1) Step-by-step derivation

• Circuit, noise model, and target state
• The circuit is applied right-to-left: 1) H2 2) CNOT21 3) H0 4) CNOT03
• Only the two CNOTs are followed by two-qubit depolarizing channels acting on the gate’s two qubits. The model states that with probability p an error occurs and, conditioned on an error, one of the 15 non-identity two-qubit Paulis is applied uniformly at random, i.e.,

where $\mathcal{P}_2=\{I,X,Y,Z\}^{\otimes 2}$. This is the “uniform Pauli” form of the 2-qubit depolarizing/Pauli channel. ([qiskit.github.io](https://qiskit.github.io/qiskit-aer/stubs/qiskit_aer.noise.depolarizing_error.html?utm_source=openai), [docs.quantum.ibm.com](https://docs.quantum.ibm.com/api/qiskit/0.42/qiskit_aer.noise.depolarizing_error?utm_source=openai), [qml-tutorial.github.io](https://qml-tutorial.github.io/chapter2/2/?utm_source=openai))

• H followed by CNOT on |00⟩ prepares a Bell state; hence each pair (2,1) and (0,3) is prepared into \(|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt{2}\). Therefore, the ideal final 4-qubit state is a product of two Bell pairs,

([en.wikipedia.org](https://en.wikipedia.org/wiki/Bell_state))

• Factorization of the noisy state
• The first error channel acts on qubits (2,1) immediately after CNOT21. No later gate touches qubits 1 or 2. The second error channel acts on (0,3) after CNOT03. Therefore, the final state factors as a tensor product of two two-qubit states:

with

• Bell-pair survival under a 2-qubit depolarizing/Pauli channel
• For a single pair (say, 21), the fidelity with the ideal Bell state is

• Using the stabilizer properties of \(|\Phi^+\rangle\),

so $|\langle\Phi^+|P|\Phi^+\rangle|^2=1$ for $P\in\{XX,YY,ZZ\}$ and 0 for the remaining 12 non-identity two-qubit Paulis (they map $|\Phi^+\rangle$ to some orthogonal Bell state, e.g., $ZI|\Phi^+\rangle=|\Phi^-\rangle$). Hence,

(The three Paulis XX, YY, ZZ leave the pure density matrix unchanged, since $P|\Phi^+\rangle\langle\Phi^+|P=|\Phi^+\rangle\langle\Phi^+|$ even when the eigenvalue is −1.)

• Four-qubit physical-state fidelity
• The target is a product state \(|\Psi_{\mathrm{ideal}}\rangle=|\Phi^+\rangle_{21}\otimes|\Phi^+\rangle_{03}\) and the noisy state factors: \(\rho_{\text{final}}=\rho_{21}\otimes\rho_{03}\). Using \(\mathrm{Tr}[(A\otimes B)(C\otimes D)]=\mathrm{Tr}[AC]\mathrm{Tr}[BD]\),

2) Mathematical details used - Depolarizing/Pauli channel acting uniformly on non-identity n-qubit Paulis (here n=2) and the standard H–CNOT Bell-state preparation. (qiskit.github.io, docs.quantum.ibm.com, en.wikipedia.org)

3) Conventions and units - Qubits labeled 0,1,2,3 as in the prompt. Gates are ideal except for the specified post-CNOT two-qubit depolarizing channels of error probability p. Fidelity is the state overlap with the ideal pure state, \(F=\langle\psi|\rho|\psi\rangle\).

Final Answer:

1) Step-by-step derivation

• Setup: code, circuit, noise, target
• [[4,2,2]] code with stabilizers S = ⟨XXXX, ZZZZ⟩. A convenient choice of logical Paulis (up to stabilizers) is X_A = X I X I, X_B = X X I I, Z_A = Z Z I I, Z_B = Z I Z I, in the physical-qubit order (0,1,2,3). The codespace basis and stabilizers are standard for this code. (errorcorrectionzoo.org, ar5iv.org)
• The state-preparation circuit acts right-to-left: H2 → CNOT21 → H0 → CNOT03. Ideal action on |0000⟩: H2+CNOT21 prepares |Φ+⟩21; H0+CNOT03 prepares |Φ+⟩03, so the ideal 4-qubit state is |Ψideal⟩ = |Φ+⟩21 ⊗ |Φ+⟩03, where |Φ+⟩ = (|00⟩+|11⟩)/√2. (qubit.guide, en.wikipedia.org)

• Noise model: After each CNOTij, a two-qubit depolarizing (uniform Pauli) channel acts on those two qubits: with probability 1−p nothing happens; with probability p a random non-identity 2-qubit Pauli from the 15 possibilities is applied uniformly (probability p/15 each). For n qubits this is the “uniform Pauli” point of the depolarizing family. (qiskit.github.io)

• One Bell pair under a two-qubit Pauli channel

• The Bell state |Φ+⟩ is stabilized by XX and ZZ, and (up to a global phase) any two-qubit Pauli maps a Bell state to some Bell state. In particular, the four operators {I, XX, YY, ZZ} leave |Φ+⟩ invariant (eigenvalue ±1), while the remaining 12 Paulis map |Φ+⟩ to the other three Bell states in 4-to-1 groupings (cosets). Therefore, for a single depolarized pair: P(Φ+) = (1−p) + 3·(p/15) = 1 − 12p/15, P(Φ−) = P(Ψ+) = P(Ψ−) = 4p/15. (This uses that Bell states are permuted by local/two-qubit Paulis and |Φ+⟩ is an eigenstate of XX and ZZ.) (en.wikipedia.org)

• Independent noise on the two disjoint pairs

• The two error channels act independently on (2,1) and (0,3), and no later gate mixes the pairs, so the joint distribution for the pair-types factorizes.

• Post-selection criterion (detectable errors rejected)

• Measuring the stabilizers XXXX and ZZZZ yields the trivial syndrome (+1,+1) iff the total Pauli error P commutes with both stabilizers, i.e., P lies in the normalizer of S. Equivalently for our two-Bell-pair picture, write the pairwise parities sXX(pair) = eigenvalue of X⊗X on that pair, sZZ(pair) = eigenvalue of Z⊗Z, which for the four Bell states are Φ+: (+,+), Φ−: (−,+), Ψ+: (+,−), Ψ−: (−,−). Since XXXX = (X2X1)(X0X3) and ZZZZ = (Z2Z1)(Z0Z3), the stabilizer outcomes are the products of the corresponding pairwise parities. Hence the run is accepted iff the two pairs have the same Bell label (their (sXX,sZZ) agree componentwise). (quantum.cloud.ibm.com)

• Acceptance probability

• Let q := P(Φ+) = 1 − 12p/15 = 1 − 4p/5 and r := P(Φ−) = P(Ψ+) = P(Ψ−) = 4p/15 for each pair. The acceptance probability is Pacc = q^2 + 3 r^2 = (1 − 4p/5)^2 + 3 (4p/15)^2 = 1 − 8p/5 + (64/75)p^2.

• Which accepted events keep the logical Bell state correct?

• The target logical state is the logical Bell state |Φ+⟩AB, which our physical product |Φ+⟩21|Φ+⟩03 realizes. Accepted events fall into four classes: 1) (Φ+,Φ+): P = P21⊗P03 with Ppair ∈ {I, XX, YY, ZZ}. Using XXXX = (X2X1)(X0X3) and ZZZZ = (Z2Z1)(Z0Z3), one checks that any such P equals a stabilizer times one of {I, X_A X_B, Z_A Z_B, Y_A Y_B}. All of these act as ±1 on |Φ+⟩AB (global phase allowed), so the logical state remains the target. (errorcorrectionzoo.org, ar5iv.org) 2) (Φ−,Φ−): A representative is P = Z2 · Z0, which equals the logical Z_B (up to stabilizers) because Z_B = Z0 Z2. This maps |Φ+⟩AB → |Φ−⟩AB, orthogonal to the target. All choices in this class are equivalent to logical Z_B (times stabilizers), hence “bad” for fidelity. (ar5iv.org) 3) (Ψ+,Ψ+): A representative is P = X2 · X0 = X_A, mapping |Φ+⟩AB → |Ψ+⟩AB (orthogonal). All such are equivalent to logical X_A (times stabilizers), so “bad.” (ar5iv.org) 4) (Ψ−,Ψ−): A representative is P = Y2 · Y0 ∝ X_A Z_B, mapping |Φ+⟩AB → |Ψ−⟩AB (orthogonal). Again “bad.” (ar5iv.org)

• Therefore, among accepted runs, only the (Φ+,Φ+) sector produces the correct logical Bell state.

• Logical fidelity conditioned on post-selection

• The (conditional) logical state fidelity is the probability mass of the (Φ+,Φ+) sector conditioned on acceptance: Flogical(p) = P[(Φ+,Φ+)] / Pacc = q^2 / (q^2 + 3 r^2) = (1 − 4p/5)^2 / [ (1 − 4p/5)^2 + 3 (4p/15)^2 ].
• Series for small p: Flogical(p) = 1 − (16/75) p^2 + O(p^3), reflecting the removal (by post-selection) of all O(p) logical errors. (quantum.cloud.ibm.com)

2) Mathematical identities and facts used - Bell-state properties and preparation by H–CNOT; |Φ+⟩ stabilized by XX and ZZ; Paulis permute the Bell basis. (en.wikipedia.org) - [[4,2,2]] code stabilizers and typical logical representatives used above. (errorcorrectionzoo.org, ar5iv.org) - Error detection accepts exactly errors commuting with all stabilizer generators (normalizer elements). (quantum.cloud.ibm.com) - Uniform two-qubit depolarizing channel (uniform Pauli at the “Pauli point”). (qiskit.github.io)

3) Conventions and units - Qubits labeled (0,1,2,3); logical qubits A,B. Gates ideal except for the specified two-qubit depolarizing channel of strength p applied after each CNOT. Fidelity is the pure-state overlap with the conditional logical state.

4) Final Answer: