Checkpoint: Run a QEC Cycle

This checkpoint puts the whole module together: take a genuine quantum superposition, protect it, damage it, and bring it back — all without ever learning what the logical state was. If your circuit works, you have implemented a complete quantum-error-correction cycle.

The cycle, end to end

A QEC cycle has four stages, and you have built each one separately in this module:

Encode. Map the data qubit onto a logical codeword with the encoder ( $H_0$ then cx(0,1); cx(0,2) for $|+_L\rangle$ ).
Error. Noise strikes — here a single $X$ on qubit 2, the kind the bit-flip code is built to survive.
Extract / decode. Run the encoder in reverse to gather the logical amplitude back onto the data qubit and expose the error as a syndrome on the partners.
Correct. Use the syndrome to apply the recovery — a Toffoli vote that flips the data qubit only when both partners disagree with it.

Why it has to be coherent

The input is $|+\rangle$ , an equal superposition. A scheme that measured an individual data qubit would collapse it and destroy the logical information. The decode-and-vote recovery uses only unitary gates, so by linearity it acts the same on $\alpha|0\rangle+\beta|1\rangle$ as on the basis states: the logical amplitudes $\alpha,\beta$ pass through untouched while the error is undone. This is the essence of QEC — repairing a state you are not allowed to look at.

What success looks like

After the cycle, qubit 0 is back in $|+\rangle$ and qubits 1 and 2 hold the leftover syndrome of the corrected error. The full three-qubit statevector is therefore deterministic: two amplitudes of $1/\sqrt{2}$ that differ only in the qubit-0 bit. That is the signature of a perfectly recovered logical $|+\rangle$ .

Try it

Implement the four-step cycle: encode $|+_L\rangle$ , inject $X$ on qubit 2, decode, and correct. The grader checks the recovered statevector — get the logical $|+\rangle$ back onto qubit 0.

Run your code to see the quantum state.

When the cycle is correct, the statevector shows exactly two populated amplitudes (indices 1 = 001 and 5 = 101), each $\approx 0.707$ , differing only in the qubit-0 bit. The logical $|+\rangle$ has survived a bit-flip error — a full QEC cycle, working.