加权Korobov空间逼近在经典和量子计算机上的可处理性
Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers
摘要：
本文研究了高维多元近似的量子复杂性、易处理性和强易处理性。我们研究了在许多应用中重要的函数空间。如果某些变量比其他变量更重要，则函数空间被加权;权重显示变量的相对重要性。在一个未加权的空间中，所有的变量都是同等重要的，多元近似是难以处理的。我们想研究多元逼近的复杂性何时与变量的个数无关并多项式依赖于1/E。主要结论如下：量子计算机上的多变量近似可以比使用随机化的经典计算机快大约（1/E）^（1+r）倍。这里，r是取决于权重的正参数，并且可以很大。这意味着量子计算机相对于经典计算机的加速可能比二次的要大得多。量子计算机上的多变量近似比经典计算机上的多变量近似快，即使权重的总和是无限的，但它们的某个幂是有限的，也有最坏情况的保证。我们设计了一个误差最大为E的量子算法，它使用了大约d+log（1/E）个量子比特。因此，我们只对维数d有线性依赖性，对1/E有对数依赖性。因此，对于某些应用，量子位的数量相当适中。
英文摘要：
The paper studies quantum complexity, tractability, and strong tractability for high dimensional multivariate approximation. We study a space of functions important in many applications. A function space is weighted if certain variables are more important than others; the weights show the relative importance of the variables. In an unweighted space all variables are equally important and multivariate approximation is intractable. We want to study when the complexity of multivariate approximation is independent of the number of variables and depends polynomially on 1/E. The main conclusions are: Multivariate approximation on a quantum computer can be solved roughly (1/E)^(1+r) times faster than on a classical computer using randomization. Here, r is a positive parameter that depends on the weights and may be large. This means that the speed-up of quantum over classical computers may be much larger than quadratic. Multivariate approximation on a quantum computer is exponentially faster than on a classical computer with a worst case assurance even if the sum of weights is infinite but a certain power of them is finite. We have designed a quantum algorithm with error at most E that uses about d+log(1/E) qubits. Hence, we have only linear dependence on the dimension d and logarithmic dependence on 1/E. Therefore, for some applications the number of qubits is quite modest.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
