Writing the Quantum State Using Polar Coordinates

The quantum state resulting from arbitrary rotations of the pentagon and triangle qubelets can, thus, be written by combining the previous equations for each respective rotation as follows:

In the example, the pentagon qubelets are rotated 20°, so is radians. Likewise, the triangle qubelets are turned 30°, or is radians. Thus, the quantum state is:

Although the quantum state is defined using complex numbers, there’s nothing imaginary about them: quantum computing works in the physical realm, albeit at tiny scales. The complex numbers are a mathematical artifact that give us a way to express 3D rotations of qubits.