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 $p$.

Checkpoint 1

Suppose that we wish to prepare a logical two-qubit GHZ state $(|00\rangle_{AB}+|11\rangle_{AB})/\sqrt{2}$ in the [[4,2,2]] code. To do so, we use the following state preparation circuit:

\[ (CNOT_{03}) (H_0) (CNOT_{21}) (H_2). \]

What is the physical state fidelity of the final physical 4-qubit state at the end of the circuit as a function of the two-qubit gate error rate $p$?

Assistant #2

To assess the physical state fidelity of the final 4-qubit state at the end of the specified circuit, we must consider the effect of the depolarizing errors after each two-qubit gate. The main steps are as follows:

Understand the Circuit: The given circuit can be broken down as follows: - Apply a Hadamard gate $H_2$ to qubit 2. - Apply a $CNOT_{21}$ with control qubit 2 and target qubit 1. - Apply a Hadamard gate $H_0$ to qubit 0. - Apply a $CNOT_{03}$ with control qubit 0 and target qubit 3.
Error Introduction: Each $CNOT$ gate is followed by a depolarizing error channel. For a single 2-qubit depolarizing error, the superoperator describing the effect on a density matrix $\rho$ is:

\[ \mathcal{E}(\rho) = (1-p) \rho + \frac{p}{15} \sum_{i=1}^{15} P_i \rho P_i, \]

where $P_i$ are the 15 non-identity two-qubit Pauli operators acting on the appropriate qubits.

Effect on Fidelity: The fidelity $F$ of the output quantum state compared to the intended ideal state is influenced by how depolarizing errors degrade the state’s purity. Specifically, if we assume each $CNOT$ gate is followed by its own independent error channel, the fidelity at each error instance can be estimated, and these fidelities can be concatenated to find the final state fidelity.
Approximation for Small Errors: For small $p$, the impact of each error can be assumed approximately independent. A commonly used approximation for the physical state fidelity of a quantum circuit subjected to independent depolarizing noise is given by:

\[ F \approx 1 - m \frac{p}{15}, \]

where $m$ is the number of errors (in this case, $m = 2$ because there are two $CNOT$ gates). This is a linear approximation valid for small $p$ where depolarizing errors are each equally likely.

Calculation for this Circuit: With $m = 2$ and each $CNOT$ affected by a depolarizing channel, we have:

\[ F \approx 1 - 2 \frac{p}{15} = 1 - \frac{2p}{15}. \]

Thus, the physical state fidelity of the final 4-qubit state, assuming depolarization errors occur as specified, is:

Final Answer: The physical state fidelity of the final 4-qubit state is approximately:

\[ F \approx 1 - \frac{2p}{15}. \]

User #3

Checkpoint 2

Suppose that we wish to prepare a logical two-qubit GHZ state $(|00\rangle_{AB}+|11\rangle_{AB})/\sqrt{2}$ in the [[4,2,2]] code. To do so, we use the following state preparation circuit:

\[ (CNOT_{03}) (H_0) (CNOT_{21}) (H_2). \]

What is the logical state fidelity of the final 2-qubit logical state at the end of the circuit as a function of the two-qubit gate error rate $p$, assuming the state is post-selected on all detectable errors in the code?

Assistant #4

To determine the logical state fidelity of the final 2-qubit logical state, we need to consider the effect of the errors on the logical state and how the error correction process (in this case, error detection through post-selection) can refine the outcome. Here’s a step-by-step analysis:

Understand the Circuit and Error Model: The given circuit is: - Apply a Hadamard gate $H_2$ to qubit 2: The effect is a superposition on qubit 2. - Apply a $CNOT_{21}$: This creates entanglement between qubits 2 and 1. - Apply a Hadamard gate $H_0$: This applies $H$ transformation on qubit 0. - Apply a $CNOT_{03}$: This entangles qubits 0 and 3.

Each $CNOT$ gate experiences a two-qubit depolarizing error, with a non-identity Pauli operator resulting with probability $p/15$.

Logical Error Detection: The [[4,2,2]] code can detect single-qubit errors through its stabilizers, $XXXX$ and $ZZZZ$. After performing the circuit, the stabilizers are measured: - If an error is detected (non-identity syndrome measurement), the state is discarded (hence post-selection).
Error Impact on Post-Selected State: - Single Non-Detectable Errors: For many errors (such as certain two-qubit errors that effectively cancel in logical space), these could still render a non-trivial impact on the logical subspace without being detectable. Thus, by post-selection, these are weeded out. - Detectability via Stabilizers: Each $CNOT$ can introduce errors corresponding to the tensor product of Pauli matrices on its controlled and target qubits, summed over the 15 possible error combinations. Yet, only combinations aligning with stabilizers or logical operators can be detected or possibly remain.
Logical State Fidelity: - Remaining errors after post-selection resemble logical errors manifesting on the logical qubits. - Probability of Non-Detectable Error Combinations: If each $CNOT$ adds a probability of $p/15$ per non-identity error, but becoming non-detectable logical errors is rare, we quantify the remaining logically undetectable error probability with post-selection.
Post-Selection Impact on Fidelity: - In post-selection, the fidelity is affected by only those errors that translate directly into logical subspace errors. - The product of individual fidelities per gate needs to exclude probabilities of detectable errors. - Approximating these small impacts gives the final valid logical fidelity:

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

due to the post-selection strategy efficiently discarding any input with detected errors (single-qubit errors as mortal), leaving only second-order in $p$ logical-level undeclared errors.

Final Answer:

The logical state fidelity of the final 2-qubit logical state is approximately:

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

This retains fidelity close to 1 as first-order error rates are mostly self-corrected by code post-selection.

User #5

Checkpoint 3

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 #6

To determine the logical state fidelity of the final 2-qubit logical state, considering both error detection in the [[4,2,2]] code and post-selection on the ancilla measurement result, we’ll need to follow these steps:

Circuit Implementation: The circuit is implemented in the following order: - Apply a Hadamard gate $H_1$ to qubit 1. - Apply $CNOT_{12}$, $CNOT_{10}$, $CNOT_{23}$, $CNOT_{34}$, and finally $CNOT_{04}$. - Measure the ancilla qubit, $M_4$.
Error Introduction: Each of the $CNOT$ gates suffers from a depolarizing error channel, as described in the problem, where each non-identity two-qubit Pauli operates with probability $p/15$.
Detection with Ancilla: The ancilla is crucial for running this circuit fault-tolerantly. The ancilla qubit acts as an additional tool for error detection: - If the ancilla measures $|0\rangle$, the error detection for state preparation is considered to have passed. - If the ancilla measures $|1\rangle$, errors causing this result have been detected and that run of the circuit is discarded.
Fault Tolerant Mechanism: - Every effective error should contribute to either activating the ancilla or affecting stabilizers/logical operators. - The ancilla acts to catch correlated errors that would otherwise escape individual stabilizer detection.
Combined Post-Selection Strategy: - By post-selecting on the ancilla being $|0\rangle$, additional errors in the circuit that might lead to undetectable logical errors become constrained, and thus lower fidelity impact is achieved. - Other errors are managed by the code’s ability to detect all single-qubit errors via its stabilizers.
Logical State Fidelity With Ancilla: - Errors resulting in logical infidelity primarily occur due to combinations inducing logical errors, which the ancilla fails to capture. However, with two layers of detection (ancilla and code’s stabilizer), the number of such cases is drastically reduced. - Most likely impactful errors need to involve multiqubit correlations exceeding error detection capability, which occurs with probability $\sim \mathcal{O}(p^2)$ following available redundancy.
Resulting Fidelity Assessment: - Applying the post-selection condition: primarily on detecting errors (either by code stabilizers or via ancilla error reporting), filters out leading order first errors, obtaining second-order residual errors.

Final Answer:

Given post-selection on the ancilla qubit measuring $|0\rangle$ and on all detectable errors:

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

The logical state fidelity remains quite high, reflecting the robustness against primary error rates which are actively mitigated through ancilla involvement and the code’s detection power.

User #7

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 #8

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