Summary: Complexifying Quantum States

Here are the key takeaways:

• To go further in QC with qubit systems that have more degrees of freedom, complex numbers must be used to satisfy the requirements of orthogonality and normalization within a change of basis. They also allow for mining information about the relative phases of quantum states.

• The complex exponential function is a fundamental building block for many quantum operators.

• The quantum Fourier transform makes use of complex exponentials to find the period of periodic functions.