To determine the physical state fidelity of the final 4-qubit state at the end of the circuit, we need to assess the impact of errors at each stage of the provided quantum circuit. The circuit is given by:

We’ll analyze the circuit step-by-step, considering the effects of the two-qubit depolarizing error channel after each CNOT gate. Recall that each two-qubit depolarizing channel acts as the identity channel with probability \(1 - p\) and applies one of the 15 non-identity two-qubit Pauli operations (over two qubits) each with probability \(p/15\).

Step-by-Step Analysis:

1. Initial State - Begin with the state \( |0000\rangle \). - This state is not affected by errors initially since no operations have been performed yet.

2. Apply Hadamard Gate \( H_2 \) - Transformation: \( |0000\rangle \xrightarrow{H_2} \frac{1}{\sqrt{2}}(|0000\rangle + |0010\rangle) \).

3. Apply \( CNOT_{21} \) and Error Channel - \( CNOT_{21} \) acts on the previously prepared superposition:

• After this, the two-qubit depolarizing channel could induce an error. Let the error operator be \( E_{21} \) which is one of the Pauli errors with probability \( p/15 \) or identity with probability \( 1 - p \).

4. Apply Hadamard Gate \( H_0 \) - Transformation due to Hadamard on qubit 0:

5. Apply \( CNOT_{03} \) and Error Channel - \( CNOT_{03} \) transforms the state:

• Apply another error channel here. Let the error operator be \( E_{03} \).

6. Account for Errors in Final State - Each error \( E_{21} \) and \( E_{03} \) has a probability \( p \) of causing deviation from the intended logical state. - The total process can thus change the prepared state’s fidelity. An error-free logical GHZ state corresponds to \( \frac{1}{\sqrt{2}}(|00\rangle_{AB} + |11\rangle_{AB}) \).

Calculating Physical State Fidelity

The fidelity \( F \) of the actual, error-affected state with respect to the target state is defined by:

where \(\text{GHZ}_\text{target}\) is the error-free final state and \(\rho_\text{actual}\) is the actual state density matrix influenced by errors.

Errors occur with probability \( p \), and the GHZ state can tolerate only one qubit-flip error. Here, intermediate errors between the actual logical states can lead to logical state transitions (logical error) or leakage outside the logical space.

When computing fidelity considering these, the major check is to think of which errors can be tolerated to remain within an acceptable state near logical GHZ:

where \(k\) is the number of \(CNOT\) gates, which have associated errors.

Final Simplified Fidelity:

Final Answer: The physical state fidelity of the final physical 4-qubit state is approximately \( (1-p)^2 \), assuming all potential errors occur independently and are uncorrelated over the circuit.

For subproblem 2, we want to calculate the logical state fidelity of the final logical 2-qubit state, given that we post-select on all detectable errors in the [[4,2,2]] code.

Steps to Determine Logical Fidelity

1. Understand the Post-Selection Strategy: - The stabilizers for this code are \( XXXX \) and \( ZZZZ \). Post-selection on detectable errors means measuring these stabilizers and discarding outcomes where the measurement indicates a deviation from the expected eigenvalues (+1) of the stabilizers. This ensures only states without detectable errors are considered.

2. Post-Selection Considerations: - Non-detectable errors within this setup are logically equivalent operations or errors correlated with the logical operators for which we have no correction capabilities. - Detectable errors (syndrome ±1) are filtered out, which ideally leaves logical operators unaffected.

3. Quantum Operations and Errors: - As before, the sequence of gates performed is:

• Each \( CNOT \) introduces an error \( E_{21} \) and \( E_{03} \) as before.

4. Logical Fidelity Calculation: - We define logical fidelity \( F_{\text{logical}} \) as the overlap between the ideal logical GHZ state \(|\text{GHZ}_\text{ideal}\rangle = \frac{1}{\sqrt{2}}(|00\rangle_{AB} + |11\rangle_{AB})\) and the actual prepared state post-selected on no detectable errors.

5. Logical Error Tolerance: - The code can, at best, detect but not correct a single error. Thus any errors altering logical information must be performed in ways that commute with stabilizers to go undetected. - At low error probability \( p \), errors producing the wrong logical state or logical errors (inducing errors in logical operators) are second order or higher in \( p \).

6. Approximating Logical Fidelity Post-Selection: - With all detected errors mitigated by post-selection, assume every logical error is due to undetectable errors. - The logical fidelity after post-selection is more resilient simply by design choice, ensuring that detectable errors are discarded.

Considering the above steps, logical fidelity primarily results from cases where any present errors are logically equivalent errors, which arise with probability on the order of \( O(p^2) \), since single CNOT-induced errors are detectable by the stabilizers and will be discarded.

Calculation Steps

The logical fidelity, after correcting for detectable errors through post-selection, largely depends on the probability of undetected errors which is much lower. We assume: - If all two-qubit gates are error-free (probability \((1-p)^2\)), then the system remains in a perfect logical state. - Logical errors undetected by post-selection arise with a probability approximately proportional to \( p^2 \).

Hence, the logical fidelity is given approximately by:

Considering logical errors are suppressed and mostly second-order correlated:

The precise contribution of \( O(p^2) \) would depend on additional correlation structures in the error model not explicitly detailed here.

Final Answer: The logical state fidelity of the final 2-qubit logical state post-selected on detectable errors is approximately \( F_{\text{logical}} \approx 1 - O(p^2) \).

To calculate the logical state fidelity for preparing a logical two-qubit \(|00\rangle_{AB}\) state in the \([[4,2,2]]\) code using the given fault-tolerant circuit with an ancilla qubit, we need to evaluate how the circuit structure and error detection strategy enhance the fidelity.

Circuit Explanation

The circuit is designed to prepare the logical state \(|00\rangle_{AB}\) and uses these steps:

This is interpreted as:

1. \( H_1 \) - Hadamard on qubit 1 to create superposition.
2. \( CNOT_{12} \) - Control qubit 1, target qubit 2.
3. \( CNOT_{10} \) - Control qubit 1, target qubit 0.
4. \( CNOT_{23} \) - Control qubit 2, target qubit 3.
5. \( CNOT_{34} \) - Control qubit 3, target qubit 4 (ancilla).
6. \( CNOT_{04} \) - Control qubit 0, target qubit 4 (ancilla).
7. \( M_4 \) - Measurement of ancilla qubit 4, which decides the success of preparation (success if measured \(|0\rangle\)).

Error Model

Each \( CNOT \) gate after execution can undergo a two-qubit depolarizing error characterized by Pauli operators with probability \( p/15 \) each, or no error with probability \( 1 - p \).

Preparation Logic

• Ancilla Strategy: The ancilla qubit 4 is used to capture error syndromes. It measures outcomes based on parities, discriminating between successful preparations (outcome \(|0\rangle\)) and failures (\(|1\rangle\)).
• Post-Selection: Only instances of outcome \(|0\rangle\) are considered for the succeeding logical fidelity, inherently filtering detectable errors.

Logical Fidelity Determination

1. Error Propagation:

   • We propagate through each \( CNOT \) gate:
   • \( CNOT_{34} \) and \( CNOT_{04} \) use ancilla to capture potential errors after propagating errors in qubits 0 and 3.

2. Detection and Correction:

   • The ancilla effectively simulates extra redundancy, aligning with stabilizer truths pre-determined for \(|00\rangle_{AB}\).

3. Measurement \( M_4 \) Post-Selection:

   • If the ancilla measures \(|0\rangle\), it implies no detectable error has occurred in qubits that could project outside \(|00\rangle_{AB}\).
   • This measurement already defines correctable errors removed logically through non-commutation avoided within the ancilla.

4. Computing Logical Fidelity:

   • Post-selection plus ancilla enhances the fidelity by reducing effective single-qubit error outcomes and ensuring fault-tolerance sustains continued success.

5. Logical Error Rate:

   • Logical fidelity is affected by undetectable errors (not caught by ancilla post-selection). The overall success assuming consecutive \((1-p)\) qualitative outcomes through logic gates significantly bolsters intended outcomes.

For small \( p \), the total logical fidelity \( F_{\text{logical}} \) upon this construction is approximately:

where higher orders comprise errors both undetected by stabilizer syndrome and ancilla measurement in tolerances beyond individual concatenated \( CNOT \) gates computation.

Final Answer: The logical state fidelity of the final 2-qubit logical state, with post-selection on all detectable errors in the code and on the ancilla qubit measuring \(|0\rangle\), is approximately \( F_{\text{logical}} \approx (1-p)^4 + O(p^2) \).

Here is the populated code based on the derived logical state fidelity.

import sympy as sp

p = sp.symbols('p')

def answer(p):
    r"""
    Return the expression of the logical state fidelity of the final 2-qubit logical state
    at the end of the circuit as a function of two-qubit gate error rate $p$.

    Inputs
    ----------
    p: sympy.Symbol, two-qubit gate error rate, $p$

    Outputs
    ----------
    F_logical: sympy.Expr, logical state fidelity of the final 2-qubit logical state
    """

    # ------------------ FILL IN YOUR RESULTS BELOW ------------------
    F_logical = (1 - p)**4  # a SymPy expression of inputs
    # ---------------------------------------------------------------

    return F_logical