Quantum Fourier Transform

The complex exponential function shows up in the quantum Fourier transform method we introduced earlier for the Shor factoring algorithm. We had used the operator $U_F (j, k)$ as a cosine function, but in actual practice it is simpler and more general to use the complex exponential form

$$U_F (j, k) = e^{i2πj k/N}.$$

The complex exponential incorporates both sine and cosine contributions and, therefore, automatically takes care of time shifts (phase shifts) between the signal being analyzed and the sines and cosines taken individually. The complex exponential is also used in operators that express how quantum states change in time. But we won’t pursue those ideas in this course.