We improve the Solovay-Kitaev theorem and algorithm for a general finite, inverse-closed generating set acting on a qudit. Prior versions of the algorithm can efficiently find a word of length $O((\log 1/\epsilon)^{3+\delta})$ to approximate an arbitrary target gate to within $\epsilon$. Using two new ideas, each of which reduces the exponent separately, our new bound on the world length is $O((\log 1/\epsilon)^{1.44042\ldots+\delta})$. Our result holds more generally for any finite set that densely generates any connected, semisimple real Lie group, with an extra length term in the non-compact case to reach group elements far away from the identity.

翻译：我们改进了作用于qudit的一般有限逆封闭生成集的Solovay-Kitaev定理及算法。该算法先前的版本能够高效地找到一个长度为$O((\log 1/\epsilon)^{3+\delta})$的单词，以在$\epsilon$精度内近似任意目标门。通过引入两种新思路（各自独立降低指数），我们得到的新单词长度上界为$O((\log 1/\epsilon)^{1.44042\ldots+\delta})$。该结果更广泛地适用于任何在连通半单实李群中稠密生成的有限集合，对于非紧致情形，需额外引入长度项以逼近远离恒等元的群元素。