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 will determine the logical state fidelity of the prepared $ |00\rangle_{AB} $ state in the [[4,2,2]] code, given the error model described. The fidelity is the overlap between the prepared state and the desired state, and we are assuming post-selection on the ancilla measuring $|0\rangle$ and detection of all detectable errors.

Step-by-Step Derivation

Determine the Error Model:

Each $ CNOT_{ij} $ gate is followed by a two-qubit depolarizing error channel: - It applies one of the 15 non-identity two-qubit Pauli errors with probability $ p/15 $. - No error occurs with probability $ 1 - p $.

Understand the Circuit:

The circuit is:

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

The circuit begins with a Hadamard gate on qubit 1, followed by a series of $CNOT$ gates and finally a measurement of ancilla qubit 4.

Logical State Initialization:

The initial state before errors can be written as:

\[ |0000\rangle_{0123} |0\rangle_4 \]

After applying the Hadamard gate:

\[ |000+\rangle_{0123} \equiv |00\rangle_{AB} |+\rangle_1 \]

 - where \( |+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \).

Effect of Errors and Ancilla Measurement:

Only post-select on ancilla measuring $|0\rangle$ and no detectable errors.
Errors detectable by the stabilizers $XXXX$ or $ZZZZ$ or ancilla are effectively discarded.

Logical Fidelity Calculation:

No Error Case ($1-p$): If no error occurs, the final state is exactly $|00\rangle_{AB}$. The contribution to the fidelity from this case is $(1-p)^5$.
Error Correction/Detection: Considering cases of errors where the stabilizers might flip or ancilla measures $|1\rangle$, these errors will be identified, and such events are post-selected out. The successful post-selection occurs only for errors that either commute with stabilizers or can be detected.

Keeping Terms for Post-Selection:

The logical fidelity, accounting for success in all gates, is thus reduced only by probabilities of errors that manage to elude detection by our code.

Final Fidelity Expression:

We seek logical fidelity $ F $ as a function of $ p $, calculated as:

\[ F(p) = (1-p)^5 + \mathcal{O}(p^2) \]

This approximation accounts for terms up to linear order in $ p $, considering post-selection rejects those easily detected errors, leading to an improved success using the error detection code.

Conclusion

In conclusion, the fidelity will express robust behavior where further analysis for more intricate sustaining scenarios may involve higher order $ \mathcal{O}(p^2) $ contributions.

Final Answer:

The logical fidelity of the final 2-qubit logical state at the end of the circuit, post-selected on all detectable errors and the ancilla qubit measuring $|0\rangle$, as a function of the two-qubit gate error rate $ p $, is approximately:

\[ F(p) = 1 - 5p + \mathcal{O}(p^2) \]

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