Reading kets · Qubits · Learn quantum computing

A shorthand worth knowing

Open any quantum computing book, paper or code library and you'll see symbols like |0⟩, |ψ⟩ and |+⟩. They look scary, but they're just labels. The physicist Paul Dirac invented them in 1939, and they caught on because they make quantum ideas short to write.

A symbol written |name⟩ is called a ket. It always means “the state called name”. You've been reading kets all along. Here's one, taken apart:

=

Nothing picked yet.

GoalTap every part of the line.

Tap each part of the line to see what it means.

Kets are columns

Behind every ket is a column of two amplitudes: how much |0⟩ on top, how much |1⟩ below. That's the vector from the warm-up, stood on its end.

|0⟩ = 10 |1⟩ = 01

So α|0⟩ + β|1⟩ is simply the column (α, β).

1 · Pick a ket

2 · Then its column

GoalMatch all four kets to their columns.

A ket is shorthand for a column of two amplitudes: the amount of |0⟩ on top, the amount of |1⟩ below. 0.71 is 1/√2.

The famous six

A handful of states turn up so often that they have their own names. Here are all six, where they live on the Bloch sphere, and what they're made of:

|+⟩ = (|0⟩ + |1⟩) ÷ √2

0.710.71

Equator, front

0: 50%1: 50%

GoalLook at all six.

These six states have their own names because they sit at the six ends of the sphere's three axes. They show up everywhere.

Where does √2 come from?

Why is there a √2 in |+⟩ = (|0⟩ + |1⟩) ÷ √2? Without it, the chances would add up to 1 + 1 = 2. Dividing both amplitudes by √2 squeezes the total back down to exactly 1. That's called normalising a state.

Your turn

A state is a|0⟩ + a|1⟩, with the same amplitude a in front of both. What must a be (to two decimal places) for this to be a real qubit state?

Gates, in ket language

Kets also make gates quick to write. A gate's letter goes in front of the state it acts on:

X|0⟩ = |1⟩ · X|1⟩ = |0⟩ · H|0⟩ = |+⟩ · H|1⟩ = |−⟩ · Z|+⟩ = |−⟩

Your turn

What is H|+⟩?

A sneak peek: bras

You'll sometimes see the bracket flipped the other way, like ⟨0|. That's a bra, the other half of a “bra-ket”. For now you only need one use of it: ⟨0|ψ⟩ means “the amplitude of 0 in ψ”.

Your turn

If |ψ⟩ = 0.6|0⟩ + 0.8|1⟩, what is ⟨1|ψ⟩?

What you now know

A ket |name⟩ means “the state called name”. ψ (psi) is the usual name for “some state”.
A ket is a column of two amplitudes: |0⟩ on top, |1⟩ below.
The famous six: |0⟩ and |1⟩ at the poles; |+⟩, |−⟩, |+i⟩, |−i⟩ around the equator.
The ÷ √2 normalises a state, so its chances add up to exactly 1.
Gates go in front of kets: H|0⟩ = |+⟩. And ⟨0|ψ⟩ is the amplitude of 0 in ψ.

Next, the fun part: a lab where you build a real quantum coin and flip it a thousand times.