Project: A Full QEC Demo

A single error-correcting gate is not error correction — a real demo runs the full cycle: encode a logical qubit, let an error strike, extract a syndrome without disturbing the encoded information, and apply the correction the syndrome names. This capstone wires all four stages together for the three-qubit bit-flip code, the simplest code that actually fixes an error.

The code and its codewords

The bit-flip code spreads one logical qubit across three physical qubits:

|0_L\rangle = |000\rangle, \qquad |1_L\rangle = |111\rangle.

A general logical state $\alpha|0_L\rangle + \beta|1_L\rangle$ is encoded from a prepared data qubit by two CNOTs. Here we encode the logical $|+_L\rangle$ :

|+_L\rangle = \frac{1}{\sqrt{2}}\bigl(|000\rangle + |111\rangle\bigr),

built by H(0); CX(0,1); CX(0,2).

Syndrome extraction without measurement collapse

The stabilizers are $Z_0Z_1$ and $Z_1Z_2$ . Rather than measuring the data qubits (which would collapse the superposition), we copy each parity onto a fresh ancilla with CNOTs:

Z_0Z_1 \longrightarrow \text{ancilla } 3, \qquad Z_1Z_2 \longrightarrow \text{ancilla } 4.

Each ancilla flips once per data qubit involved in its check. The encoded superposition is left intact because the parity of a valid codeword is identical across both branches $|000\rangle$ and $|111\rangle$ — the ancilla learns only the error, not the logical amplitude.

A worked round

Inject $X_1$ on the middle data qubit. It participates in both checks, so both ancillas flip, giving syndrome $(1,1)$ . The decoder for this code maps $(1,1) \mapsto$ "flip qubit 1," and applying $X_1$ again restores the codeword:

X_1\, X_1\, |+_L\rangle = |+_L\rangle.

Try it

Build the four stages — encode, inject $X_1$ , extract the syndrome, correct — on the 5-qubit register. The grader checks the final statevector: the data block back in $|+_L\rangle$ and the ancillas holding the syndrome $(1,1)$ .

Run your code to see the quantum state.