- Step-by-Step Derivation
\[
H_1 \;\xrightarrow{\;CNOT_{1\to 2}\;}\;\xrightarrow{\;CNOT_{1\to 0}\;}\;\xrightarrow{\;CNOT_{2\to 3}\;}\;\xrightarrow{\;CNOT_{3\to 4}\;}\;\xrightarrow{\;CNOT_{0\to 4}\;}\; M_4,
\]
with data qubits 0–3 and ancilla 4. A two-qubit depolarizing channel acts after each CNOT. Thus there are five independent error locations, one after each CNOT. After a given CNOT on qubits $(i,j)$, the error is one of the 15 non-identity two-qubit Paulis on $(i,j)$, each with probability $p/15$; with probability $1-p$ it is the identity.
-
Because all gates are Clifford, any sequence of Pauli faults through the circuit is equivalent to a single Pauli \(P_{\text{tot}}\) on the 5 qubits immediately before measurement.
-
Acceptance (post-selection) consists of:
1) Ancilla measurement in \(Z\) with outcome “0”: this occurs if and only if the ancilla component of \(P_{\text{tot}}\) is in \(\{I,Z\}\) (since \(X\) or \(Y\) on the ancilla flips the \(Z\) measurement outcome).
2) “No code-syndrome”: the data (qubits 0–3) component of \(P_{\text{tot}}\) must commute with both stabilizers \(XXXX\) and \(ZZZZ\). In Pauli-vector terms, this is equivalent to requiring that the total number of \(X/Y\) on data is even and the total number of \(Z/Y\) on data is even.
-
Among accepted events, the data operator lies in the normalizer of the stabilizer, i.e. it is equivalent to some logical Pauli (times possibly a stabilizer). With the given logicals
\[
X_A=X\,I\,X\,I,\quad X_B=X\,X\,I\,I,\quad Z_A=Z\,Z\,I\,I,\quad Z_B=Z\,I\,Z\,I,
\]
the prepared logical state $|00\rangle_{AB}$ is stabilized by $Z_A$ and $Z_B$; hence any accepted error equivalent to a logical $Z$ (or product of logical $Z$’s, possibly times stabilizers) leaves the state unchanged, while any accepted error whose logical part contains $X_A$ and/or $X_B$ maps $|00\rangle_{AB}$ to an orthogonal logical basis state.
- Therefore, the logical fidelity conditioned on acceptance equals the conditional probability that the net (accepted) logical operator contains no \(X\) on either \(A\) or \(B\):
\[
F_L(p)=\frac{P\{\text{accepted and no logical }X\}}{P\{\text{accepted}\}}.
\]
\[
X_c\mapsto X_c X_t,\quad Z_t\mapsto Z_c Z_t,\quad X_t\mapsto X_t,\quad Z_c\mapsto Z_c,
\]
and $Y=iXZ$ follows from these.
- Ancilla acceptance depends only on the ancilla component of \(P_{\text{tot}}\):
\[
\text{accept ancilla} \iff P_{\text{tot}}|_4\in\{I,Z\}.
\]
- Code acceptance depends on parities on data qubits:
\[
\text{accept code} \iff \#(X/Y)\equiv 0\ (\text{mod }2)\ \text{and}\ \#(Z/Y)\equiv 0\ (\text{mod }2)\ \text{on qubits }0\text{–}3.
\]
- Presence of a logical \(X\) component can be detected by commutation with \(Z_A\) and \(Z_B\). For any accepted Pauli \(P\) on data, \(P\) has a logical \(X\) on \(A\) iff it anticommutes with \(Z_A\); similarly for \(B\):
\[
\text{$X_A$ present} \iff \{P,Z_A\}=0,\qquad \text{$X_B$ present} \iff \{P,Z_B\}=0.
\]
\[
\begin{aligned}
&(1,\ XX)\ \to\ XXXX I,\\
&(1,\ YY)\ \to\ XYYX I,\\
&(1,\ ZZ)\ \to\ IZZII,\\
&(2,\ ZZ)\ \to\ ZZIII,\\
&(3,\ ZZ)\ \to\ IIZZI,\\
&(4,\ ZZ)\ \to\ ZIIZZ,\\
&(5,\ IZ)\ \to\ IIIIZ,
\end{aligned}
\]
where the string shows the resulting $P_{\text{tot}}$ on qubits 0–4. All 7 correspond to a stabilizer or a product of logical $Z$’s (possibly times stabilizers), hence are benign for $|00\rangle_{AB}$. Therefore there are no accepted malignant single faults:
\[
\Rightarrow\ \text{no }O(p)\text{ logical infidelity}.
\]
-
Second order (two-fault) analysis and beyond
-
Two-fault patterns (errors on two distinct CNOTs) occur with probability \(\sim (p/15)^2\) and can combine to produce accepted logical-\(X\) errors. We exhaustively enumerated all \(10\times 15^2=2250\) ordered two-fault patterns and found:
\[
\#\{\text{accepted}\}=282,\qquad \#\{\text{accepted and benign}\}=138,\qquad \#\{\text{accepted and malignant}\}=144.
\]
Thus the leading (quadratic) logical infidelity is governed by the 144 malignant, accepted pairs.
- To obtain the exact dependence on \(p\) (not only to leading order), we counted, for \(k=0,\dots,5\), the number of \(k\)-fault patterns that are accepted, and among those, how many are benign. Denote these counts by c_acc[k] and c_good[k]. The exhaustive enumeration yields:
\[
\begin{aligned}
&\text{Total patterns at }k\text{ faults: }[1,75,2250,33750,253125,759375],\\
&\text{Accepted counts }c_{\text{acc}}[k]=[1,7,282,4222,31637,94923],\\
&\text{Accepted and benign }c_{\text{good}}[k]=[1,7,138,1198,7397,24027].\\
\end{aligned}
\]
- From these counts, the acceptance probability and the “accepted-and-benign” probability are exact polynomials of degree 5 in \(p\):
\[
\begin{aligned}
P_{\text{acc}}(p)
&=\sum_{k=0}^{5} c_{\text{acc}}[k]\,(1-p)^{5-k}\Big(\frac{p}{15}\Big)^k\\
&=1-\frac{68}{15}p+\frac{704}{75}p^2-\frac{32768}{3375}p^3+\frac{253952}{50625}p^4-\frac{262144}{253125}p^5,\\[4pt]
P_{\text{good}}(p)
&=\sum_{k=0}^{5} c_{\text{good}}[k]\,(1-p)^{5-k}\Big(\frac{p}{15}\Big)^k\\
&=1-\frac{68}{15}p+\frac{656}{75}p^2-\frac{29312}{3375}p^3+\frac{223232}{50625}p^4-\frac{229376}{253125}p^5.\end{aligned}
\]
- The logical state fidelity conditioned on acceptance is therefore the exact rational function
\[
F_L(p)=\frac{P_{\text{good}}(p)}{P_{\text{acc}}(p)}.
\]
- Small-\(p\) expansion: Since there are no accepted malignant single faults, the linear term cancels in the ratio. Expanding,
\[
\begin{aligned}
F_L(p)&=1-\frac{16}{25}p^2-\frac{704}{375}p^3-\frac{10496}{3375}p^4-\frac{216064}{84375}p^5+O(p^6)\\
&=1-0.64\,p^2-1.877\overline{3}\,p^3-3.11037\,p^4-2.56076\,p^5+O(p^6).
\end{aligned}
\]
In particular, the leading coefficient is
\[
\frac{16}{25}=\frac{\#\{\text{malignant accepted pairs}\}}{15^2\binom{5}{2}}=\frac{144}{225}.
\]
-
Physical interpretation
-
The ancilla parity check (CNOTs from qubits 3 and 0 into the ancilla) “flags” the dangerous single “hook” faults that would otherwise produce undetected weight-2 \(X\)-type errors on the cat state used for \(|00\rangle_{AB}\) preparation. Consequently, all accepted single faults are purely \(Z\)-type (logical \(Z\)’s or stabilizers), leaving \(|00\rangle_{AB}\) unchanged. The first harmful events require two faults, so the conditional logical infidelity scales as \(O(p^2)\) with coefficient \(16/25\).
- Mathematical Typesetting
All equations above are in LaTeX.
- Conventions and Units
- Code qubits indexed 0–3, ancilla 4.
- Stabilizers: \(XXXX\) and \(ZZZZ\).
- Logical operators: \(X_A=X\,I\,X\,I\), \(X_B=X\,X\,I\,I\), \(Z_A=Z\,Z\,I\,I\), \(Z_B=Z\,I\,Z\,I\).
- Two-qubit depolarizing error after each CNOT: non-identity two-qubit Pauli with probability \(p/15\).
- Final Answer:
The logical state fidelity of the post-selected output (conditioning on both “no code syndrome” and ancilla outcome \(|0\rangle\)) is the exact rational function
\[
F_L(p)
=\frac{1-\frac{68}{15}p+\frac{656}{75}p^2-\frac{29312}{3375}p^3+\frac{223232}{50625}p^4-\frac{229376}{253125}p^5}
{1-\frac{68}{15}p+\frac{704}{75}p^2-\frac{32768}{3375}p^3+\frac{253952}{50625}p^4-\frac{262144}{253125}p^5}.
\]
For small \(p\) this expands to
\[
F_L(p)=1-\frac{16}{25}p^2+O(p^3)=1-0.64\,p^2+O(p^3).
\]
In particular, there is no \(O(p)\) logical error (the circuit is fault-tolerant for this state preparation), and the leading logical infidelity is \(\frac{16}{25}p^2\).