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.

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

Step-by-Step Derivation

Ideal (error-free) output state
Assume the physical qubits start in |0000⟩.
Apply H2 then CNOT21: this prepares a Bell state on qubits (2,1):

\[ \]

Apply H0 then CNOT03: this prepares a Bell state on qubits (0,3):

\[ \]

Hence the ideal final 4-qubit state factorizes:

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

Noise model and factorization
Each CNOT is followed by a two-qubit depolarizing Pauli channel on its two qubits:

\[ \mathcal{E}(\rho)=(1-p)\rho+\frac{p}{15}\sum_{P\in\mathcal{P}_2\setminus\{I\}}P\rho P, \]

where the sum is over the 15 non-identity two-qubit Paulis on that pair.

The first noise channel acts on (2,1) after CNOT21; the second acts on (0,3) after CNOT03. Since subsequent gates act on disjoint qubits, the final state is

\[ \rho_{\mathrm{final}}=\mathcal{E}_{21}\big(|\Phi^+\rangle\langle\Phi^+|_{21}\big)\ \otimes\ \mathcal{E}_{03}\big(|\Phi^+\rangle\langle\Phi^+|_{03}\big). \]

Therefore, the physical state fidelity to the ideal pure state factorizes:

\[ \]

where

\[ F_{\Phi^+}(p)=\langle\Phi^+|\mathcal{E}(|\Phi^+\rangle\langle\Phi^+|)|\Phi^+\rangle. \]

Fidelity of a Bell state under the two-qubit Pauli depolarizing channel
For a Bell state, among the 15 non-identity two-qubit Paulis, exactly three map |Φ+⟩ to ±|Φ+⟩ (namely XX, YY, ZZ); the other 12 map it to a Bell state orthogonal to |Φ+⟩. Thus,

\[ F_{\Phi^+}(p)=(1-p)\cdot 1+\frac{p}{15}\big(3\cdot 1+12\cdot 0\big)=1-\frac{4p}{5}. \]

Equivalently, using the identity

\[ \]

we get

\[ F_{\Phi^+}(p)=\Big(1-\frac{16p}{15}\Big)+\frac{4p}{15}\cdot\frac{1}{4}=1-\frac{4p}{5}. \]

Combine the two independent pairs

\[ \]

Mathematical Typesetting

All expressions above are given in LaTeX.

Conventions and Units

Standard quantum information conventions; no physical units are involved.

Final Answer: F_phys(p) = (1 - 4p/5)^2 = 1 - (8/5)p + (16/25)p^2.

User #3

The correct answer is: $ F_{\rm{physical}}=(1-\frac{12}{15}p)^2 $

Now let’s solve the next Checkpoint.

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-by-Step Derivation

Ideal output and factorization
The circuit prepares two independent Bell pairs on disjoint qubit pairs:

\[ \]

A two-qubit depolarizing channel follows each CNOT, one on pair (2,1) and one on pair (0,3), acting independently.
Action of the two-qubit depolarizing channel on a Bell state
For a single pair, the channel is

\[ \mathcal{E}(\rho)=(1-p)\rho+\frac{p}{15}\sum_{P\in\mathcal{P}_2\setminus\{I\}}P\rho P. \]

Among the 15 non-identity two-qubit Paulis, exactly 3 stabilize $|\Phi^+\rangle$ (up to phase): $XX,YY,ZZ$; the other 12 map it uniformly onto the other 3 Bell states. Therefore, for each pair independently,

\[ \]

Post-selection on the [[4,2,2]] code stabilizers
The stabilizers are $S_X=XXXX$ and $S_Z=ZZZZ$.
Writing $S_X=(X\!\otimes\!X)_{03}(X\!\otimes\!X)_{21}$ and $S_Z=(Z\!\otimes\!Z)_{03}(Z\!\otimes\!Z)_{21}$, the eigenvalue on the 4-qubit state is the product of the eigenvalues on each Bell pair.
Each Bell state is labeled by its $(XX,ZZ)$ eigenvalues:

\[ \Phi^+:(+,+),\ \Phi^-:(-,+),\ \Psi^+:(+,-),\ \Psi^-:(-,-). \]

Hence a 4-qubit product $|B_j\rangle_{21}\otimes|B_k\rangle_{03}$ has

\[ S_X\text{-eigenvalue}=a_j a_k,\qquad S_Z\text{-eigenvalue}=b_j b_k, \]

where $(a_j,b_j)$ are the $(XX,ZZ)$ eigenvalues of $|B_j\rangle$.

Post-selection (acceptance) requires both be $+1$, i.e., $a_j=a_k$ and $b_j=b_k$, which is equivalent to $j=k$. Therefore, the only accepted 4-qubit states are

\[ |B_j\rangle_{21}\otimes|B_j\rangle_{03},\qquad j\in\{\Phi^+,\Phi^-,\Psi^+,\Psi^-\}. \]

Accepted ensemble and logical fidelity
Because the two pairs are independent, the unnormalized weights of the accepted states are $a^2$ for $j=\Phi^+$ and $b^2$ for each of the other three $j$’s. The acceptance probability is

\[ P_{\mathrm{acc}}=a^2+3b^2. \]

These four accepted code states are orthonormal and correspond to the four logical Bell states; only $j=\Phi^+$ equals the target logical GHZ state. Therefore, the logical state fidelity conditioned on acceptance is

\[ \]

Closed form

\[ {\big(1-\tfrac{12}{15}p\big)^2+3\big(\tfrac{4}{15}p\big)^2} =\frac{(1-\tfrac{4}{5}p)^2}{(1-\tfrac{4}{5}p)^2+\tfrac{16}{75}p^2} \]

Small-$p$ expansion (showing distance-2 behavior under post-selection):

\[ F_{\mathrm{logical}}(p)=1-\frac{16}{75}p^2+O(p^3). \]

Mathematical Typesetting

All equations are presented in LaTeX as required.

Conventions and Units

Quantum information setting; no physical units.

Final Answer: F_logical(p) = [(1 - 12p/15)^2] / [(1 - 12p/15)^2 + 3(4p/15)^2] = (1 - 8p/5 + 16p^2/25) / (1 - 8p/5 + 64p^2/75).

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 - sp.Rational(4, 5)*p)**2) / ((1 - sp.Rational(4, 5)*p)**2 + 3*(sp.Rational(4, 15)*p)**2)
    # ---------------------------------------------------------------

    return F_logical