Invited Talks 17

One of the most challenging questions of today, in the overlap of computer science, mathematics, and physics, is the exploration of potential capabilities of quantum computers. Milestones which intensified and enlarged research considerably were the algorithm of Shor [1], who showed that quantum computers could factor large integers efficiently (which is widely believed to be infeasible on classical computers) and the quantum search algorithm of Grover [2], which provides a quadratic speedup over deterministic and randomized classical algorithms of searching a database. So far research was mainly concentrated on discrete problems like the above and many others one encounters in computer science. Much less is known about computational problems of analysis, including such typical field of application of Monte Carlo methods as high dimensional integration. We seek to understand how well these problems can be solved in the quantum model of computation (that is, on a – hypothetical – quantum computer) and how the outcome compares to the efficiency of deterministic or Monte Carlo algorithms on a classical (i. e. non-quantum) computer. First steps were made by Novak [3], who considered integration of functions from Hölder spaces. This line of research was continued by the author [4], where quantum algorithms for the integration of Lp-functions and, as a key prerequisite, for the computation of the mean of p-summable sequences were constructed. In [4] a rigorous model of quantum computation for numerical problems was developed, as well. The case of integration of functions from Sobolev spaces is considered in [5], and more on the computation of the mean will be presented in [6]. These papers also established matching lower bounds. A short survey of first results can be found in [7]. Combining these results with previous ones of information-based complexity theory about the best possible ways of solving the respective problems deterministically or by Monte Carlo on classical computers, we are now in a position to fairly well answer the question where quantum computation can provide a speedup in high dimensional integration and where not. We know cases among the above where quantum algorithms yield an exponential speedup over deterministic algorithms and a quadratic speedup over randomized ones (on classical computers). In the present talk I will give an overview about the state of the art in this field.