1. Step-by-Step derivation
We denote the five CNOTs that appear (from right to left in the circuit) by
\[
\begin{aligned}
g_1 &= C_{12},\qquad
g_2 &= C_{10},\qquad
g_3 &= C_{23},\qquad
g_4 &= C_{34},\qquad
g_5 &= C_{04},
\end{aligned}
\]
where the first index is the control, the second the target.
After every CNOT a two-qubit depolarising channel acts
\[
\mathcal E_{ct}(\rho)= (1-p)\rho+\frac{p}{15}\!
\sum_{(P,Q)\neq(I,I)}
\bigl(P_c\otimes Q_t\bigr)\rho\bigl(P_c\otimes Q_t\bigr),\qquad
P,Q\in\{I,X,Y,Z\}.
\]
Because we post-select
- on the ancilla measurement outcome \(|0\rangle\), and
- on the two stabilisers \(S_X=XXXX,\;S_Z=ZZZZ\) both giving eigenvalue \(+1\),
only those Pauli faults that commute with both stabilisers and leave the
ancilla in an eigenstate of \(Z\) with eigenvalue \(+1\) can survive.
All other faults are detected and the run is discarded.
A single–gate fault is already order \(p\); probabilities with two or
more faults are \(\mathcal O(p^{2})\) and can be neglected for leading-order
fidelity. Hence we examine, for every gate \(g_k\), the 15 Pauli pairs that
may appear immediately after that gate and propagate them through the
remaining circuit.
Useful CNOT conjugation rules
(control \(c\), target \(t\)):
\[
\begin{aligned}
X_c &\;\longrightarrow\; X_cX_t, &\qquad Z_c &\;\longrightarrow\; Z_c,\\
X_t &\;\longrightarrow\; X_t, & Z_t &\;\longrightarrow\; Z_cZ_t,\\
Y_c &\;\longrightarrow\; Y_cX_t, & Y_t &\;\longrightarrow\; Z_cY_t .
\end{aligned}
\]
Because weight-1 faults are always caught by the
distance-2 code, we only have to look at the three weight-2 same-Pauli
errors
\(XX,\;YY,\;ZZ\) on each pair; all other 12 possibilities are detected.
Gate–by–gate analysis (Heisenberg propagation)
-
\(g_{1}=C_{12}\):
• \(XX_{12}\;\Rightarrow\;XXXX = S_X\) (stabiliser – accepted, no
logical error).
• \(YY_{12},\,ZZ_{12}\;\Rightarrow\;
X_0Y_1Y_2X_3\equiv Z_1Z_2 = Z_AZ_B\)
(undetected logical \(Z_AZ_B\)).
-
\(g_{2}=C_{10}\):
• \(ZZ_{10}\;\Rightarrow\;Z_0Z_1=Z_A\) (undetected logical \(Z_A\)).
• \(XX_{10},YY_{10}\) contain \(X_4\) and flip the ancilla; detected.
-
\(g_{3}=C_{23}\):
• \(ZZ_{23}\;\Rightarrow\;Z_0Z_1=Z_A\) (undetected logical \(Z_A\)).
• \(XX_{23},YY_{23}\) propagate an \(X_4\); detected.
-
\(g_{4}=C_{34}\):
• \(ZZ_{34}\;\Rightarrow\;Z_0Z_3\equiv Z_AZ_B\) (undetected logical \(Z_AZ_B\)).
• \(XX_{34},YY_{34}\) act on the ancilla; detected.
-
\(g_{5}=C_{04}\):
All three same-Pauli errors leave a single-qubit Pauli on qubit 0,
which flips at least one stabiliser and is therefore detected.
Summary of single-gate faults that survive post-selection
\[
\begin{array}{c|c}
\text{gate}&\text{undetected logical Pauli}\\ \hline
g_1&YY_{12},\;ZZ_{12}\;(\;Z_AZ_B\;)\\
g_2&ZZ_{10}\;(\;Z_A\;)\\
g_3&ZZ_{23}\;(\;Z_A\;)\\
g_4&ZZ_{34}\;(\;Z_AZ_B\;)\\
g_5&\text{none}
\end{array}
\]
That is five logically harmful Pauli pairs altogether.
Probability bookkeeping to first order in \(p\)
• Probability that exactly one gate suffers one of the five bad
Paulis:
\[
P_{\text{log}}^{(1)} = 5 \times\frac{p}{15}= \frac{p}{3}.
\]
• Total acceptance probability (no detected fault) is
\[
P_{\text{acc}} = 1-\underbrace{\Bigl(\tfrac{69}{15}\Bigr)}_{\text{detected pairs}}\,p
+\mathcal O(p^{2})
= 1+\mathcal O(p).
\]
Logical fidelity conditioned on acceptance
\[
F_{\text{logical}}(p)=
1-\frac{P_{\text{log}}^{(1)}}{P_{\text{acc}}}
=1-\frac{p}{3}+\mathcal O(p^{2}).
\]
(The \(\mathcal O(p^{2})\) term comes from two-fault events as well as the
product \(P_{\text{log}}^{(1)}/P_{\text{acc}}\); it is suppressed here
because the problem only asks for the functional dependence.)
2. Mathematical typesetting
All equations above are typeset in LaTeX.
3. Conventions and Units
No physical units occur; probabilities are dimensionless.
Final Answer:
To leading order in the physical two-qubit gate error rate \(p\), the logical two-qubit state fidelity after post-selection is
\[
\boxed{\,F_{\text{logical}}(p)=1-\dfrac{p}{3}+\mathcal O\!\bigl(p^{2}\bigr)\,}.
\]
Hence the logical infidelity scales linearly with slope \(1/3\) in the small-\(p\) regime.