Implementation of phase estimation and quantum counting algorithms on an NMR quantum-information processor

The quantum Fourier transform is a key factor in achieving exponential speedup relative to classical algorithms. We implemented the phase estimation algorithm, which is the very basic application example of the quantum Fourier transform, on a three-bit nuclear magnetic resonance quantum-information processor. The algorithm was applied to one-bit Grover operators to estimate eigenvalues. We also demonstrated the counting algorithm with the same operators to obtain the number of the marked states. The results of projective measurements required by the algorithms were obtained from the split peak positions in the spectra.