= Shor-code phase-flip syndrome
{c}
{title2=$(X^{\otimes3}_1X^{\otimes3}_2,\ X^{\otimes3}_2X^{\otimes3}_3)$}
Group the nine physical <qubits> into three blocks and write $B_r$ for the product of three <Pauli X gates> in block $r$. The commuting <stabilizer generators> $B_1B_2$ and $B_2B_3$ have eigenvalue $+1$ on both logical codewords. A <Pauli Z gate> in blocks one, two or three gives syndromes $(-,+)$, $(-,-)$ or $(+,-)$ respectively. Apply a Z gate to any one qubit of the indicated block to recover the logical state. Z errors at distinct locations within a block act identically on the code, so this correction is degenerate.
Back to article page