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Paper 1
Distillation with sublogarithmic overhead
M. B. Hastings, J. Haah
- Year
- 2017
- Journal
- arXiv preprint
- DOI
- arXiv:1709.03543
- arXiv
- 1709.03543
It has been conjectured [1] that for any distillation protocol for magic states for the $T$ gate, the number of noisy input magic states required per output magic state at output error rate $ε$ is $Ω(\log(1/ε))$. We show that this conjecture is false. We find a family of quantum error correcting codes of parameters $[[\sum_{i=w+1}^m \binom{m}{i}, \sum_{i=0}^{w} \binom{m}{i}, \sum_{i=w+1}^{r+1} \binom{r+1}{i}]]$ for any integers $ m > 2r$, $r > w \ge 0$, by puncturing quantum Reed-Muller codes. When $m > νr$, our code admits a transversal logical gate at the $ν$-th level of Clifford hierarchy. In a distillation protocol for magic states at the level $ν= 3$ ($T$-gate), the ratio of input to output magic states is $O(\log^γ(1/ε))$ where $γ= \log(n/k)/\log(d)< 0.678$ for some $m,r,w$. The smallest code in our family for which $γ< 1$ is on $\approx 2^{58}$ qubits.
Open paperPaper 2
Efficient magic state cultivation with lattice surgery
Yutaka Hirano, Riki Toshio, Tomohiro Itogawa, Keisuke Fujii
- Year
- 2025
- Journal
- arXiv preprint
- DOI
- arXiv:2510.24615
- arXiv
- 2510.24615
Magic state distillation plays a crucial role in fault-tolerant quantum computation and represents a major bottleneck. In contrast to traditional logical-level distillation, physical-level distillation offers significant overhead reduction by enabling direct implementation with physical gates. Magic state cultivation is a state-of-the-art physical-level distillation protocol that is compatible with the square-grid connectivity and yields high-fidelity magic states. However, it relies on the complex grafted code, which incurs substantial spacetime overhead and complicates practical implementation. In this work, we propose an efficient cultivation-based protocol compatible with the square-grid connectivity. We reduce the spatial overhead by avoiding the grafted code and further reduce the average spacetime overhead by utilizing code expansion and enabling early rejection. Numerical simulations show that, with a color code distance of 3 and a physical error probability of $10^{-3}$, our protocol achieves a logical error probability for the resulting magic state comparable to that of magic state cultivation ($\approx 3 \times 10^{-6}$), while requiring about half the spacetime overhead. Our work provides an efficient and simple distillation protocol suitable for megaquop use cases and early fault-tolerant devices.
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