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

Step 1: Understand the given problem and the circuit for state preparation

The problem involves preparing a logical two-qubit $|00\rangle_{AB}$ state in the [[4,2,2]] code using a specific circuit and assessing its logical fidelity as a function of the physical infidelity $p$ of two-qubit gates. The circuit given is $M_4 (CNOT_{04}) (CNOT_{34}) (CNOT_{23}) (CNOT_{10}) (CNOT_{12}) (H_1)$, where the operations are applied from right to left.

Step 2: Analyze the circuit and its operation

First, let’s understand the circuit’s operation step by step. The initial state is $|0000\rangle$ (assuming all qubits are initialized to $|0\rangle$), and an ancilla qubit $|0\rangle$ is introduced, making the initial state $|00000\rangle$. The circuit applies $H_1$, creating a superposition on qubit 1.

Step 3: Apply the Hadamard gate $H_1$ to the initial state

Applying $H_1$ to $|00000\rangle$ gives $\frac{1}{\sqrt{2}}(|00000\rangle + |01000\rangle)$.

Step 4: Understand the effect of subsequent $CNOT$ gates

The $CNOT_{12}$ gate applies to $\frac{1}{\sqrt{2}}(|00000\rangle + |01000\rangle)$, resulting in $\frac{1}{\sqrt{2}}(|00000\rangle + |01100\rangle)$. Then, $CNOT_{10}$ is applied, giving $\frac{1}{\sqrt{2}}(|00000\rangle + |01100\rangle)$ because $CNOT_{10}$ does not change the state since the control qubit (1) is $|1\rangle$ only in the second term, but it targets qubit 0 which is $|0\rangle$, so it remains unchanged. $CNOT_{23}$ then gives $\frac{1}{\sqrt{2}}(|00000\rangle + |01110\rangle)$, and $CNOT_{34}$ results in $\frac{1}{\sqrt{2}}(|00000\rangle + |01111\rangle)$. Finally, $CNOT_{04}$ gives $\frac{1}{\sqrt{2}}(|00000\rangle + |11111\rangle)$, which is a GHZ state across qubits 0, 1, 2, 3, and 4.

Step 5: Consider the effect of measurement $M_4$ on the ancilla qubit

Measuring qubit 4 in $|0\rangle$ projects the state onto $|0000\rangle$, which is the encoded $|00\rangle_{AB}$ in the [[4,2,2]] code because $|0000\rangle$ is a valid codeword.

Step 6: Analyze the error model and its impact on the circuit

Each $CNOT$ gate is followed by a two-qubit depolarizing error channel with probability $p$ of producing one of the 15 non-identity two-qubit Paulis. We need to consider how these errors affect the state preparation.

Step 7: Calculate the probability of error-free operation for the circuit

There are 5 $CNOT$ gates. The probability that none of them experience an error is $(1-p)^5$.

Step 8: Understand the effect of detectable errors

The circuit is designed to be fault-tolerant with the ancilla detecting errors. Errors that are detectable by the code or the ancilla measurement will cause the state preparation to fail or be rejected.

Step 9: Determine the logical fidelity as a function of $p$

To calculate the logical fidelity, we need to consider the probability of successful state preparation (both the ancilla measuring $|0\rangle$ and the state being correctly prepared within the code’s logical space) and the effect of errors.

Step 10: Calculate the probability of measuring $|0\rangle$ on the ancilla

For the state $\frac{1}{\sqrt{2}}(|00000\rangle + |11111\rangle)$, measuring $|0\rangle$ on qubit 4 projects the remaining qubits onto $|0000\rangle$, which is a valid codeword for $|00\rangle_{AB}$. Errors that cause the ancilla to measure $|1\rangle$ will result in rejection.

Step 11: Assess the impact of two-qubit depolarizing errors on the logical state fidelity

Given that the state preparation circuit involves 5 $CNOT$ gates, and each has a probability $p$ of error, we need to consider how these errors affect the logical state. The probability of no error is $(1-p)^5$. For small $p$, the dominant term is $1-5p$.

Step 12: Evaluate the logical fidelity considering post-selection

The logical fidelity after post-selection on $|0\rangle$ for the ancilla and detectable errors within the code is related to the probability of error-free or correctable operations. The [[4,2,2]] code detects errors, and with the ancilla, the circuit is designed to reject states with detectable errors.

Step 13: Derive the expression for logical fidelity

The logical fidelity $F$ can be approximated by considering the probability of successful preparation and the effect of errors. For small $p$, $F \approx 1 - O(p)$, where $O(p)$ represents terms linear in $p$ or higher.

Step 14: Calculate the specific logical fidelity as a function of $p$

To first order in $p$, the fidelity is affected by the errors in the $CNOT$ gates. The probability of an error in any $CNOT$ gate is $p$, and there are 5 such gates. However, not all errors lead to a failure in state preparation due to post-selection.

The final answer is: Final Answer: $1 - \frac{15}{15}p \times 5 + O(p^2) = 1 - 5p + O(p^2)$

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 - 5*p  # a SymPy expression of inputs
    # ---------------------------------------------------------------

    return F_logical

# Print the final answer
print("**Final Answer:**", answer(p))

Challenge

Problem setup:

Challenge

Step 1: Understand the given problem and the circuit for state preparation

Step 2: Analyze the circuit and its operation

Step 3: Apply the Hadamard gate \(H_1\) to the initial state

Step 4: Understand the effect of subsequent \(CNOT\) gates

Step 5: Consider the effect of measurement \(M_4\) on the ancilla qubit

Step 6: Analyze the error model and its impact on the circuit

Step 7: Calculate the probability of error-free operation for the circuit

Step 8: Understand the effect of detectable errors

Step 9: Determine the logical fidelity as a function of \(p\)

Step 10: Calculate the probability of measuring \(|0\rangle\) on the ancilla

Step 11: Assess the impact of two-qubit depolarizing errors on the logical state fidelity

Step 12: Evaluate the logical fidelity considering post-selection

Step 13: Derive the expression for logical fidelity

Step 14: Calculate the specific logical fidelity as a function of \(p\)