Quantum Gate Geometry

A generalized visualization of applying single-qubit quantum gates. We begin with our initial pair of amplitudes, a₀ and a₁. When applying a quantum gate, each amplitude is split into two components. We can view these two components as the sides to a triangle, with the hypotenuse being our original amplitude. The components are vectors whose sum is, again, the original amplitude. The angle between the components is determined by the gate. Importantly, the sum of the angles between the pairs of broken components is 180°. As shown in the image, we recombine the components to create two new amplitudes (labelled b₀ and b₁). Each pair uses complimentary probability terms as color coded in the previous figure.