Title: Efficient magic state factories with a catalyzed |CCZ> to 2|T> transformation Show Chinese title

Title: 高效的魔术态工厂与催化 |CCZ> 到 2|T> 的转换

Abstract: We present magic state factory constructions for producing $|CCZ\rangle$ states and $|T\rangle$ states. For the $|CCZ\rangle$ factory we apply the surface code lattice surgery construction techniques described by Fowler et al. to the fault-tolerant Toffoli. The resulting factory has a footprint of $12d \times 6d$ (where $d$ is the code distance) and produces one $|CCZ\rangle$ every $5.5d$ surface code cycles. Our $|T\rangle$ state factory uses the $|CCZ\rangle$ factory's output and a catalyst $|T\rangle$ state to exactly transform one $|CCZ\rangle$ state into two $|T\rangle$ states. It has a footprint 25% smaller than the factory of Fowler et al. but outputs $|T\rangle$ states twice as quickly. We show how to generalize the catalyzed transformation to arbitrary phase angles, and note that the case $\theta=22.5^\circ$ produces a particularly efficient circuit for producing $|\sqrt{T}\rangle$ states. Compared to using the $12d \times 8d \times 6.5d$ $|T\rangle$ factory of Fowler et al., our $|CCZ\rangle$ factory can quintuple the speed of algorithms that are dominated by the cost of applying Toffoli gates, including Shor's algorithm and the chemistry algorithm of Babbush et al.. Assuming a physical gate error rate of $10^{-3}$, our CCZ factory can produce $\sim 10^{10}$ states on average before an error occurs. This is sufficient for classically intractable instantiations of the chemistry algorithm, but for more demanding algorithms such as Shor's algorithm the mean number of states until failure can be increased to $\sim 10^{12}$ by increasing the factory footprint ~20%. Show Chinese abstract

Abstract: 我们提出了用于生成$|CCZ\rangle$状态和$|T\rangle$状态的魔术态工厂构造。 对于$|CCZ\rangle$工厂，我们将 Fowler 等人描述的表面码晶格手术构造技术应用于容错 Toffoli。 该工厂的占地面积为$12d \times 6d$（其中$d$是码距），并且每$5.5d$个表面码周期产生一个$|CCZ\rangle$。 我们的$|T\rangle$状态工厂使用$|CCZ\rangle$工厂的输出和一个催化剂$|T\rangle$状态，将一个$|CCZ\rangle$状态精确地转换为两个$|T\rangle$状态。 它的占地面积比 Fowler 等人的工厂小 25%，但输出$|T\rangle$状态的速度快两倍。 我们展示如何将催化转换推广到任意相位角，并注意到情况$\theta=22.5^\circ$会产生一个特别高效的电路，用于生成$|\sqrt{T}\rangle$状态。 与使用 Fowler 等人的$12d \times 8d \times 6.5d$ $|T\rangle$ 工厂相比，我们的$|CCZ\rangle$工厂可以使以应用 Toffoli 门成本为主的算法速度提高五倍，包括 Shor 算法和 Babbush 等人的化学算法。 假设物理门错误率为$10^{-3}$，我们的 CCZ 工厂在出现错误前平均可以生成$\sim 10^{10}$状态。 这足以应对化学算法的经典不可行实例，但对于像Shor算法这样更复杂的算法，通过将工厂占地面积增加约20%，可以将失败前的平均状态数提高到$\sim 10^{12}$。