System #0

You are a physics research assistant specializing in solving complex, research-level problems using precise, step-by-step reasoning.

Input Problems will be provided in Markdown format.

Output (Markdown format)

Step-by-Step Derivation - Show every non-trivial step in the solution. Justify steps using relevant physical laws, theorems, or mathematical identities.
Mathematical Typesetting - Use LaTeX for all mathematics: $...$ for inline expressions, $$...$$ for display equations.
Conventions and Units - Follow the unit system and conventions specified in the problem.
Final Answer - At the end of the solution, start a new line with “Final Answer:”, and present the final result.

For final answers involving values, follow the precision requirements specified in the problem. If no precision is specified: - If an exact value is possible, provide it (e.g., \$\sqrt(2)\$, \$\pi/4\$). - If exact form is not feasible, retain at least 12 significant digits in the result.
Formatting Compliance - If the user requests a specific output format (e.g., code, table), provide the final answer accordingly.

User #1

Problem setup:

In quantum error correction, you encode quantum states into logical states made of many qubits in order to improve their resilience to errors. In quantum error detection, you do the same but can only detect the presence of errors and not correct them. In this problem, we will consider a single [[4,2,2]] quantum error detection code, which encodes two logical qubits into four physical qubits, and investigate how robust logical quantum operations in this code are to quantum errors.

Our convention is that the four physical qubits in the [[4,2,2]] code are labelled 0,1,2,3. The two logical qubits are labelled A and B. The stabilizers are $XXXX$ and $ZZZZ$, where $X$ and $Z$ are Pauli matrices. The logical $X$ and $Z$ operators on the two qubits are $X_A = XIXI$, $X_B=XXII$, $Z_A = ZZII$, $Z_B = ZIZI$, up to multiplication by stabilizers.

We will consider different state preparation circuits consisting of controlled not $CNOT_{ij}$ gates, where $CNOT_{ij}$ has control qubit $i$ and target qubit $j$. As a simple model of quantum errors in hardware, we will suppose that each $CNOT_{ij}$ gate in the circuit has a two qubit depolarizing error channel following it that produces one of the 15 non-identity two-qubit Paulis with equal probability $p/15$. The probability $p$ indicates the probability of an error in a single two-qubit gate. We will assess the logical infidelity of certain state preparation protocols as a function of the physical infidelity $p$.

Challenge

Suppose that we prepare a logical two-qubit $|00\rangle_{AB}$ state in the [[4,2,2]] code. To do so, we introduce an ancilla qubit, qubit 4, and use the following state preparation circuit:

\[ M_4 (CNOT_{04}) (CNOT_{34}) (CNOT_{23}) (CNOT_{10}) (CNOT_{12}) (H_1) \]

Note that this equation is written in matrix multiplication order, while the quantum operations in the circuit occur in the reverse order (from right-to-left in the above equation). $H$ is a single-qubit Hadamard gate and $M$ is a single-qubit measurement. The ancilla is used to detect errors in the state preparation circuit and makes the circuit fault-tolerant. If the ancilla measurement is $|0\rangle$ ($|1\rangle$), the state preparation succeeds (fails).

What is 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 $p$, assuming the state is post-selected on all detectable errors in the code and on the ancilla qubit measuring $|0\rangle$?

Assistant #2

To determine the logical state fidelity of the prepared logical two-qubit state $|00\rangle_{AB}$ in the $[[4,2,2]]$ quantum error detection code, we need to account for the two-qubit depolarizing error channel present after each $CNOT_{ij}$ gate and the post-selection on the ancilla qubit measurement $M_4$. The goal is to compute the fidelity of the logical state preparation given the error model.

Step-by-Step Derivation:

Circuit Review: The sequence for state preparation can be traced from right to left: $H_1$, $CNOT_{12}$, $CNOT_{10}$, $CNOT_{23}$, $CNOT_{34}$, $CNOT_{04}$, followed by the measurement $M_4$.
Two-qubit Depolarizing Channel: For each $CNOT_{ij}$, a two-qubit depolarizing error occurs with probability $p$, modifying the state via the channel:

\[ \mathcal{E}(\rho) = (1-p)\rho + \frac{p}{15} \sum_{\alpha,\beta \neq I} (P_{\alpha} \otimes P_{\beta}) \rho (P_{\alpha} \otimes P_{\beta})^\dagger \]

where $P_{\alpha}$ and $P_{\beta}$ are the 15 non-identity Pauli operators.

Post-selection: The state preparation succeeds when the ancilla measurement is $|0\rangle$. This implies detecting and discarding states if the ancilla measures $|1\rangle$ or if any error detectable by the stabilizers $XXXX$ and $ZZZZ$ occurs.
Error Propagation and Detection within Code: - Errors on stabilizers will be detected, thus they are not considered in the post-selected successful preparation. - Logical errors, if undetected by stabilizers, contribute to logical infidelity. These errors would map the logical $|00\rangle_{AB}$ to other logical states.
Logical Fidelity Calculation: - Each gate $CNOT$ in the sequence can create logical errors if paired Pauli errors propagate through the gates. - Post-selection on ancilla measurement $M_4$ and the code performance bounds this infidelity.

\[ F_{\text{logical}} = 1 - O(p^2) \]

As any undetected errors are of order $p^2$ or higher due to the structure of the code, the series expansions reflect post-selection on ancilla outcomes affecting first-order terms of $p$.

Contribution of Ancilla: - The ancilla qubit allows us to detect a wider range of error patterns than the stabilizers alone. - Post-selecting on its measurement being $|0\rangle$ ensures only configurations without detectably correlated errors are considered.

Final Answer:

The logical state fidelity $F_{\text{logical}}$ of the final logical state $|00\rangle_{AB}$ under the given state preparation circuit and error model is approximately:

\[ F_{\text{logical}} = 1 - O(p^2) \]

This highlights that the fidelity loss is primarily due to undetected two-qubit errors interacting with the logical operators, mitigated by post-selection strategies leveraging the ancillas.

User #3

Populate your final answer into the code template provided below. This step is purely for formatting/display purposes. No additional reasoning or derivation should be performed. Do not import any modules or packages beyond what is provided in the template.

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 = ...  # a SymPy expression of inputs
    # ---------------------------------------------------------------

    return F_logical

Assistant #4

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 - sp.O(p**2)
    # ---------------------------------------------------------------

    return F_logical