One Bit-Flip and the Limits of Repetition

The three-bit repetition code encodes one classical bit into three identical copies. If a single bit flips during transmission, the original value remains the majority among the three. Decoding simply selects the value appearing twice. This majority vote corrects isolated errors effectively, restoring the intended bit without complex algorithms. Two or three flipped bits defeat the vote and return the wrong value.

For a binary symmetric channel flipping each bit independently with probability p, the code reduces error likelihood when p is below one-half. The probability of two or three flips becomes 3p² - 2p³. Since this value is smaller than p for small p, the receiver’s chance of obtaining the wrong bit decreases, demonstrating noise reduction through redundancy.

Applying this logic to qubits introduces complications. While the classical majority vote works for discrete bit-flips, quantum states exist in superpositions. The source notes that repetition codes do not protect quantum information against every error type. In fact, attempting to use them may make some specific quantum errors more likely, rendering the simple classical approach insufficient for general quantum error correction.

The repetition code is a useful teaching step, not a complete protection scheme. IBM uses it to explain error correction before moving to the nine-qubit Shor code. The lesson shows that the quantum version can correct one bit-flip under its stated assumptions, but does not protect against phase-flips. More physical qubits do not automatically mean better protection.

For an eventual Quantum Ireland test, the question would be specific: which errors does the encoding detect, and which does it miss? A useful report would state the error model, assumptions and recovery results, rather than treating a larger qubit count as proof. This is a proposed way to evaluate a test, not a report of an implementation.