- Step-by-Step Derivation
- Ideal target state and circuit structure
Starting from |0000⟩ on data qubits (0,1,2,3) and ancilla |0⟩ on qubit 4, the noiseless circuit
H1 → CNOT12 → CNOT10 → CNOT23 → CNOT34 → CNOT04 → M4
prepares the 4-qubit GHZ codeword
\[
|\psi_{\mathrm{id}}\rangle = \frac{|0000\rangle+|1111\rangle}{\sqrt2}
\]
(this is the [[4,2,2]] code’s logical |00⟩AB) and yields ancilla outcome 0 deterministically.
- Noise model and acceptance
After each CNOT, a two-qubit depolarizing channel acts on that gate’s qubit pair:
\[
\mathcal{E}(\rho)=(1-p)\rho+\frac{p}{15}\sum_{P\in\mathcal{P}_2\setminus\{II\}}P\rho P.
\]
There are 5 CNOTs (12,10,23,34,04), so 5 such error locations. We post-select on:
1) ancilla measurement Z4=+1 (i.e., no X4/Y4 on the final error), and
2) code-detectability: the final data Pauli commutes with both XXXX and ZZZZ.
- Pauli propagation and acceptance invariants
Let a,b,c be the three binary invariants of a Pauli at the end of the circuit:
- a := x4 (does the final Pauli anticommute with Z4? a=1 flips ancilla result ⇒ reject),
- b := x0⊕x1⊕x2⊕x3 (anticommutation with ZZZZ),
- c := z0⊕z1⊕z2⊕z3 (anticommutation with XXXX).
Acceptance requires a=b=c=0.
For logical action, define the logical-Z flip indicators
- cA := x0⊕x1 = anticommutes with Z_A=ZZII,
- cB := x0⊕x2 = anticommutes with Z_B=ZIZI.
Conditioned on code acceptance, a nontrivial logical error occurs iff (cA,cB) ≠ (0,0).
- Single-fault analysis (propagation and classification per gate)
Conjugating a general two-qubit Pauli inserted after each gate through the suffix of the circuit yields (a,b,c,cA,cB) as functions of the local exponents. One finds:
1) After CNOT12 (gate 1 on qubits 1,2):
\[
a=x_1\oplus x_2,\quad b=0,\quad c=z_1\oplus z_2,\quad c_A=0,\quad c_B=a.
\]
Among the 15 non-identity two-qubit Paulis, the counts per (a,b,c) class are:
- b=0, a=0, c=0: 3 elements (XX,YY,ZZ),
- b=0, a,c otherwise: 4 elements each;
- b=1: 0.
Single-fault acceptance requires (a,b,c)=(0,0,0), i.e., XX/YY/ZZ only (3/15). These give only stabilizers (no logical flip since cA=cB=0).
2) After CNOT10 (gate 2 on 1,0):
\[
a=x_0,\quad b=x_0\oplus x_1,\quad c=z_0\oplus z_1,\quad c_A=b,\quad c_B=a.
\]
For every (a,b,c) except (0,0,0), there are 2 Paulis; for (0,0,0) there is 1 (ZZ). Single-fault acceptance: only ZZ (1/15), no logical flip.
3) After CNOT23 (gate 3 on 2,3):
\[
a=x_3,\quad b=x_2\oplus x_3,\quad c=z_2\oplus z_3,\quad c_A=0,\quad c_B=a\oplus b.
\]
Same counting as gate 2; single-fault acceptance: only ZZ (1/15), no logical flip.
4) After CNOT34 (gate 4 on 3,4):
\[
a=x_4,\quad b=x_3,\quad c=z_3\oplus z_4,\quad c_A=0,\quad c_B=0.
\]
Same counting; single-fault acceptance: only Z3Z4 (1/15), which induces Z0Z3 on data (GHZ stabilizer), no logical flip.
5) After CNOT04 (gate 5 on 0,4):
\[
a=x_4,\quad b=x_0,\quad c=z_0,\quad c_A=b,\quad c_B=b.
\]
Same counting; single-fault acceptance: only Z4 (1/15), no logical flip.
Conclusion: Every accepted single fault yields only a stabilizer on the target state (cA=cB=0), so first-order logical errors are fully eliminated by the ancilla + code post-selection (fault tolerance).
- Linear term in acceptance probability
Let gℓ be the number of accepted non-identity Paulis at location ℓ:
g1=3, and g2=g3=g4=g5=1. Then, to first order,
\[
=1-\left[\sum_{\ell}\left(1-\frac{g_\ell}{15}\right)\right]p+O(p^2)
\]
- Two-fault accepted events and malignant pairs
For two faults at locations ℓ<m, acceptance requires their (a,b,c) classes be identical; the number of accepted pairs for the pair (ℓ,m) is
\[
S_{\ell m}=\sum_{(a,b,c)} k_\ell(a,b,c)\,k_m(a,b,c),
\]
where kℓ(a,b,c) is the non-identity count in that class for gate ℓ (given above). Summing over all unordered pairs gives
\[
\sum_{\ell<m} S_{\ell m}=4\times 27+6\times 29=282.
\]
The logical effect for a two-fault accepted event is determined by
\[
\]
which depend only on (a,b) and the gate types. A pair is malignant if (f_A,f_B)≠(0,0). Evaluating all 10 gate pairs yields the malignant pair counts
\[
(M_{\ell m}) = (0,0,16,16,16,24,16,16,24,16),\quad \sum_{\ell<m}M_{\ell m}=144.
\]
- Quadratic terms (acceptance and logical error probabilities)
Keeping terms through O(p^2):
- Acceptance probability
\[
\]
- Probability of accepted logical error
\[
\]
- Logical fidelity conditioned on acceptance
The output logical state fidelity (post-selected on acceptance) is
\[
=1-\frac{P_{\mathrm{log\,err}\;\cap\;\mathrm{acc}}(p)}{P_{\mathrm{acc}}(p)}
\]
As expected for a fault-tolerant, post-selected preparation, the first nonzero term in the logical infidelity is O(p^2).
-
Mathematical Typesetting
- All expressions are in LaTeX.
-
Conventions and Units
- Standard qubit/Pauli conventions; no physical units.
-
Final Answer:
\(F_{\mathrm{logical}}(p)=1-\dfrac{\frac{16}{25}\,p^2}{1-\frac{68}{15}\,p+\frac{844}{75}\,p^2}\;+\;O(p^3)\,.\)