1. Step–by–Step Derivation
- Ideal encoded state
Exactly as in the previous checkpoint, in the absence of faults the circuit
\[
(CNOT_{03})\;H_0\;(CNOT_{21})\;H_2
\]
prepares the four–qubit state
\[
|\psi_{\rm id}\rangle
=|\Phi^{+}\rangle_{21}\otimes|\Phi^{+}\rangle_{03},\qquad
|\Phi^{+}\rangle\equiv\frac{|00\rangle+|11\rangle}{\sqrt2},
\]
which is the logical Bell state
\[
|\Psi^{+}\rangle_{AB}
=\frac{|00\rangle_{AB}+|11\rangle_{AB}}{\sqrt2}.
\]
The code stabilisers are
\[
S_X= X_0X_1X_2X_3,\qquad
S_Z= Z_0Z_1Z_2Z_3 .
\]
The logical Pauli’s (up to stabilisers) are
\[
X_A=X_0X_2,\;X_B=X_0X_1,\quad
Z_A=Z_0Z_1,\;Z_B=Z_0Z_2 .
\]
Hence
\[
|\Psi^{+}\rangle_{AB}
\text{ is stabilised by }
\{\,S_X,\;S_Z,\;X_AX_B,\;Z_AZ_B\,\}.
\]
- Noise on the two CNOT gates
Each CNOT is followed by an independent two-qubit depolariser on the same
qubits:
\[
\mathcal D^{(p)}(\rho)
=(1-p)\rho+\frac{p}{15}\!\!\sum_{P\neq I\!\otimes\! I}\!
P\rho P,\qquad P\in\mathcal P_2 .
\]
• After the first CNOT (\(21\)) an error \(E_{21}\) acts on qubits 2 and 1.
• After the second CNOT (\(03\)) an error \(E_{03}\) acts on qubits 0 and 3.
The final physical state before syndrome measurement is
\[
\rho(p)=
\sum_{E_{21},E_{03}}
p(E_{21})\,p(E_{03})\;
E_{03}E_{21}\,
|\psi_{\rm id}\rangle\langle\psi_{\rm id}|\,
E_{21}E_{03}.
\]
- Classifying the two-qubit Paulis
For a single qubit Pauli \(P\) set
\[
c_X(P)=
\begin{cases}
1 & P\in\{Z,Y\}\\
0 & P\in\{I,X\}
\end{cases},\qquad
c_Z(P)=
\begin{cases}
1 & P\in\{X,Y\}\\
0 & P\in\{I,Z\}.
\end{cases}
\]
\(c_X(P)\,[c_Z(P)]\) tells whether \(P\) anticommutes with the all-\(X\)
\([\,\)all-\(Z]\) stabiliser.
For a two-qubit operator \(P\otimes Q\) define the parity vector
\[
\mathbf b(P\!\otimes\!Q)=
\bigl(c_X(P)\oplus c_X(Q),\;
c_Z(P)\oplus c_Z(Q)\bigr)
\in\{(0,0),(0,1),(1,0),(1,1)\}.
\]
Enumerating the 16 two-qubit Paulis on any fixed pair gives
• category \(\mathcal C_0\;(0,0)\) : \(\{II,\,XX,\,YY,\,ZZ\}\)
• category \(\mathcal C_1\;(0,1)\) : \(\{XI,I X,\,YZ,\,ZY\}\)
• category \(\mathcal C_2\;(1,0)\) : \(\{ZI,I Z,\,YX,\,XY\}\)
• category \(\mathcal C_3\;(1,1)\) : \(\{YI,I Y,\,XZ,\,ZX\}\)
with probabilities
\[
p_0 = 1-\frac{4p}{5},\qquad
p_1=p_2=p_3=\frac{4p}{15}.
\]
- Syndrome measurement and post-selection****
Measuring \(S_X,S_Z\) and accepting only the \(+1,+1\) outcome projects
onto the codespace.
The outcome is trivial iff the parity vectors of the two pairs coincide,
i.e. both errors belong to the same category \(\mathcal C_k\).
Hence the success probability
\[
P_{\rm succ}(p)=p_0^{\,2}+p_1^{\,2}+p_2^{\,2}+p_3^{\,2}
=(1-\tfrac{4}{5}p)^2+3\Bigl(\tfrac{4}{15}p\Bigr)^{\!2}
=(1-\tfrac{4}{5}p)^2+\tfrac{16}{75}p^{2}.
\]
- Logical action of the undetected errors
• If both pairs are in \(\mathcal C_0\) the combined operator is one of
\[
\{\,I,\;S_X,\;S_Z,\;S_XS_Z,\;
X_AX_B,\;Z_AZ_B,\;Y_AY_B,\;
{\rm (those\;multiplied\;by\;}S_X\!,S_Z)\},
\]
i.e. an element of the 16-element stabiliser of
\(|\Psi^{+}\rangle_{AB}\).
**It leaves the logical state unchanged.**
• If the shared category is \(\mathcal C_1,\mathcal C_2\) or \(\mathcal
C_3\) the resulting operator is respectively a logical
\(X_{A/B},\,Z_{A/B}\) or \(Y_{A/B}\) (up to \(S_X,S_Z\)),
all of which map \(|\Psi^{+}\rangle_{AB}\) to an orthogonal Bell
state. The logical fidelity is then zero.
Therefore
\[
F_{\rm logical}(p)
=\Pr[\,\text{category } \mathcal C_0\mid\text{succ}\,]
=\frac{p_0^{\,2}}
{p_0^{\,2}+3p_1^{\,2}}
=\frac{\bigl(1-\frac{4}{5}p\bigr)^{2}}
{\bigl(1-\frac{4}{5}p\bigr)^{2}+\frac{16}{75}p^{2}} .
\]
- Small-\(p\) expansion
\[
F_{\rm logical}(p)=1-\frac{16}{75}p^{2}+O(p^{3}),
\]
i.e. all undetected faults are second-order in the physical error
probability, as expected for a distance-2 detection code.
2. Mathematical Typesetting
All equations have been given in LaTeX.
3. Conventions and Units
The gate error rate \(p\) is dimensionless, as is the fidelity.
4. Final Answer
Final Answer:
\[
\boxed{\,F_{\text{logical}}(p)=
\dfrac{\left(1-\dfrac{4}{5}p\right)^{2}}
{\left(1-\dfrac{4}{5}p\right)^{2}
+\dfrac{16}{75}\,p^{2}} }\,
\approx 1-\dfrac{16}{75}p^{2}+O(p^{3}).
\]