We show that when the gravitational field is treated quantum-mechanically, it induces fluctuations - noise - in the lengths of the arms of gravitational wave detectors. The characteristics of the noise depend on the quantum state of the gravitational field and can be calculated exactly in several interesting cases. For coherent states, the noise is very small, but it can be greatly enhanced in thermal and (especially) squeezed states. Detection of this fundamental noise would constitute direct evidence for the quantization of gravity and the existence of gravitons.