Superposition
Amplitudes, not maybes: what it really means to be 0 and 1.
Lesson 2 of 7 about 11 min
Not what the headlines say
You've probably read that “a qubit can be 0 and 1 at the same time”. It's a catchy line, but it confuses almost everyone who hears it. Here's a better picture, and you already know it from every map.
Think of directions on a compass. North-east is not “north and east at the same time”. It isn't secretly north or secretly east, either. It's a direction of its own, made of some north and some east.
The recipe
Every qubit state has a recipe: how much |0⟩ and how much |1⟩ goes into it. Physicists write it like this:
The two numbers α (alpha) and β (beta) are the amplitudes you met in the warm-up. Square them and you get the chances: α² for 0 and β² for 1. The arrow has length 1, so the two chances always add up to exactly 1.
0.87|0⟩ + 0.5|1⟩
Your turn
Is it secretly 0 or 1?
Here's a fair suspicion. Maybe a superposition just means “we don't know yet”, like a coin under a cup. The coin is already heads or tails. We just haven't looked.
We can actually test this. Below are two machines that look identical when you measure them: both give 0 half the time. But try doing one thing first: an H gate, then measure.
A · coin under a cup
Secretly 0 or 1, half the time each.
No runs yet.
B · a qubit, 50-50
An equal superposition of 0 and 1.
No runs yet.
Your turn
Same odds, different states
Here's a puzzle. These two states give exactly the same odds, 50-50. Are they the same state?
The only difference is one minus sign. Measure them straight away and you can't tell them apart. But their arrows point different ways, so a gate treats them differently.
|+⟩
0.71|0⟩ + 0.71|1⟩= |+⟩|−⟩
0.71|0⟩ − 0.71|1⟩= |−⟩Amplitudes can point anywhere
In the warm-up, amplitudes grew into complex numbers: arrows of their own, which can point any way. That's true for qubits too. α and β can be any complex numbers, as long as their sizes squared add up to 1.
For example, 0.6|0⟩ + 0.8i|1⟩ is a perfectly good qubit. The size of 0.8i is 0.8, so 1 has a 64% chance, and 0 has 36%.
Your turn
Make one yourself
You don't need anything exotic to make a superposition. One gate does it: the Hadamard gate, or H for short. It turns a certain |0⟩ into |+⟩, a perfect 50-50.
|0⟩
Your turn
What you now know
- A superposition is one definite state made of some 0 and some 1, like north-east is made of north and east.
- Its recipe is α|0⟩ + β|1⟩. The chances are α² and β², and they add up to 1.
- A superposition is not “secretly 0 or 1”: an H gate tells them apart.
- |+⟩ and |−⟩ have the same odds but different phases, and gates can tell the difference.
- Amplitudes can be complex. The H gate makes a superposition from |0⟩.
Next: the Bloch sphere, a globe where every possible qubit is a point, and the phase is something you can see.