 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 encodesMeasure the commuting stabilizer generators $Z_1{Z}_2$ and $Z_2{Z}_3$ using ancillary qubits. Their outcomes reveal the parity relations, with the three-qubit bit-flip syndrome tableBoth codewords have the same parity checks, so the syndrome does not reveal $\alpha$ or $\beta$. For example, the $X_2$ error produces $\alpha |010⟩+\beta |101⟩$; 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 thatIt 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.