The Bloch Sphere

The numbers $\theta$ and $\varphi$ in equation (2) define a point on the unit three-dimensional sphere. This sphere is called the Bloch Sphere. It provides a useful means of visualizing the state of a single qubit. However, there is no simple generalization of the Bloch sphere known for multiple qubits.

The north pole of the sphere is assigned the state $\vert\rangle$ and the south pole the state $\vert 1\rangle$. A classical bit would be represented on the bloch sphere as being either at the north pole of the sphere or at the south pole. A qubit however, can be a point anywhere on the surface of the sphere. The bloch sphere is not a precise indicator of where the qubit lies on the unit sphere, it merely shows the latitude of the qubit. The latitude defines how close the qubit is to the poles, depending on the probability amplitudes.

The sphere has 3 axes; x, y and z. The number $\theta$ defined in equation (2) gives the angle of the qubit from the vertical z axis. This angle is easily found in the following way: we have already seen that equation (1) can be written as equation (2). By comparing the $\vert\rangle$ parts of both equations we get;

$$\alpha\vert\rangle = cos\frac{\theta}{2}\vert\rangle$$

Since $\alpha$ is known, $\theta$ can be found by;

$$\theta = (\cos^{-1}\alpha)*2$$

It is impossible to solve the equation to find out where the qubit exists on the semicircle defined by the longitude. The qubit exists on every point on that semicircle, so in the bloch sphere implementation for the Quantum Computing Language, the qubit is assigned an arbitrary longitude of 70 pixels from the vertical z axis.