11.7 Repetition codes

In order to give a sense of how quantum error correction actually works, let us begin with a classical example of a repetition code. Suppose a transmission channel flips each bit in transit with probability p. If this error rate is considered too high then it can be decreased by encoding each bit into, say, three bits: \begin{aligned} 0 &\mapsto 000 \\1 &\mapsto 111. \end{aligned} That is, each time we want to send logical 0, we send three physical bits, all in state 0; each time we want to send logical 1, we send three physical bits, all in state 1. The receiver decodes the bit value by a “majority vote” of the three bits. If only one error occurs, then this error correction procedure is foolproof. In general, the net probability of error is just the likelihood that two or three errors occur, which is 3p^2(1-p) + p^3 < p. Thus the three bit code improves the reliability of the information transfer. The quantum case, however, is more complicated, because we have both bit-flip and phase-flip errors.