Quantum information processing promises to speed up certain computational tasks. The expectations are high, but many technological hurdles have to be overcome before we can build a quantum computer. Nevertheless, already today quantum information allows for novel insights into physics. In this talk, I will summarize our progress towards building an ion trap quantum computer as well as a surprising application of quantum information to fundamental physics. Using the precise experimental control over individual ions, we verify Lorentz invariance at a level of 10-18, improving the limits set by traditional Michelson-Morley experiments by a factor of five.