sing Quantum error correction Table 24.1 Outcome possibilities for the basis states |0000), |1111) on passing through a bit-flip channel, assuming up to one flip. Input Output Event |0000), |1111) |0000), |1111) No bit flip [1000), |0111) Bit flip on 1st qubit |0100), |1011) Bit flip on 2nd qubit |0010), |1101) Bit flip on 3rd qubit |0001), |1110) Bit flip on 4th qubit q)=a(0)+B1) = a/0000)+6|1111 Figure 24.1 Quantum circuit for encoding a qubit |q) into a 4-qubit repetition codeword la) before transmission to a bit-flip channel. the following projector set: Po=0000)(0000+|1111) (1111| |0111) (0111| P2=10100)(0100| + |1011) (1011| (24.4) P3 [0010) ( <0010| + |1101) (1101| |0001) <0001 + |1110) (1110|. Assume that the received qubit is (9')=a0010)+B1101).It is clear that a pro- jection measurement through P3 yields the certain outcome, or "expectation value" (P3) = (q'|P3/q') = 1, with the post-measurement state being lq*). We note that the measurement does not yield any information about the amplitudes a, B, thus, leaving the codeword unknown. Yet the measurement tells which type of error is to be corrected, and the original codeword can be restored to its full integrity by flipping again the errored qubit i should any Pi=1,2,3,4 measurement yield ('/Pilq') = 1, while the measurement (q'|Polq') = 1 indicates with certainty that there are no errors. The effectiveness of a quantum error-correction code can be qualified through fidelity, a notion which was introduced in Chapter 22. Assuming a noisy channel E with the encoded originator message p=lq (l and the recipient message E(p) = p', the fidelity can be defined as follows: (24.5) 24.1 Quantum repetition code In the case of the bit-flip channel, we have (Eq. (23.11)): p'==(p) =pxpx+(1-p)p (24.6) From the definition in Eq. (24.5), the corresponding fidelity of the code is + (1 -p)(919)(919) (24.7) Consider now some cases of interest. If we use no error correction, the two basic sets of pure-state symbols we can use are lq) = |0), |1) and 1->. As we know, the actio