Qubit Visualization

Visualizing a single qubit

We’ve looked at multiple examples of a quantum state being shown as a vector on a 2-D plane for the sake of simplicity. However, that is not accurate since we ignored one important fact. Each of the two complex numbers that make up the quantum state themselves have two dimensions, a real part and an imaginary part.

The examples we had visually seen so far only had a real component in them. To fully visualize a qubit with all possible superposition states, we need another dimension for the imaginary part. Then, we will see how to represent a qubit in real 3-D space.

The Bloch Sphere

The Bloch Sphere is a mathematical tool to represent all possible states of a single qubit.

$$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle ;where\space\alpha,\beta\in \mathbf{C}$$

To visualize a quantum state in a sphere we need to represent it with three real numbers instead of two complex numbers. Without going much into the details for the proof, here’s what the conversion looks like:

$$|\psi\rangle = \alpha|0\rangle + e^{i\phi}\beta|1\rangle ;where\space\alpha, \beta, \phi \in \mathbf{R}$$

In case you feel intimidated by this sudden transformation and the surprise appearance by the Euler’s Number $e$, a simple explanation is that we want to represent our state on a sphere. So utilizing polar coordinates would be a concise way, and using Euler’s Formula, a complex number in polar coordinates, such as $\cos \theta + i\sin \theta$, can be written simply as $e^{i\phi}$. Don’t worry if you don’t quite get this. It’s unimportant for the purposes of this lesson.

We know that all states are normalized. And, thus applying some trigonometry magic (that you don’t need to worry about) we can now write our arbitrary state as:

$$|\psi\rangle = \cos\frac{\theta}2 |0\rangle + e^{i\phi}\sin\frac{\theta}2|1\rangle ;where\space\theta, \phi \in \mathbf{R}$$

Now, what are these new angles $\theta$ and $\phi$ in terms of our sphere? Let’s take a look.