Three-qubit repetition and phase-flip codes

Define the logical basis as $|0_L\rangle=|000\rangle$ and $|1_L\rangle=|111\rangle$. A controlled encoding maps an unknown input and two initialized ancillas to $\alpha|000\rangle+\beta|111\rangle$. It does not produce three copies of the unknown qubit: the encoded state generally has entanglement, and an individual output qubit has lost the input coherence when viewed alone. The information resides in correlations across the block and can remain entangled with a remote reference.

Measure the commuting parities $Z_1Z_2$ and $Z_2Z_3$, rather than measuring each Z separately. Both logical basis states have positive parity, so the check outcomes are independent of the unknown amplitudes. Measuring individual qubits would distinguish the logical basis states and destroy their superposition. An ancilla-assisted parity circuit must therefore record only agreement or disagreement. Its ideal mathematical description projects onto an entire parity sector without resolving states inside that sector.

Report each parity eigenvalue in the fixed order of the two checks. No error gives $(+,+)$; a flip on qubit one gives $(-,+)$; a flip on qubit two gives $(-,-)$; and a flip on qubit three gives $(+,-)$. Applying X on the inferred qubit returns the full encoded superposition. The correction works for any amplitudes because it reverses the location error rather than estimating the unknown logical state.

Two bit flips are already ambiguous. Flips on qubits one and two have the same syndrome as a single flip on qubit three. The single-error decoder then applies a third flip, leaving $X_1X_2X_3$, which exchanges the logical basis states. More fundamentally, a single Z on any physical qubit changes the logical relative phase without changing either parity. The unrestricted quantum distance is therefore one, even though the code has distance three against the restricted X-error family.

Conjugating every physical qubit by Hadamard exchanges X and Z. The transformed code has basis $|+++\rangle,|---\rangle$ and checks $X_1X_2,X_2X_3$. It corrects one Z error with the same syndrome table. The reason is algebraic: Z exchanges the plus and minus X eigenstates. Calling this a phase-flip code refers to the computational basis, while its parity checks operate in the complementary basis.

A single physical X now acts as a logical phase operation, so changing basis exchanges the weakness rather than removing it. Protecting both types requires a larger construction with compatible complementary checks. Also, the binomial calculation excludes the gates used to collect parities: an ancilla fault can propagate to several data qubits. The next step in code design is to preserve the clean syndrome logic while accounting for the new errors introduced by carrying it out physically.