AI 中文总结
研究可变维度下索洛维-基塔耶夫算法，通过引入良好指数基避免搜索指数规模预计算网络。对含特定基的指令集，改进算法在d和1/$\epsilon$上复杂度更优；对任意集，隔离d^2指数依赖为预处理成本，还基于微分几何提出扩展。
AI 中文摘要
索洛维-基塔耶夫算法描述了如何使用任何固定通用门集将特殊酉群SU(d)中的矩阵近似到任意精度。尽管该算法的规模为O(poly(log(1/$\epsilon$)))，但其运行时间对量子比特维度d呈指数依赖。我们研究了可变维度情况下的算法问题，展示了如何避免为每个目标酉矩阵搜索指数规模的预计算网络。引入了良好指数基的概念，可取代常规的深度零网络搜索例程。对于已包含或能高效构建良好指数基的指令集，改进后的在线合成算法在d上是多项式的，在1/$\epsilon$上是多项对数的。对于任意通用指令集，虽未消除对d^2的指数依赖，但将其隔离为一次性加法预处理成本。我们的技术利用微分几何方法设计了一种整数化的 Trotterization版本，用建设性的局部合成例程取代深度零网络查询。相同框架还基于其他离散数值积分方案提出了可能的扩展。
英文摘要
The Solovay-Kitaev algorithm describes how to approximate, to arbitrary precision, a matrix in the special unitary group SU(d) using any fixed universal gate set. Although the algorithm scales as O(poly(log(1/$ε$))), where $ε$ is the maximum targeted approximation error, its running time depends exponentially on the qudit dimension d. This bad dependence can be traced to its explicit use of an $ε$_0-net of size 2 $Ω$(d^2) , which is queried O(poly(log(1/$ε$))) times throughout the execution. For this reason, the standard Solovay-Kitaev theorem is usually stated for fixed d, with the base net and its lookup cost absorbed into the constants. We study the algorithmic problem in the variabledimension regime and show how to avoid searching an exponentially large precomputed net for each target unitary. In particular, we introduce the notion of a good exponential basis and show that such a basis can replace the usual depth-zero net-search routine. This yields a modification of the algorithm in which the use of an explicit net is fully moved to a preprocessing step. For instruction sets that already contain, or allow the efficient construction of, a good exponential basis, the resulting online synthesis algorithm is polynomial in d and polylogarithmic in 1/$ε$. For arbitrary universal instruction sets, the exponential dependence on d^2 is not removed, but is isolated into a one-time additive preprocessing cost. Our technique uses differential-geometric methods to devise an integerized version of trotterization that replaces the depth-zero net query by a constructive local synthesis routine. The same framework also suggests possible extensions based on other discretized numerical integration schemes.
