The Bit-Flip Code
The three-qubit bit-flip code stores one logical qubit as α|000⟩ + β|111⟩, detects any single X error by a majority vote, and undoes it exactly — recovering the original amplitudes, not just their probabilities. This chapter runs the entire encode–error–correct cycle coherently on three qubits, with every prediction traced by hand first.
What is the three-qubit bit-flip code?
The oldest trick in classical communication is repetition: send every bit three times, and if one arrives flipped, the other two out-vote it. Last chapter proved we cannot do that literally — an unknown qubit state cannot be copied. But it also showed the escape: spread the state across several qubits with entanglement instead of copying it.
The three-qubit bit-flip code is that escape made concrete, and it is small enough to run end to end right here. One logical qubit — the unit of information we are protecting — is stored in three physical qubits (the distinction is drawn carefully in physical vs logical qubits): the state α|0⟩ + β|1⟩ is encoded as α|000⟩ + β|111⟩. Here α and β are the state's amplitudes — the weights on 0 and 1 whose squared magnitudes are the measurement probabilities.
The full mission for this chapter: encode → inject a bit-flip error → correct it → verify that the original amplitudes come out exactly. Not approximately — exactly. And "verify" means checking run counts you predicted by hand.
You’ve read the opening of chapter 4 — 7 more sections follow, with worked examples and circuits you can run on the page. A free account unlocks every chapter of every course (paid plans aren’t live yet — early readers get everything free).