  The Quantum Fourier Transformation ($QFT$) is a key building block for a
whole wealth of quantum algorithms.
  Despite its proven efficiency, only a few proof-of-principle demonstrations
have been reported.
  Here we utilize $QFT$ to enhance the performance of a quantum sensor.
  We implement the $QFT$ algorithm in a hybrid quantum register consisting of a
nitrogen-vacancy (NV) center electron spin and three nuclear spins.
  The $QFT$ runs on the nuclear spins and serves to process the sensor - NV
electron spin signal.
  We demonstrate $QFT$ for quantum (spins) and classical signals (radio
frequency (RF) ) with near Heisenberg limited precision scaling.
  We further show the application of $QFT$ for demultiplexing the nuclear
magnetic resonance (NMR) signal of two distinct target nuclear spins.
  Our results mark the application of a complex quantum algorithm in sensing
which is of particular interest for high dynamic range quantum sensing and
nanoscale NMR spectroscopy experiments.
