Assessing the impact of glaciation on Earth's surface requires understanding glacial erosion processes. Developing erosion theories is challenging because of the complex nature of the erosion processes and the difficulty of examining the ice/bedrock interface of contemporary glaciers. We demonstrate that the glacial erosion rate is proportional to the ice-sliding velocity squared, by quantifying spatial variations in ice-sliding velocity and the erosion rate of a fast-flowing Alpine glacier. The nonlinear behavior implies a high erosion sensitivity to small variations in topographic slope and precipitation. A nonlinear rate law suggests that abrasion may dominate over other erosion processes in fast-flowing glaciers. It may also explain the wide range of observed glacial erosion rates and, in part, the impact of glaciation on mountainous landscapes during the past few million years.