= Solution
<Quantum error correction> encodes a <logical qubit> into a larger <Hilbert space>, diagnoses a specified error family with an <error syndrome>, and reverses it without learning the protected logical amplitudes. It must preserve arbitrary superpositions, not just distinguish two classical codewords.
The <bit-flip repetition code> encodes
$$
\alpha|0\rangle+\beta|1\rangle\longmapsto\alpha|000\rangle+\beta|111\rangle.
$$
Measure the commuting <stabilizer generators> $Z_1Z_2$ and $Z_2Z_3$ using ancillary qubits. Their outcomes reveal the parity relations, with the <three-qubit bit-flip syndrome table>
$$
\boxed{\begin{array}{c|c|c}
\text{error}&(Z_1Z_2,Z_2Z_3)&\text{recovery}\\\hline
I&(+1,+1)&I\\
X_1&(-1,+1)&X_1\\
X_2&(-1,-1)&X_2\\
X_3&(+1,-1)&X_3
\end{array}}
$$
Both codewords have the same parity checks, so the syndrome does not reveal $\alpha$ or $\beta$. For example, the $X_2$ error produces $\alpha|010\rangle+\beta|101\rangle$; applying $X_2$ restores the encoded superposition. A coherent combination of correctable errors is also recoverable after the syndrome is recorded in an ancilla, by linearity.
This three-qubit code corrects one bit flip, not every one-qubit error. A physical $Z_1$ changes the relative logical phase without a nontrivial bit-flip syndrome. General single-qubit noise is a linear combination of $I,X,Y,Z$; protecting arbitrary single-qubit errors therefore needs a code such as the <Shor code>, the <Steane code> or the <five-qubit code>, with the corresponding enlarged syndrome structure.
The general exact-recovery criterion is the <Knill--Laflamme condition>. For code projector $P$ and error operators $E_r$, there must be constants $C_{rs}$ such that
$$
\boxed{PE_r^\dagger E_sP=C_{rs}P\quad\text{for every }r,s.}
$$
It expresses that the syndrome information can depend on the error but not on the logical state. Encoding entangles several physical qubits; it does not create independent copies of an unknown input, so it respects the <no-cloning theorem>. Fault-tolerant implementations must also prevent noisy syndrome extraction and recovery gates from spreading an initially correctable error into an uncorrectable one.
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