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 $p$.

Checkpoints

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:

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

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.

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 determine the physical state fidelity of the final 4-qubit state after the state preparation circuit, we need to consider the effect of the two-qubit depolarizing channels and how they influence the final prepared state. We will evaluate this in the following way:

Circuit Description and Initial State:

The circuit to prepare the logical two-qubit GHZ state $(|00\rangle_{AB} + |11\rangle_{AB})/\sqrt{2}$ is given by the sequence of operations $ (CNOT_{03}) (H_0) (CNOT_{21}) (H_2) $. The initial state before the circuit is applied is $|0000\rangle$.

Gate Operations:

Step 1: Apply Hadamard gate $H_2$ on qubit 2:
- $|0000\rangle \rightarrow \frac{1}{\sqrt{2}}(|0000\rangle + |0010\rangle)$.
Step 2: Apply $CNOT_{21}$:
- The $CNOT_{21}$ flips the state of qubit 1 when qubit 2 is in the state $|1\rangle$:
- $\frac{1}{\sqrt{2}} (|0000\rangle + |0010\rangle) \rightarrow \frac{1}{\sqrt{2}} (|0000\rangle + |0011\rangle)$.
Step 3: Apply Hadamard gate $H_0$ on qubit 0:
- $\frac{1}{\sqrt{2}} (|0000\rangle + |0011\rangle) \rightarrow \frac{1}{2} (|0000\rangle + |1000\rangle + |0011\rangle + |1011\rangle)$.
Step 4: Apply $CNOT_{03}$:
- This operation flips qubit 3 if qubit 0 is in state $|1\rangle$:
- $|1000\rangle \rightarrow |1001\rangle$ and $|1011\rangle \rightarrow |1010\rangle$.
- Final state: $\frac{1}{2} (|0000\rangle + |1001\rangle + |0011\rangle + |1010\rangle)$.

Consider Two-Qubit Depolarizing Errors:

Each $CNOT$ gate is followed by a two-qubit depolarizing error channel. Such a channel transforms a two-qubit density matrix $\rho$ as:

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

where $P_i, P_j$ range over the non-identity Pauli matrices $X, Y, Z$.

Fidelity Calculation:

The fidelity of the final physical state with the target state can be computed as:

\[ F = \text{Tr} \left(\rho_{\text{ideal}} \rho_{\text{noisy}} \right) \]

Here $\rho_{\text{ideal}}$ is the ideal density matrix for the desired state $|\psi_{\text{ideal}}\rangle = \frac{1}{2} (|0000\rangle + |1001\rangle + |0011\rangle + |1010\rangle)$:

\[ \rho_{\text{ideal}} = |\psi_{\text{ideal}}\rangle \langle \psi_{\text{ideal}}| \]

The fidelity reduction from each depolarizing noise after $CNOT$ gates can be analyzed in terms of $p/15$ contributions from each error. The total physical fidelity, considering error perturbations on one state affecting fidelity by a factor of $(1 - p) + p/15$ for each non-identity error.

Extending to Logical Fidelity:

The logical fidelity at the end depends on obtaining the correct logical state and you need to account for possible Pauli replacements that preserve the logical state.
Each $CNOT$ introduces errors with probability $p$, and thus fidelity scales approximately as $(1 - p)$ per gate for small $p$.

Considering two gates and the sequence of operations:

\[ F \approx (1 - p)^{2} \]

This assumes the dominant effect of depolarizing channels per gate.

Conclusion:

Considering the degenerate coding space and corrections for logical operations specified in terms of stabilizers, and adopting the common instability scaling, the physical fidelity to leading order of $p$ becomes:

Final Answer:

\[ F_\text{physical} = (1 - p)^2 \approx 1 - 2p \]

This estimation of infidelity comes from the straightforward application of two $CNOT$ gates with depolarizing noise, each responsible for diminishing fidelity.

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 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 $p$.

    Inputs
    ----------
    p: sympy.Symbol, two-qubit gate error rate $p$

    Outputs
    ----------
    F_physical: sympy.Expr, the physical state fidelity of the final physical 4-qubit state
    """

    # ------------------ FILL IN YOUR RESULTS BELOW ------------------
    F_physical = ...  # a SymPy expression of inputs
    # ---------------------------------------------------------------

    return F_physical

Assistant #4

import sympy as sp

p = sp.symbols('p')

def answer(p):
    r"""
    Return the expression of 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 $p$.

    Inputs
    ----------
    p: sympy.Symbol, two-qubit gate error rate $p$

    Outputs
    ----------
    F_physical: sympy.Expr, the physical state fidelity of the final physical 4-qubit state
    """

    # ------------------ FILL IN YOUR RESULTS BELOW ------------------
    F_physical = (1 - p)**2  # a SymPy expression of inputs
    # ---------------------------------------------------------------

    return F_physical