9 / 13 Gates & Phase

Cannot Just Make Three Copies

An engineer proposes protecting a qubit in the unknown state α∣0⟩+β∣1⟩\alpha|0\rangle + \beta|1\rangle against bit flips by copying it into two spare qubits and majority-voting the three at the end. Why does the three-qubit bit-flip code not work this way, and what does it do instead?

A

No-cloning forbids duplicating entangled states. A single unentangled qubit in a pure state can be cloned, so the scheme fails only once the qubit is part of a larger register.

B

No unitary can take ∣ψ⟩∣0⟩∣0⟩|\psi\rangle|0\rangle|0\rangle to ∣ψ⟩∣ψ⟩∣ψ⟩|\psi\rangle|\psi\rangle|\psi\rangle for arbitrary unknown ∣ψ⟩|\psi\rangle. The code instead entangles one qubit into α∣000⟩+β∣111⟩\alpha|000\rangle + \beta|111\rangle, which is not three copies, and measures parity checks that reveal which qubit flipped without revealing α\alpha or β\beta.

C

CX does copy an unknown qubit onto a fresh ∣0⟩|0\rangle. The real obstacle is that measuring the three qubits to take the majority collapses the superposition and destroys the state.

D

Copying works, but all three copies decohere at the same rate and in a correlated way, so the majority vote carries no more information than the original qubit.