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 solve this problem, we need to analyze the given quantum state preparation circuit and compute the logical fidelity of the final two-qubit logical state. This involves understanding the effects of errors introduced by the quantum gates, especially the two-qubit errors following the $CNOT$ gates, and assessing the impact of these errors on the encoded logical state when post-selected successfully.

Step-by-Step Derivation:

Identify the Nature of Errors and Post-Selection: - Each $CNOT_{ij}$ gate in the circuit is followed by a two-qubit depolarizing error channel, which can apply one of the 15 non-identity two-qubit Pauli errors ($XX, XY, XZ, YX, \ldots, ZZ$) with equal probability $p/15$. This means error occurs with total probability $p$ after each gate.
Logical State Preparation Circuit: - Initial state is $|0000\rangle := |0\rangle^{\otimes 4}$. - Apply the circuit:

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

The ancilla (qubit 4) starts in $|0\rangle_4$.

Error-Free State Preparation: - Prepare logical state $|00\rangle_{AB}$ if no errors have occurred:
- Hadamard $H_1$ transforms qubit 1: $|0\rangle \rightarrow \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$.
- $CNOT_{12}$ creates entanglement between qubits 1 and 2.
- Subsequent $CNOT$ gates create specific multi-qubit entanglement as directed by the gate sequence.
Understanding Error Detection Post-Selection: - We post-select on measurements that indicate no detectable error — ancilla measurement $|0\rangle$ and stabilizers $XXXX, ZZZZ$ not detecting errors. - Detectable errors are those that flip the ancilla measurement or violate stabilizers, hence certain errors on qubits 0-3 can remain undetected if they do not affect the stabilizer checks or ancilla outcome.
Calculate the Probability of No Error Impact on Logical State: - Since each $CNOT$ is followed by an error channel, the probability no errors occur after a gate is $1-p$. - Given ancilla error detection, some errors might be detectable and lead to post-selection failure, thereby being excluded from successful state preparation.
Logical Fidelity Calculation: - Logical fidelity $F_{\text{logical}}$ post-selects on no detectable error. Successful post-selection implies reduced error impact. - Depolarizing channel: Reduces state fidelity by $1 - \frac{4p}{15}$ if one error occurs per gate. - If $n$ gates have errors, the final fidelity in post-selected cases roughly follows:

\[ F_{\text{logical}} \approx (1 - 4p/15)^n \]

Given that errors compound across $m$ gates but most undetected errors (that pass post-selection) happen with tiny probabilities, the post-selection on unmeasured ancilla state reduces likelihood of impacting the logical state.

Assuming perfect error detection capability, post-selection results in: - The probability one error is detected (and state discarded) and undetected errors’ impact diminish the logical state fidelity based on $n$ gates with probability $p$, - Applying error detection (ancilla and stabilizers) we refine our fidelity function into consideration of detectable and non-detectable errors.

\[ F_{\text{logical}} = 1 - \kappa p \]

where $\kappa$ is an effective parameter capturing influence of error strategies and post-selection, different for no gate-specific error, relying on accurate post-selection.

Final Answer: The logical state fidelity $F_{\text{logical}}$ as a function of two-qubit gate error rate $p$, assuming all detectable errors are post-selected out, is:

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

In practice, small $\mathcal{O}(p^2)$ terms might develop due to concatenated impacts in quantum coding, detected specifically via rigorous stabilizer checks and projection post-selection. This estimate assumes the detection protocol can correctly identify any anomalous transformations of the logical basis, conditioned on detecting ancilla $|0\rangle$.

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 ------------------
    # Expression for the logical state fidelity
    F_logical = 1 - sp.O(p**2) 
    # ---------------------------------------------------------------

    return F_logical