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

Error-model and post‐selection
– We have five CNOT gates in the preparation circuit (in execution order):
g₁≔CNOT₁₂ (ctrl 1→tgt 2)
g₂≔CNOT₁₀ (ctrl 1→tgt 0)
g₃≔CNOT₂₃ (ctrl 2→tgt 3)
g₄≔CNOT₃₄ (ctrl 3→tgt 4)
g₅≔CNOT₀₄ (ctrl 0→tgt 4)
– After each gᵢ we apply the two‐qubit depolarizing channel

\[ \mathcal{D}_p(\rho)= (1-p)\,\rho\;+\;\frac p{15}\sum_{P\in\{I,X,Y,Z\}^{\otimes2}\setminus I\otimes I}P\,\rho\,P\,. \]

– We then measure the ancilla (qubit 4) in the Z-basis and post-select on “0,” and (separately) measure the two code stabilizers

\[ S_X=X_0X_1X_2X_3,\qquad S_Z=Z_0Z_1Z_2Z_3 \]

 post-selecting on “+1” for each.  Any error which anticommutes with any of \{Z₄ (ancilla), S_X, S_Z\} is discarded.

Classification of single‐fault events
To leading order in p, the only way to get a logical error is for exactly one gate to suffer a Pauli fault which
a) goes undetected by ancilla (i.e. has no X or Y on any wire that reaches qubit 4),
b) commutes with both code stabilizers (so is undetected at the end), and
c) acts non-trivially within the code space (i.e. is in N(S) but ∉ S).

(i) g₁=CNOT₁₂ errors on (qubit 1,qubit 2). The only fault which (a) has no X/Y on 1 or 2 that can propagate to 4, (b) commutes with S_X,S_Z, and (c) is nontrivial, is
E₁=Z_1 Z_2 → final error = Z_1Z_2 ∈N(S)\setminus S.

(ii) g₂=CNOT₁₀ errors on (1,0). Again the only choice P₁⊗P₀ with P₀∈{I,Z} that survives ancilla and code checks is
E₂=Z_1 Z_0 → final error = Z_0Z_1.

(iii) g₃=CNOT₂₃ errors on (2,3). The only undetected logical fault is
E₃=Z_2 Z_3 → final error = Z_2Z_3.

No single fault on g₄ or g₅ can produce an undetected logical error on qubits 0–3.

Probability of a single‐fault logical error
– Each two-qubit gate errors with any particular non-identity Pauli occurs with probability p/15.
– We have exactly three single‐fault events {E₁,E₂,E₃} that survive all post-selections and act logically.
⇒ to first order in p the logical‐error probability is

\[ P_L^{(1)} \;=\;\frac{3}{15}\,p \;=\;\frac15\,p\,. \]

Logical‐state fidelity
By definition the fidelity of the post-selected output w.r.t. the ideal |00⟩ₗ is

\[ F(p)\;=\;1 \;-\;P_L\;=\;1\;-\;\frac15\,p\;+\;O(p^2)\,. \]

One can check that all two-fault combinations either (i) get detected by ancilla or by the stabilizers or (ii) multiply together to give a trivial stabilizer, so the next non-zero contribution is O(p²).

Final Answer:

\[ \boxed{F_{\rm logical}(p)=1-\frac15\,p+\mathcal O(p^2)\,.} \]

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 - p/5  # logical fidelity to first order in p
    # ---------------------------------------------------------------

    return F_logical