New Developments in Computer Algebra using Techniques from Quantum Computing

我们开发了基于量子计算技术的新方法，以有效地处理具有代数结构的计算问题。在我们的主要成果，我们构建了新的算法区分，在属性测试的框架，类组，和新的量子算法有效地计算两个布尔矩阵的产品。作为这些新方法的应用，我们成功地证明了量子通信中出现的解码问题的困难性，并为诸如隐私保护通信等几项任务构建了新的协议。

We developed new approaches based on techniques from quantum computing to efficiently handle computational problems with an algebraic structure. Among our main results, we constructed new algorithms for distinguishing, in the framework of property testing, classes of groups, and new quantum algorithms for computing efficiently the product of two Boolean matrices. As applications of these novel approaches, we succeeded in proving the hardness of decoding problems arising in quantum communication, and in constructing new protocols for several tasks such as privacy-preserving communication.