Step 1: Model of the faulty encoding circuit
We have 4 data qubits (labelled 0,1,2,3) and one ancilla qubit 4. The ideal encoding sequence is
H₁ → CNOT₁→₂ → CNOT₁→₀ → CNOT₂→₃ → CNOT₃→₄ → CNOT₀→₄ → M₄
and immediately after each two‐qubit CNOT₍i→j₎ we insert an i.i.d. two‐qubit depolarizing error on qubits (i,j):
with probability 1–p no error,
with probability p/15 each one of the 15 non-identity Pauli pairs.
We post‐select on (a) the ancilla measuring “0” (i.e. the measured Z₄‐eigenvalue is +1, which equivalently enforces the data‐parity Z₀Z₃=+1), and
(b) the two code‐stabilizers S_z=Z₀Z₁Z₂Z₃ and S_x=X₀X₁X₂X₃ both measuring +1.
Because this [[4,2,2]] code has distance 2, any single two-qubit error anywhere in the circuit either (i) flips the ancilla‐parity or (ii) flips one of the final stabilizer measurements. Hence all single-gate faults are caught by our post-selection. The only way to get a logical mis-preparation and yet pass all checks is to have two faults which (a) conspire to commute with S_z, S_x and with the ancilla‐parity Z₀Z₃, and (b) act as a non-trivial logical operator on the 2-qubit codespace.
Step 2: Logical infidelity arises at order p²
Let
P_success(p) = Prob{ no detected fault }
P_no-logical(p) = Prob{ no detected fault and net data error ∈ stabilizer }
P_logical-error(p) = Prob{ no detected fault and net data error ∈ normalizer∖stabilizer }
The logical fidelity is
\[
F(p)\;=\;\frac{\;P_{\rm no\!-\!logical}(p)\;}{\;P_{\rm success}(p)\;}\;=\;1-\frac{P_{\rm logical\!-\!error}(p)}{P_{\rm success}(p)}\,.
\]
Since single faults are always detected,
\[
P_{\rm logical\!-\!error}(p)=O(p^2),\qquad
P_{\rm success}(p)=1-\underbrace{5\,p}_{\rm single\hbox{-}faults}+O(p^2)\,,
\]
and one checks by series‐expansion that to second order
\[
F(p)\;=\;1 - P_{\rm logical\!-\!error}(p)\;+\;O(p^3)\,.
\]
Step 3: Counting the leading two-fault logical events
A little bookkeeping shows that exactly three pairs of CNOT‐locations can combine to give an undetected logical action (one such pair for each of the four weight-2 logical generators \(X_A,X_B,Z_A,Z_B\), up to stabilizer). Moreover one finds by explicit Pauli‐propagation that for each of those three location‐pairs there is exactly one choice of the two Pauli‐errors (out of the \(15\times15\)) which
• commutes with \(S_z,\;S_x,\;Z_0Z_3\),
• does implement one non-trivial logical (\(X_A\), \(X_B\), or \(Z_A\)) on the data,
and all other two-fault patterns either get caught or collapse to a purely stabilizer action.
Hence the total probability of logical error to second order is
\[
P_{\rm logical\!-\!error}(p)\;=\;
\underbrace{3}_{\rm pairs}\,\times\,
\underbrace{\Bigl(\tfrac{p}{15}\Bigr)^2
}_{\displaystyle p^2/225}
\;+\;O(p^3)
\;=\;\frac{3}{225}\,p^2+O(p^3)
\;=\;\frac{p^2}{75}+O(p^3)\,.
\]
Step 4: Final result
Substituting into \(F(p)=1-P_{\rm logical\!-\!error}(p)+O(p^3)\) gives, to leading non‐trivial order,
Final Answer:
\[
\boxed{
F_{\rm logical}(p)\;=\;1 \;-\;\frac{p^2}{75}\;+\;O(p^3)\,.
}
\]
In other words, after post‐selection the logical infidelity is suppressed to \(\;O(p^2)\), with leading coefficient \(1/75\approx0.0133\,\).