Visualize The Invisible Qubit Phase

And learn how to tell the difference between the relative and the global phase

The phase of a qubit is the decisive factor when we work with amplitudes. Even though we can’t measure the phase directly, it is not invisible. Once, we can use our simulator to visualize it. Second, we can use math to describe it precisely.

When we look at the Bloch Sphere, states with a positive amplitude reside at the front-side of the X- and Z-axes.

By contrast, states with a negative amplitude reside at the back-side of the Bloch Sphere.

Let’s consider some simple states, first. If we rotate the qubit state vector around the Y-axis by the angle θ, we get the following states. For 0<θ<π, we can say the phase is positive (blue). For π<θ<2π, the phase is negative (red).

The figure shows that these states lie on the plane the X- and the Z-axes span. The vector θ specifies the rotation around the Y-axis. Therefore, we call the corresponding gate RY-gate.

Two vectors mirroring each other on the Z-axis have the same measurement probability, such as the two vectors depicted in the figure. They share the same measurement probability, but their phase differs.