To determine the physical state fidelity of the final physical 4-qubit state, we need to understand how errors from each \(CNOT\) gate contribute to the overall fidelity. Here, we will derive the fidelity through each step of the preparation circuit.

Step 1: Circuit Operations and Error Model

1. Circuit Description: - The circuit from right to left is: \(H_2\), \(CNOT_{21}\), \(H_0\), \(CNOT_{03}\).

2. Hadamard Gate Operations: - Before analyzing errors, consider how the \(H\) transforms the state.

   • \(H_0\) converts \(|0\rangle\) to \((|0\rangle + |1\rangle)/\sqrt{2}\).
   • \(H_2\) converts \(|0\rangle\) to \((|0\rangle + |1\rangle)/\sqrt{2}\).
   • Apply \(H_2\) first, then \(CNOT_{21}\), followed by \(H_0\), and finally \(CNOT_{03}\).

3. Error Model for \(CNOT\) gates: - After each \(CNOT_{ij}\), there’s a two-qubit depolarizing error channel which introduces one of 15 non-identity Pauli errors with equal probability \(p/15\).

Step 2: Calculate the Fidelity

1. Initial State: - Begin with the product state of 4 qubits: \(|0000\rangle_{0123}\).

2. Ideal Final State: - The target is the logical GHZ state:

• Multiple \(CNOT\) gates and \(H\) create entangled states acting upon physical qubits.

3. Fidelity Calculation: - Chain of Operations: Apply each operation, with its associated errors. - Fidelity decreases with each \(CNOT\) due to depolarizing error. Assume perfect single-qubit gates (ideal \(H\)). - Total fidelity \(F\) is calculated as the product of fidelity contributions from each \(CNOT\):

• Explain the formula: With each \(CNOT\), successful operation probability is \((1-p)\); square this since two \(CNOT\) gates are involved:

Final Answer:

The physical state fidelity \(F\) of the final 4-qubit state after the circuit given the two-qubit gate error rate \(p\) is:

This quantifies the efficiency of executing this specific circuit under the error model described.

To calculate the logical state fidelity of the final 2-qubit logical state after post-selecting on all detectable errors using the [[4,2,2]] code, we’ll consider how errors within the circuit affect the detection via stabilizers and impact the logical qubits.

Step 1: Understand Error Detection Capabilities

1. Error Detection by Stabilizers: - The stabilizers for the [[4,2,2]] code are \(XXXX\) and \(ZZZZ\). These stabilizers can detect any single-qubit error. - The errors that occur due to the depolarizing channel on the \(CNOT\) gates will be detected if they involve a single-qubit error. The code cannot detect correlated two-qubit errors on different pairs of qubits simultaneously if they do not commute with stabilizers.

2. Circuit Errors: - Each \(CNOT\) introduces a two-qubit depolarizing error with probability \(p\), introducing one of the 15 non-identity Paulis on those qubits. - The probability of introducing an undetectable error is smaller, as most single errors will be detected.

Step 2: Logical Fidelity Calculation

1. Initial State: - Start with \(|0000\rangle_{0123}\).

2. Logical State Preparation: - The desired logical GHZ state is \(\frac{1}{\sqrt{2}}(|00\rangle_{AB} + |11\rangle_{AB})\).

3. Effect of \(H\) and \(CNOT\) with Errors: - \(H\) gates are assumed error-free. - Each \(CNOT\) gate is followed by an error model, producing non-identity two-qubit Pauli errors with probability \(p/15\) for each specific pair.

4. Calculate Post-Selection Probability: - The probability of having no detectable error after both \(CNOT\) operations (corresponding to undetectable errors) is calculated assuming independence:

   • Odds of introducing any detectable error per \(CNOT\): \(14/15 \times p\) for 14 detected two-qubit Pauli errors.
   • Probability of no detectable error (post-selected):

5. Logical Fidelity: - Post-selection eliminates states with detectable errors: Trivial physical errors are detected. - Remaining errors can still affect logical state fidelity. - Assuming perfect correction of all detectable single-qubit errors: logical errors are due to undetected errors not causing detectable syndrome results.

• Logical fidelity of state, approximates:

• Dominated by the lack of error formation or error correction.

Conclusion:

The logical state fidelity \(F_{\text{logical}}\) of the final 2-qubit logical state in the circuit after post-selection, as a function of the error rate \(p\), is approximated as:

Final Answer:

To determine the logical state fidelity of the final 2-qubit logical state in preparation for the \(|00\rangle_{AB}\) state in the [[4,2,2]] code, we must account for the fault-tolerant aspect introduced by the ancilla qubit. This setup uses post-selected success, meaning the ancilla measurement outcome must be \(|0\rangle\), and the logical state must be free from detectable errors.

Step-by-Step Analysis

1. Initial Setup:

   • We initialize qubits 0, 1, 2, 3 to \(|0000\rangle\) and add an ancilla qubit, qubit 4, initialized to \(|0\rangle\).

2. Circuit Description and Operation:

   • Let’s revisit the operations in their application order:
   • \(H_1\): A Hadamard gate applied to qubit 1 transforms it into \((|0\rangle + |1\rangle)/\sqrt{2}\).
   • Apply \(CNOT\) gates sequentially:
     1. \(CNOT_{12}\): Control qubit 1, target qubit 2.
     2. \(CNOT_{10}\): Control qubit 1, target qubit 0.
     3. \(CNOT_{23}\): Control qubit 2, target qubit 3.
     4. \(CNOT_{34}\): Control qubit 3, target qubit 4 (ancilla).
     5. \(CNOT_{04}\): Control qubit 0, target qubit 4 (ancilla).
   • Measure qubit 4 with \(M_4\).

3. Ancilla Error Detection:

   • The ancilla qubit is used to detect correlated errors between the system qubits. \(|0\rangle\) outcome indicates successful detection (no detected errors).

4. Error Model:

   • After each \(CNOT\), apply a depolarizing channel: 15 non-identity two-qubit Pauli errors with probability \(p/15\) each.
   • Ancilla facilitates detecting correlated errors related to \(CNOT_{04}\) and \(CNOT_{34}\).

5. Error Detection and Post-Selection:

   • Measuring the ancilla qubit in the \(|0\rangle\) state guarantees the absence of certain detectable errors.
   • Logical fidelity, \( F_{\text{logical}} \):
   • Probability of no undetectable error being introduced.
   • Include error detection/filter through the stabilizers.

6. Logical State Fidelity:

   • Let’s suppose the optimal functioning of the error-correcting capabilities, as each \(CNOT_{ij}\) operation either succeeds or fails (post-selection filters failure):
   • Probability that each gate does not introduce an error is \((1-p)\).
   • All detectable errors being corrected via the ancilla and the stabilizer becomes:

Here, \((1-p)^3\) accounts for each occurrence of depolarizing events that are undetected and reduces error likelihood. Ancilla eliminates some detectable errors, leading to: - Explanation: This derivation focuses on successful execution without introducing catastrophic errors across the gates.

Conclusion:

By leveraging post-selection on errors detectable through the ancilla qubit and logical code in conjunction with circuit design, the fidelity represents an approximation of how robustly the logical \(|00\rangle_{AB}\) state can be prepared under the described physical error model.

Final Answer:

This calculation aligns with the probability of successful, error-free state preparation as facilitated by error detection through ancillary measurement.

Here’s the completed code template with your specified requirements:

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)**3  # a SymPy expression of inputs
    # ---------------------------------------------------------------

    return F_logical

This code specifies the logical state fidelity calculation in terms of the error rate \( p \) using the symbolic mathematics library SymPy, resulting in an expression \( (1 - p)^3 \) that reflects the logical state fidelity in the described scenario.