Meet the qubit

A classical bit has two possible values: $0$ and $1$. A qubit has two basic states too — we call them $|0\rangle$ and $|1\rangle$, pronounced “ket zero” and “ket one.” Those funny brackets are just notation. The important part is what’s inside.

If the qubit’s only tricks were $|0\rangle$ and $|1\rangle$, it would be a regular bit. What makes it different is that a qubit can be in a combination of both at once:

$|\psi\rangle = \alpha \, |0\rangle + \beta \, |1\rangle$

Here $\alpha$ and $\beta$ are complex numbers called amplitudes. They say how much of $|0\rangle$ and how much of $|1\rangle$ are in the mix. The state $|\psi\rangle$ (pronounced “psi”) is the qubit’s full description.

There’s one rule: the amplitudes have to be sized so that $|\alpha|^2 + |\beta|^2 = 1$. That’s called the normalization condition. We’ll see why in the measurement lesson.

Seeing a qubit

Here’s a picture. The Bloch sphere represents every possible state of a single qubit as a point on its surface. The arrow is the current state. $|0\rangle$ is the north pole; $|1\rangle$ is the south pole. Everything in between is a superposition.

(You can click and drag to rotate the camera.)

The state shown above has roughly equal parts $|0\rangle$ and $|1\rangle$, tilted a bit toward one side of the equator. If you measured it right now, you’d get $|0\rangle$ or $|1\rangle$ — we’ll see exactly how the probabilities work out in the next two lessons.

The two ways to write a state

In the formula above, we wrote the state as a weighted sum: $\alpha |0\rangle + \beta |1\rangle$. That’s called Dirac notation, and it’s the way physicists prefer to think.

You can also write it as a column vector:

$|\psi\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}$

These are the same thing, said two different ways. The column vector is what you’d feed into code; the Dirac form is what you’d write on a whiteboard. You’ll see both on this site — toggle between them in the display below:

This particular state is $\frac{1}{\sqrt{2}}|0\rangle + \frac{1}{\sqrt{2}}|1\rangle$ — equal parts of both. It’s special enough to have its own name: $|+\rangle$ (pronounced “ket plus”). It’s the quantum version of perfect ignorance about a bit, but with a twist we’ll see in Lesson 4.

What’s next

We’ve introduced the qubit but we’ve only shown it to you. In the next lesson, you’ll get to move it around: drag the sliders to explore every possible single-qubit state on the Bloch sphere, and see the amplitudes change in real time.