One qubit needs 2 numbers. Two qubits need 4. Not 2 + 2, 2 × 2.
The operation that does this multiplying is written ⊗, and it has a scary name and an easy job.
| Qubits | Amplitudes | |
|---|---|---|
| 1 | 2 | a single qubit |
| 2 | 4 | |00⟩ |01⟩ |10⟩ |11⟩ |
| 10 | 1,024 | a laptop shrugs |
| 50 | ~10¹⁵ | a serious supercomputer |
| 300 | ~10⁹⁰ | more than atoms in the observable universe |
How it works
Multiply every number in the first list by the whole second list:
(a, b) ⊗ (c, d) = (ac, ad, bc, bd)
So |0⟩ ⊗ |0⟩ is (1,0) ⊗ (1,0) = (1,0,0,0), which we shorten to |00⟩. Those four slots are the amounts of 00, 01, 10 and 11, in that order.
Gates stack the same way. Doing H to the first qubit and nothing to the second is one 4×4 grid, written H ⊗ I.
Now try to pull a Bell pair apart
Here is where it gets interesting. Take the entangled pair from Fundamentals and try to write it as "this first qubit ⊗ that second qubit":
(|00⟩ + |11⟩)/√2 = (a,b) ⊗ (c,d)?
You would need ac and bd to be big, and both ad and bc to be zero. But if ad = 0 then either a or d is zero, and either one wrecks one of the first two.
There is no answer. It genuinely cannot be done.
Entangled, properly defined
A pair that cannot be written as one qubit's description times another's. That failure is not a limitation of our notation. It is the definition.
Worth remembering
- Combining qubits multiplies the number of slots rather than adding them.
- n qubits need 2ⁿ numbers.
- Some pairs simply cannot be pulled apart. Those are the entangled ones.
- That impossibility is what makes ordinary simulation blow up.