing

Abstract

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 $\epsilon \rightarrow 0$ for each input magic state, the number of input magic states consumed on average to yield an output state with infidelity $O(\epsilon^{2^r})$ 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 $\epsilon_{\mathrm{in}} = 0.01$ 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, and they provide insight into the limitations of nearly resource-optimal quantum error correction.Comment: 10 pages, 4 figure

Composite Toffoli gate with two-round error detection

“…Still, the distinction is important because some operations are hard, meaning they require substantially more overhead to perform. Often, the hard operations invoke many easy operations to perform some distillation procedure [22][23][24][25][26].…”

Trading Classical and Quantum Computational Resources

“…Furthermore, topological quantum codes, such as the surface code, enable a direct measurement of certain logical Pauli operators by measuring a properly chosen subset of physical qubits [12]. Several fault-tolerant protocols for preparing encoded magic states such as jHi have been developed [13][14][15][16][17]. PBCs implicitly appeared in the previous work on quantum fault tolerance.…”

Dual-code quantum computation model

“…Therefore, resource efficient magic state injection and distillation methods are essential. More efficient magic state distillation protocols have recently been proposed [8,27]. Code teleportation scheme [35,37,47].…”

“…As a result, we focus on the resource reduction of such gates on a specific code. Several cost-efficient magic state methods have been proposed by using better codes [8] or by using the concatenation of magic states [27].…”