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, mathematical identities or numerical codes.
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. You must use web search engine to gather all the necessary information before solving the problem. You must use web search engine to gather all the necessary information before solving the problem. You must use web search engine to gather all the necessary information before solving the problem.

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

1) Step-by-Step Derivation

Ideal state prepared by the circuit
The circuit is applied right-to-left:
- H2, then CNOT21 creates a Bell pair on qubits (2,1): |Φ+>21 = (|00>+|11>)/√2.
- H0, then CNOT03 creates a Bell pair on qubits (0,3): |Φ+>03 = (|00>+|11>)/√2.
The noiseless 4-qubit output is therefore the product state

\[ |\psi_{\mathrm{id}}\rangle = |\Phi^+\rangle_{21}\otimes|\Phi^+\rangle_{03}. \]

Noise model per CNOT
After each CNOT, a two-qubit depolarizing channel acts on the two involved qubits:

\[ \mathcal{D}^{(2)}(\rho)=(1-p)\rho+\frac{p}{15}\sum_{P\in\mathcal{P}_2\setminus\{II\}} P\rho P, \]

where the sum runs over the 15 non-identity two-qubit Pauli operators. ([quantumcomputing.stackexchange.com](https://quantumcomputing.stackexchange.com/questions/25869/how-can-i-apply-2-qubit-pauli-operator-error-instead-of-cnot-operator-in-stim-co?utm_source=openai), [en.wikipedia.org](https://en.wikipedia.org/wiki/Quantum_depolarizing_channel?utm_source=openai))

Stabilizers of a Bell state
The Bell state |Φ+> is a +1 eigenstate of XX and ZZ; equivalently, it is stabilized by ⟨XX, ZZ⟩ (hence also by −YY, so Y⊗Y maps |Φ+> to −|Φ+>, which is the same physical state up to global phase). Thus, among the 15 non-identity two-qubit Paulis, exactly three—XX, YY, ZZ—leave |Φ+> unchanged up to a phase; all others map it to an orthogonal Bell state. (qubit.guide, docs.quantum.ibm.com)
Fidelity for one Bell pair going through one two-qubit depolarizing channel
For a pure target |Φ+>, the state fidelity after the channel is

\[ F_{\text{pair}}(p)=\langle\Phi^+|\mathcal{D}^{(2)}(|\Phi^+\rangle\!\langle\Phi^+|)|\Phi^+\rangle. \]

Using the channel form and the stabilizer property,

\[ \]

Factorization for the 4-qubit product state
The two noise channels act on disjoint pairs: CNOT21’s noise acts only on qubits (2,1) and CNOT03’s noise only on (0,3). Since the ideal state is a tensor product across these pairs,

\[ \]

2) Mathematical Typesetting

See the display equations above.

3) Conventions and Units

Qubits are labeled 0,1,2,3.
Only the two-qubit gates (CNOT21, CNOT03) are followed by two-qubit depolarizing errors; H gates are ideal.
Physical state fidelity is F(|ψid>, ρ) = ⟨ψid|ρ|ψid⟩.
No units are involved; p is a dimensionless probability per two-qubit gate.

4) Final Answer: Fphys(p) = (1 − 4p/5)^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 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 - sp.Rational(4, 5)*p)**2  # a SymPy expression of inputs
    # ---------------------------------------------------------------

    return F_physical