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:

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

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

Step 1: Understand the given problem and the circuit for preparing a logical two-qubit GHZ state.

The problem asks us to calculate the physical state fidelity of the final 4-qubit state after applying a specific circuit to prepare a logical two-qubit GHZ state in the [[4,2,2]] code. The circuit given is $(CNOT_{03}) (H_0) (CNOT_{21}) (H_2)$, and we need to consider the effect of two-qubit depolarizing error channels after each $CNOT$ gate.

Step 2: Analyze the circuit and the effect of the gates on the initial state.

First, let’s understand the initial state and the operations. Assuming we start with $|0000\rangle$, applying $H_2$ gives $|0\rangle_0|0\rangle_1|+\rangle_2|0\rangle_3 = \frac{1}{\sqrt{2}}(|0\rangle_0|0\rangle_1|0\rangle_2|0\rangle_3 + |0\rangle_0|0\rangle_1|1\rangle_2|0\rangle_3)$. Then, $CNOT_{21}$ results in $\frac{1}{\sqrt{2}}(|0\rangle_0|0\rangle_1|0\rangle_2|0\rangle_3 + |0\rangle_0|1\rangle_1|1\rangle_2|0\rangle_3)$. Next, $H_0$ gives $\frac{1}{2}(|+\rangle_0|0\rangle_1|0\rangle_2|0\rangle_3 + |+\rangle_0|1\rangle_1|1\rangle_2|0\rangle_3) = \frac{1}{2\sqrt{2}}(|0\rangle_0|0\rangle_1|0\rangle_2|0\rangle_3 + |1\rangle_0|0\rangle_1|0\rangle_2|0\rangle_3 + |0\rangle_0|1\rangle_1|1\rangle_2|0\rangle_3 + |1\rangle_0|1\rangle_1|1\rangle_2|0\rangle_3)$. Finally, $CNOT_{03}$ yields $\frac{1}{2\sqrt{2}}(|0\rangle_0|0\rangle_1|0\rangle_2|0\rangle_3 + |1\rangle_0|0\rangle_1|0\rangle_2|1\rangle_3 + |0\rangle_0|1\rangle_1|1\rangle_2|0\rangle_3 + |1\rangle_0|1\rangle_1|1\rangle_2|1\rangle_3)$.

Step 3: Calculate the ideal final state.

Let’s simplify the expression for the final state: $\frac{1}{2\sqrt{2}}(|0000\rangle + |1001\rangle + |0110\rangle + |1111\rangle)$. This is the ideal state without errors.

Step 4: Understand the effect of the depolarizing error channel on the $CNOT$ gates.

Each $CNOT$ gate is followed by a two-qubit depolarizing error channel that applies one of the 15 non-identity two-qubit Paulis with probability $p/15$. The probability of no error is $1 - p$.

Step 5: Calculate the fidelity of the final state considering the errors.

To calculate the physical state fidelity, we need to consider how the errors affect the state. The circuit has two $CNOT$ gates, so we have two error channels. The probability that no error occurs in either $CNOT$ gate is $(1-p)^2$. For each $CNOT$ gate, the error channel can introduce one of 15 possible errors.

Step 6: Determine the effect of errors on the fidelity.

The fidelity $F$ of the final state with respect to the ideal state is given by the probability of obtaining the ideal state (or a state that is equivalent to it up to a global phase) after the noisy operations. For a depolarizing channel, the fidelity after one $CNOT$ gate is $1 - p + p/15 \cdot (\text{number of Paulis that leave the state unchanged or change it to an equivalent state})$. However, since we’re dealing with a specific initial state and circuit, we need to directly consider how errors propagate.

Step 7: Analyze the error propagation through the circuit.

For $CNOT_{21}$, errors can occur on qubits 1 and 2. For $CNOT_{03}$, errors can occur on qubits 0 and 3. We need to consider how these errors affect the final state’s fidelity.

Step 8: Compute the fidelity based on the error model.

The fidelity of the state after the circuit, considering the depolarizing error model, can be derived by analyzing the effect of the error channels on the density matrix of the state. However, a simpler approach is to directly calculate the probability of the state being correct.

Step 9: Calculate the probability of the ideal outcome.

Given that each $CNOT$ is followed by a depolarizing error channel with probability $p$, the overall fidelity can be related to the probability of no error or errors that do not affect the logical state. However, the direct calculation of fidelity involves considering the effect of errors on the state’s density matrix.

Step 10: Derive the expression for the physical state fidelity.

The physical state fidelity $F$ can be expressed as $F = \langle\psi|\rho|\psi\rangle$, where $|\psi\rangle$ is the ideal state and $\rho$ is the density matrix after the noisy circuit. For a depolarizing error channel after each $CNOT$, the fidelity can be simplified by considering the probability of error.

Step 11: Simplify the calculation by directly using the properties of the depolarizing channel.

For a two-qubit depolarizing channel with error probability $p$, the output density matrix $\rho'$ after applying the channel to a density matrix $\rho$ is given by $\rho' = (1-p)\rho + p/15 \sum_{i=1}^{15} P_i \rho P_i$, where $P_i$ are the 15 non-identity two-qubit Paulis.

Step 12: Apply the error model to the given circuit.

Since there are two $CNOT$ gates, we apply the error model twice. The fidelity after one $CNOT$ gate with error is $1 - p + p/15$ for the identity component, but we need to consider the overall effect on the state.

The final answer is: $\boxed{(1-p)^2}$

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

Step 1: Understand the given problem and the circuit for preparing a logical two-qubit GHZ state.

The problem asks us to calculate the logical state fidelity of the final 2-qubit logical state after applying a specific circuit to prepare a logical two-qubit GHZ state in the [[4,2,2]] code, assuming post-selection on all detectable errors.

Step 2: Recall the stabilizers and logical operators of the [[4,2,2]] code.

The stabilizers are $XXXX$ and $ZZZZ$, and the logical operators are $X_A = XIXI$, $X_B = XXII$, $Z_A = ZZII$, $Z_B = ZIZI$. The logical GHZ state is $(|00\rangle_{AB} + |11\rangle_{AB})/\sqrt{2}$.

Step 3: Understand the effect of the circuit on the logical state.

The circuit prepares a physical state that encodes the logical GHZ state. We need to consider how errors affect this preparation and how post-selection on detectable errors impacts the logical fidelity.

Step 4: Analyze the error detection capability of the [[4,2,2]] code.

The [[4,2,2]] code can detect one-qubit errors because any single-qubit Pauli error will anti-commute with at least one of the stabilizers ($XXXX$ or $ZZZZ$), thus being detectable.

Step 5: Consider the effect of two-qubit depolarizing errors after $CNOT$ gates.

Each $CNOT$ gate is followed by a depolarizing error channel. The probability of an error is $p$, and it can be one of 15 non-identity two-qubit Paulis.

Step 6: Determine the conditions under which errors are detectable.

Errors are detectable if they anti-commute with either $XXXX$ or $ZZZZ$. Most two-qubit errors will be detectable because they will not commute with both stabilizers.

Step 7: Calculate the probability of undetectable errors.

For the [[4,2,2]] code, undetectable errors are those that commute with both $XXXX$ and $ZZZZ$. This includes the identity and certain combinations of two-qubit Paulis that are equivalent to logical operators or stabilizers.

Step 8: Analyze the logical fidelity after post-selection.

After post-selecting on detectable errors, the remaining states are either correct or have undetectable errors. The logical fidelity is related to the probability of having an undetectable error that affects the logical state.

Step 9: Compute the logical state fidelity.

To compute the logical fidelity, we need to consider the probability that the state is correctly prepared logically after post-selection.

Step 10: Simplify the expression for logical fidelity.

Given that the circuit and error model are known, and assuming that post-selection removes states with detectable errors, the logical fidelity can be related to the probability of errors that are not detectable and affect the logical state.

The final answer is: $\boxed{1}$

User #5

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 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 $p$ in Sympy format.

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

    Outputs
    ----------
    F_logical: sympy.Expr, the logical state fidelity as a function of $p$
    """

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

    return F_logical

Assistant #6

import sympy as sp

p = sp.symbols('p')

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

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

    Outputs
    ----------
    F_logical: sympy.Expr, the logical state fidelity as a function of $p$
    """

    # ------------------ FILL IN YOUR RESULTS BELOW ------------------
    F_logical = 1  # a SymPy expression of inputs
    # ---------------------------------------------------------------

    return F_logical

# Execute the function to get the final answer
F_logical = answer(p)
print("Final Answer: ", F_logical)

Checkpoint-2

Problem setup:

Checkpoint 1

Step 1: Understand the given problem and the circuit for preparing a logical two-qubit GHZ state.

Step 2: Analyze the circuit and the effect of the gates on the initial state.

Step 3: Calculate the ideal final state.

Step 4: Understand the effect of the depolarizing error channel on the \(CNOT\) gates.

Step 5: Calculate the fidelity of the final state considering the errors.

Step 6: Determine the effect of errors on the fidelity.

Step 7: Analyze the error propagation through the circuit.

Step 8: Compute the fidelity based on the error model.

Step 9: Calculate the probability of the ideal outcome.

Step 10: Derive the expression for the physical state fidelity.

Step 11: Simplify the calculation by directly using the properties of the depolarizing channel.

Step 12: Apply the error model to the given circuit.

Checkpoint 2

Step 1: Understand the given problem and the circuit for preparing a logical two-qubit GHZ state.

Step 2: Recall the stabilizers and logical operators of the [[4,2,2]] code.

Step 3: Understand the effect of the circuit on the logical state.

Step 4: Analyze the error detection capability of the [[4,2,2]] code.

Step 5: Consider the effect of two-qubit depolarizing errors after \(CNOT\) gates.

Step 6: Determine the conditions under which errors are detectable.

Step 7: Calculate the probability of undetectable errors.

Step 8: Analyze the logical fidelity after post-selection.

Step 9: Compute the logical state fidelity.

Step 10: Simplify the expression for logical fidelity.