= Logical failure of the three-qubit bit-flip repetition code
{title2=$p_L=3p^2-2p^3$}
Assume that each physical <qubit> undergoes an independent <quantum bit-flip channel> with <probability> $p$ during one storage interval, and that encoding, <error syndrome> extraction, recovery and decoding are all ideal. Use the <three-qubit bit-flip syndrome table> to apply the indicated minimum-weight recovery. Zero or one bit flips are corrected. Two bit flips have the same <error syndrome> as the complementary one-site error: for example, $X_1X_2$ gives $01$, so recovery $X_3$ leaves $\overline X=X_1X_2X_3$. Three bit flips give $00$ and also leave $\overline X$. This exchanges $|000\rangle$ and $|111\rangle$, and therefore acts as a logical <Pauli X gate> after decoding.
The number of physical bit flips has a <binomial distribution> with three trials, hence the logical-error <probability> is
$$
p_L=\binom32p^2(1-p)+p^3=3p^2-2p^3.
$$
For a logical <density operator> $\rho$, the decoded <quantum channel> is exactly $\rho\mapsto(1-p_L)\rho+p_LX\rho X$. Here logical failure means a nontrivial logical <Pauli X gate> error; a particular input such as $|+\rangle$ can be invariant under that error. At $p=0.1$, exactly two flips have <probability> $3(0.1)^2(0.9)=0.027$ and three flips have <probability> $(0.1)^3=0.001$, giving $p_L=0.028$. The block succeeds with <probability> $0.972$ compared with $0.9$ for an unencoded <qubit>. At $p=0.01$, $p_L=0.000298$, illustrating the quadratic suppression of rare independent bit flips.
Factoring the improvement gives
$$
p-p_L=p(1-p)(1-2p).
$$
Thus one level reduces logical-error <probability> for $0<p<1/2$, is equal at $p=0,1/2,1$, and makes it worse for $1/2<p<1$ under this same decoder. The suppression arises because a logical error requires at least two physical bit flips. With independent disjoint inner blocks, the same noise model and ideal correction at every level, a <concatenated quantum error-correcting code> has $p_{\ell+1}=3p_\ell^2-2p_\ell^3$, with $p_0=p$. At $p_0=0.1$, two levels use nine data <qubits> and give $p_2=3(0.028)^2-2(0.028)^3=0.002308096$.
\Image[/quantum-bit-flip-error-suppression.png]
{title=Ideal repetition-code correction suppresses independent bit flips below one half}
This calculation describes ideal correction of stored information with the specified <quantum bit-flip channel>. Repeated correction in time additionally requires a fresh specified noise model for each interval; imperfect <quantum gates>, faulty measurements and correlated errors change the calculation. The <timed bit flip during repetition-code syndrome extraction> and <exact encoded-state failure with a faulty parity check> show concretely why this storage calculation alone is not a fault-tolerance result.
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