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 dis-tillation. 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-l implementation in it to capitalize on the small footprint of zero-level distillation. Under conditions of a physical error probability of Pphys = 10<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-4</sup> (10<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup>) and targeting an error rate for the magic state within [5 × 10<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-17</sup>,10<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-11</sup>] ([5 × 10<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-11</sup>,10<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-8</sup>]), (0+1)-level distillation reduces the spatiotemporal overhead by more than 63% (61%) compared to the (15-to-l)x(15-to-l) protocol and more than 43% (44%) compared to the (15-to-l)x(20-to-4) protocol, offering a substantial efficiency gain over the traditional protocols.