Phase-flip repetition code

The $n$-qubit phase-flip repetition code is stabilized by $X_j{X}_{j+1}$ and has codewords $|+{⟩}^{\otimes n}$ and $|-{⟩}^{\otimes n}$. It has logical operators $\overline{Z}=X_1$ and $\overline{X}=Z_1\cdots Z_n$, so its distance against phase flips is $n$.

The codewords $|+++⟩,|---⟩$ have residual logical Pauli X gate after an uncorrectable two- or three-site phase error. Precede their encoding by a logical Hadamard gate and follow decoding by its inverse. Since $HXH=Z$, the same error now acts as a logical Pauli Z gate. Explicitly the encoded coefficients are $(\alpha +\beta )/\sqrt{2}$ and $(\alpha -\beta )/\sqrt{2}$ for an input $\alpha |0⟩+\beta |1⟩$. This convention matters when translating physical phase errors into a claimed decoded phase-error channel.

The phase-flip repetition code with codewords $|+++⟩,|---⟩$ corrects any one physical Pauli Z gate error. Its two adjacent-pair error syndromes cannot distinguish an error pattern from its three-bit complement. Minimum-weight recovery therefore leaves no residual error at weight zero or one, and leaves $Z_1{Z}_2{Z}_3$ at weight two or three. This residual operator exchanges the codewords, so inverse encoding gives a logical Pauli X gate. For independent physical phase errors of probability $ϵ$, the decoded quantum channel is $(1-p_L)\rho +p_{L}X\rho X$, where $p_L=3{ϵ}^2(1-ϵ)+{ϵ}^3$. For $0<ϵ<1/2$, $ϵ-p_L=ϵ(1-ϵ)(1-2ϵ)>0$.

Insert Hadamard gates on every data qubit immediately after encoding by a bit-flip repetition code and immediately before its bit-flip detection and recovery. The intervening physical error is thereby conjugated by $H^{\otimes n}$, and $HZH=X$ turns a single Pauli Z gate error into a single bit flip for the original correction circuit. The encoded logical codewords are now $|+{⟩}^{\otimes n}$ and $|-{⟩}^{\otimes n}$. The syndrome ancillas and the final inverse encoding stay unchanged. This protects against phase flips in the new basis; it does not simultaneously correct arbitrary bit flips in the physical basis.

For independent errors of probability $p<1/2$, maximum-likelihood decoding of a repetition code chooses the lower-weight error compatible with the syndrome. This is majority voting, with threshold $p_c=1/2$ and exponentially vanishing logical-error probability below threshold.