"Quantum computing combines computer science, physics and mathematics to fundamentally speed up computation using effects from quantum physics. Starting in the early 1980s with Feynman and Deutsch, and gaining momentum in the 1990s with the algorithms of Shor and Grover, this very interdisciplinary area has potentially far reaching consequences. While a large-scale quantum computer has not been built yet, experimenters are getting more optimistic: a recent prediction is that it will take another 10-15 years.However, the tasks where such a quantum computer would be able to significantly outperform classical computers are still quite limited, which lends urgency to finding new applications. This proposal will find more such tasks, and produce new insights into the strengths and weaknesses of quantum computing. It is divided into three workpackages:1. Algorithms & complexity. Find new quantum algorithms that are more efficient than the best classical algorithms, for example for matrix multiplication and graph problems. Extend our knowledge of the ultimate limitations of quantum algorithms, and possible parallelization (which has barely been studied so far).2. Quantum communication. Communication complexity analyzes the amount of communication needed to solve distributed computational tasks, where separate parties each hold part of the input. Find newdistributed problems where quantum communication outperforms classical communication, and explore links with fundamental physics issues like the role of entanglement and Bell-inequality violations.3. Classical applications. Apply the newly developed mathematical tools of quantum computing to analyze problems in other areas, as we recently did for linear programs for the traveling salesman problem. Thisthird workpackage will have impact regardless of progress in building a quantum computer.The PI is one of the world’s top researchers in each of these three areas."