确定性、随机化和量子算法的最优求和与积分
Optimal Summation and Integration by Deterministic, Randomized, and Quantum Algorithms
AI总结：
本文综述了有限序列求和与Hölder或Sobolev空间函数积分的最优算法，并引入量子计算新视角，比较了确定性、随机化和量子三种设置下的算法性能。
摘要：
我们综述了关于有限序列求和以及来自Hölder或Sobolev空间函数积分的最优算法的旧结果和新结果。首先讨论最优确定性和随机化算法。然后我们添加一个新的方面，这在之前的（拟）蒙特卡洛方法会议上尚未涉及：量子计算。我们简要介绍这一设置，并展示作者在求和与积分的最优量子算法方面的最新结果。我们讨论三种设置之间的比较。对于蒙特卡洛和量子积分，最有趣的情况是中等光滑度k和大维度d，这实际上出现在许多重要的应用问题中。在这种情况下，确定性指数可以忽略，因此n^{-1/2}的蒙特卡洛和n^{-1}的量子加速基本上构成了整个收敛速度。我们观察到：——如果k/d趋于零，则量子算法相对于确定性（经典）算法存在指数加速；——如果k/d较小，则量子算法相对于随机化经典算法存在（大致）二次加速。
英文摘要：
We survey old and new results about optimal algorithms for summation of finite sequences and for integration of functions from Hoelder or Sobolev spaces. First we discuss optimal deterministic and randomized algorithms. Then we add a new aspect, which has not been covered before on conferences about (quasi-) Monte Carlo methods: quantum computation. We give a short introduction into this setting and present recent results of the authors on optimal quantum algorithms for summation and integration. We discuss comparisons between the three settings. The most interesting case for Monte Carlo and quantum integration is that of moderate smoothness k and large dimension d which, in fact, occurs in a number of important applied problems. In that case the deterministic exponent is negligible, so the n^{-1/2} Monte Carlo and the n^{-1} quantum speedup essentially constitute the entire convergence rate. We observe that -- there is an exponential speed-up of quantum algorithms over deterministic (classical) algorithms, if k/d tends to zero; -- there is a (roughly) quadratic speed-up of quantum algorithms over randomized classical algorithms, if k/d is small.
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