Everything you can actually test about a qubit comes down to a single rule, and it fits in one line:
Square the amplitude, and that is the chance of that result.
It has a name — the Born rule — and Max Born won a Nobel Prize for it in 1954.
P(0) = |0.6|² = 0.36 · P(1) = |0.8|² = 0.64 · they sum to 1
Using it
For a qubit holding α of 0 and β of 1:
- Chance of getting 0:
|α|² - Chance of getting 1:
|β|²
The "must add to 1" rule from two lessons ago is what makes these behave like real chances. A column of numbers that does not add up properly is not describing anything real.
For lots of qubits it is identical — there are just more numbers, and you pick out the one you care about.
Amplitudes are not chances
This is the distinction the whole field rests on, so it is worth being blunt.
| | Amplitudes | Chances | | --- | --- | --- | | Can be negative? | yes | never | | Two routes meeting | can cancel | only pile up | | Can you see one? | no | yes |
Two routes arriving as +½ and −½ give you nothing at all. Two routes each with a one-in-four chance give you one in two. Different arithmetic. Different universe.
Averages, and why they are expensive
Often you do not want one result — you want an average over many runs. Chemistry simulations are built entirely out of these averages.
You get one by running the circuit again and again and taking the mean. The catch is statistics: to get ten times more precise, you need a hundred times more runs.
That single fact is why some very promising quantum algorithms are impractically slow, and it comes up again in the QML track.
Worth remembering
- Square the amplitude to get the chance. That is the Born rule.
- The squares adding to 1 is what makes the numbers describe something real.
- Amplitudes can cancel; chances cannot. That is the whole advantage.
- Ten times the precision costs a hundred times the runs.
- Any probability is relative to how you chose to measure.