Leveraging Zero-Level Distillation to Generate High-Fidelity Magic States

Magic state distillation plays an important role in universal fault-tolerant quantum computing, and its overhead is one of the major obstacles to realizing fault-tolerant quantum computers. Hence, many studies have been conducted to reduce this overhead. Among these, Litinski has provided a concrete assessment of resource-efficient distillation protocol implementations on the rotated surface code. On the other hand, recently, Itogawa et al. have proposed zero-level distillation, a distillation protocol offering very small spatial and temporal overhead to generate relatively low-fidelity magic states. While zero-level distillation offers preferable spatial and temporal overhead, it cannot directly generate high-fidelity magic states since it only reduces the logical error rate of the magic state quadratically. In this study, we evaluate the spatial and temporal overhead of two-level distillation implementations generating relatively high-fidelity magic states, including ones incorporating zero-level distillation. To this end, we introduce (0+1) -level distillation, a two-level distillation protocol which combines zero-level distillation and the 15-to-1 distillation protocol. We refine the second-level 15-to-1 implementation in it to capitalize on the small footprint of zero-level distillation. Under conditions of a physical error probability of p₏₇ₘₒ = 10^-4 (10^-3) and targeting an error rate for the magic state within 5 10^-17, 10^-11 (5 10^-11, 10^-8), (0+1) -level distillation reduces the spatiotemporal overhead by more than 63% (61%) compared to the (15-to-1) (15-to-1) protocol and more than 43% (44%) compared to the (15-to-1) (20-to-4) protocol, offering a substantial efficiency gain over the traditional protocols.