Let me work through this problem step by step.

First, let me understand the [[4,2,2]] code: - 4 physical qubits (labeled 0,1,2,3) - 2 logical qubits (labeled A,B) - Stabilizers: \(XXXX\) and \(ZZZZ\) - Logical operators: - \(X_A = XIXI\) - \(X_B = XXII\) - \(Z_A = ZZII\) - \(Z_B = ZIZI\)

The logical computational basis states are: - \(|00\rangle_{AB}\): stabilized by both \(XXXX\) and \(ZZZZ\), with \(Z_A|00\rangle_{AB} = |00\rangle_{AB}\) and \(Z_B|00\rangle_{AB} = |00\rangle_{AB}\) - \(|01\rangle_{AB}\): \(Z_A|01\rangle_{AB} = |01\rangle_{AB}\) and \(Z_B|01\rangle_{AB} = -|01\rangle_{AB}\) - \(|10\rangle_{AB}\): \(Z_A|10\rangle_{AB} = -|10\rangle_{AB}\) and \(Z_B|10\rangle_{AB} = |10\rangle_{AB}\) - \(|11\rangle_{AB}\): \(Z_A|11\rangle_{AB} = -|11\rangle_{AB}\) and \(Z_B|11\rangle_{AB} = -|11\rangle_{AB}\)

Since \(Z_A = ZZII\) and \(Z_B = ZIZI\), the logical computational basis states are: - \(|00\rangle_{AB} = \frac{1}{2}(|0000\rangle + |0101\rangle + |1010\rangle + |1111\rangle)\)

• \(|01\rangle_{AB} = \frac{1}{2}(|0001\rangle + |0100\rangle + |1011\rangle + |1110\rangle)\)
• \(|10\rangle_{AB} = \frac{1}{2}(|0010\rangle + |0111\rangle + |1000\rangle + |1101\rangle)\)
• \(|11\rangle_{AB} = \frac{1}{2}(|0011\rangle + |0110\rangle + |1001\rangle + |1100\rangle)\)

I see how these basis states are constructed with equal superposition of specific four-qubit states, each representing a different logical qubit combination.

Let me verify this. \(XXXX\) should stabilize all logical states. Checking \(|00\rangle_{AB}\) reveals the stabilizer property, where the operator returns the same state with a potential phase change. The computation shows the state remains invariant under this transformation, confirming the stabilizer’s consistency.

The stabilizer group emerges from \(XXXX\) and \(ZZZZ\), defining the logical subspace through their +1 eigenspace. I’ll focus on selecting computational basis states that satisfy two key constraints: stabilization by both operators and specific eigenstate properties for local \(Z\) operators.

The circuit’s structure involves strategic \(CNOT\) and Hadamard gates applied to the initial \(|0000\rangle\) state, suggesting a carefully designed quantum transformation that likely encodes quantum information across the system’s qubits.

I’ll trace the quantum state evolution step by step, noting how each gate modifies the quantum amplitudes and basis states systematically.

The Hadamard gate on the second qubit creates a superposition, splitting the initial state into two equal amplitude components. The subsequent controlled-NOT operation then entangles these states, progressively building quantum correlations across different qubits.

The controlled-NOT gate at step 4 selectively flips the third qubit’s state based on the first qubit’s value. This results in precise state transformations where some basis states remain unchanged while others experience targeted bit flips.

The quantum circuit demonstrates how strategic gate applications can manipulate quantum states with remarkable precision, creating intricate superpositions and entangled configurations through controlled quantum operations.

I’m investigating the code space’s fundamental structure, focusing on identifying simultaneous eigenstates that satisfy the \(XXXX\) and \(ZZZZ\) constraints. The code’s dimensionality suggests a robust two-qubit logical encoding mechanism.

The logical operators reveal intriguing symmetries across the four-qubit system. These operators strategically distribute quantum information, enabling controlled transformations while preserving the code’s fundamental properties.

I’m exploring how the joint eigenstates of \(\bar{Z}_A\) and \(\bar{Z}_B\) define the logical computational basis, which will help characterize the quantum error correction scheme’s core computational subspace.

The standard construction approach promises to illuminate the underlying algebraic structure of these stabilizer generators, potentially revealing deeper insights into the quantum code’s design principles.

Specifically, the logical \(|00\rangle_L\) state emerges through a precise combination of stabilizer projections, yielding a superposition state that encapsulates the code’s inherent quantum correlations. The computational basis states are constructed by systematically applying stabilizer operators to the initial reference state, creating entangled logical representations that encode quantum information robustly.

I’ll verify the state construction by carefully tracking the projective transformations and their resulting quantum configurations. The key insight involves understanding how the stabilizer generators manipulate the initial computational basis state to generate the desired logical encoding.

The [[4,2,2]] code defines four logical states using symmetric superpositions across a 4-qubit register. Each logical qubit state represents a carefully constructed quantum configuration that maintains specific symmetry properties under stabilizer operations.

I’ll systematically check the stabilizer behavior by applying \(XXXX\) across different basis states, tracking how the quantum amplitudes transform while preserving the overall quantum state structure.

For the \(|00\rangle_L\) logical state, the \(XXXX\) operation generates a symmetric superposition that maps precisely back to the original logical state, confirming its stabilizer-invariant nature.

The \(ZZZZ\) operation similarly demonstrates fascinating quantum symmetry, with phase transformations that preserve the fundamental quantum configuration while introducing controlled sign modifications across different basis state components.

I need to carefully analyze the code space constraints, examining how the stabilizer generators \(S_1\) and \(S_2\) interact to define the permissible quantum states. The computational basis representations reveal intricate relationships between bit flips and phase shifts that determine the logical qubit’s structural integrity.

The stabilizer conditions require a nuanced exploration of how superposition states must transform under these symmetry operations, suggesting my initial encoding requires fundamental reconsideration. I’ll complete the partial thought from the previous rewritten thinking by copying over the next thinking to finish the sentence, and then summarize:

\(\bar{b}_4} \beta = \alpha + \beta\), which involves the bit-flipped states and their amplitudes. By noting that the complement of each bit satisfies \(\bar{b}_i = 1 - b_i\), the total complement of bits sums to \(4 - (b_1 + b_2 + b_3 + b_4)\). This leads to a symmetric condition where the amplitudes of complementary states are related through a phase factor that ensures the symmetry of \(S_2\)’s action.

The key observation is how the bit-wise complementation creates a precise relationship between state amplitudes, maintaining a specific phase structure that allows the symmetry operation to act trivially across the quantum states.

The code space emerges from bit strings with an even number of 1’s, forming symmetric pairs like \(|0000\rangle\) and \(|1111\rangle\), which are equally weighted in the quantum representation. These paired states create a 4-dimensional subspace with specific symmetry properties.

I’m investigating how logical \(Z\) operators interact with these basis states. By applying \(\bar{Z}_A = ZZII\) to different bit configurations, I’m checking which states remain invariant under this transformation. So far, the first two basis vectors \(|v_1\rangle\) and \(|v_2\rangle\) appear unchanged by this logical \(Z\) operation.

The systematic analysis continues, examining how these logical operators preserve or transform the carefully constructed quantum code space states. I notice \(\bar{Z}_B\) behaves differently from \(\bar{Z}_A\). Some basis states are +1 eigenstates, while others are -1 eigenstates. For \(v_1\), the eigenstate is +1, but for \(v_2\) and \(v_3\), the eigenstates are -1. The pattern suggests a complex interaction between the Z and I operators in this transformation.

I’m seeing potential inconsistencies in how these logical states map to computational basis states. The eigenvalue relationships hint at a non-trivial encoding strategy that requires careful analysis of the symmetries and transformations.

The current classification of \(|v_1\rangle\) and \(|v_4\rangle\) as \(|00\rangle_L\) seems premature. I need to investigate the precise mapping and symmetry properties more rigorously before drawing definitive conclusions about the logical state representations.

The detailed eigenvalue checks for \(\bar{Z}_A\) reveal a pattern of sign changes for states with certain bit configurations, suggesting a more complex stabilizer code structure than initially assumed.

I’ll systematically analyze the logical operator \(\bar{Z}_A\) by examining its action on different basis states, noting which states get negated and which remain unchanged. This careful examination helps understand the code’s quantum error correction properties.

The stabilizer generators \(S_1 = XXXX\) and \(S_2 = ZZZZ\) provide key insights into the [[4,2,2]] code’s symmetries, with logical operators \(\bar{X}_A, \bar{X}_B, \bar{Z}_A, \bar{Z}_B\) defining the logical qubit transformations.

I’ll start by using \(|0000\rangle\) as a reference state to explore the code space’s structure, carefully projecting to find the logical basis states.

The normalization process reveals an interesting superposition state proportional to \(|0000\rangle + |1111\rangle\), which appears to represent a logical zero state with specific symmetry properties.

I verify this state is an eigenstate for both \(\bar{Z}_A\) and \(\bar{Z}_B\) by demonstrating that applying these operators leaves the state unchanged. This suggests the state maintains its fundamental quantum configuration under these transformations.

To construct the next logical basis state, I plan to apply \(\bar{X}_B\) to the current state, which will likely generate a new logical qubit representation.

The resulting state involves a superposition of two basis states, specifically \(|1100\rangle + |0011\rangle\), indicating a controlled quantum state transition.

I then extend this approach to generate \(|10\rangle_L\) by applying \(\bar{X}_A\), which produces another superposition state of \(|1010\rangle + |0101\rangle\).

The process continues with an intention to create the final logical basis state by applying both \(\bar{X}_A\) and \(\bar{X}_B\) simultaneously, suggesting a systematic method for constructing a complete logical qubit basis.

Normalizing these transformations yields four distinct logical basis states, each represented as an equal superposition of two computational basis states. These normalized states span the logical qubit space, with specific transformations mapping between them through controlled quantum operations.

The ultimate goal appears to be constructing a logical GHZ state, which represents a key entanglement configuration for quantum error correction and robust quantum computation.

By carefully manipulating these logical basis states, the circuit successfully prepares a specific four-qubit quantum state with precise amplitude relationships, demonstrating controlled quantum state preparation.

I’ll now analyze the noise impact by examining the two-qubit depolarizing channel’s effect. This probabilistic quantum channel introduces controlled randomness across 15 possible two-qubit Pauli transformations, with a parameter \(p\) governing noise intensity. The circuit’s two sequential CNOT gates, each followed by this depolarizing process, will probabilistically perturb the initial quantum state.

The channel’s mathematical structure suggests a systematic way to model quantum decoherence, where the original state gets mixed with random two-qubit Pauli transformations proportional to the noise parameter.

I’ll continue tracking how this noise progressively modifies the quantum state through each gate operation, starting from the initial \(|0000\rangle\) basis state.

After applying \(CNOT_{21}\), the state becomes \(\frac{1}{\sqrt{2}}(|0000\rangle + |0110\rangle)\). When the depolarizing channel \(\mathcal{E}_{21}\) is applied, the state transforms probabilistically. I’ll represent this as a mixed state \(\rho_1\), where \((1-p)\) preserves the original pure state and \(p\) introduces random noise across possible two-qubit transformations.

The key observation is how the depolarizing channel introduces controlled randomness, mixing the original quantum information with a maximally mixed state across the two-qubit subspace. This probabilistic transformation captures the fundamental quantum noise model.

When Hadamard \(H_0\) is subsequently applied, the state undergoes a superposition transformation. The Hadamard gate creates quantum interference, spreading the initial state across the computational basis in a way that preserves the overall quantum coherence.

The controlled-NOT gate \(CNOT_{03}\) then entangles specific qubits, followed by an error channel \(\mathcal{E}_{03}\) that introduces potential quantum noise. I calculate the fidelity by comparing the resulting state \(\rho_3\) against the logical GHZ state, which represents a key entangled quantum configuration. This measurement reveals how closely the quantum circuit maintains its intended quantum information integrity after these transformative operations.

I’ll systematically analyze error propagation through the quantum circuit, focusing on how Pauli errors interact and transform during unitary evolution. By leveraging quantum mechanics’ linear properties, I can track how noise impacts the quantum state’s coherence and entanglement.

The ideal unitary evolution \(U\) and its noisy counterpart \(\tilde{U}\) provide a framework for understanding how quantum errors emerge and potentially compound across different quantum gates and error channels.

The depolarizing channel’s mathematical structure reveals a probabilistic model where quantum states can randomly transition between different Pauli error configurations, with each transition weighted by specific probability parameters.

I’ll complete the partial circuit operation by incorporating the CNOT and Hadamard gates across different qubit indices, tracking the quantum state’s transformation through a complex sequence of unitary and error-induced operations. The process involves probabilistic error injection and gate transformations that systematically modify the initial quantum state.

The fidelity calculation will quantify how closely the output state matches the desired logical GHZ state, integrating the probabilistic error model with the quantum circuit’s specific gate sequence.

I’ll systematically evaluate different error scenarios, starting with the no-error case where \(k_1 = k_2 = 0\), which yields a baseline fidelity of \((1-p)\), representing the circuit’s performance under minimal noise conditions.

When considering non-trivial Pauli errors on qubits 0 and 3, I need to analyze how these specific quantum operations interact with the logical GHZ state. The key challenge involves determining which Pauli transformations preserve or orthogonally map the state’s symmetry.

The GHZ state’s intricate structure suggests that not all Pauli errors will have equivalent impacts, requiring a careful examination of each potential transformation’s effect on the quantum information. I’ll systematically list the 16 two-qubit Pauli operators on qubits 0 and 3, showing how they transform the logical GHZ state. I’ll track the transformations carefully, noting the specific changes in state amplitudes and signs.

The first few Pauli operators reveal interesting state modifications. The identity operator leaves the state unchanged, while X and Y operators on the third qubit significantly alter the state’s amplitudes and complex phases. Each operator demonstrates a unique transformation of the logical GHZ state’s basis components.

I’ll continue examining how these Pauli operators modify the quantum state, tracking the precise changes in amplitudes and complex phases across different two-qubit Pauli combinations. \(Y_0I_3|\text{GHZ}_L\rangle\) continues the quantum state transformation, showing how different Pauli operators modify the GHZ state. The pattern reveals symmetric transformations with complex coefficients and sign variations across quantum basis states.

The subsequent quantum operations demonstrate intricate state manipulations, with \(Y_0X_3\) and \(Y_0Y_3\) exhibiting specific phase and sign changes. Notably, \(Y_0Y_3\) results in a negated GHZ state, highlighting the non-trivial nature of these quantum transformations.

The calculations reveal precise quantum state interactions, where each Pauli operator induces unique modifications to the initial quantum configuration, preserving underlying structural symmetries.

The preserved Paulis include identity and specific tensor product operators across different qubit indices. These operators maintain the core quantum state characteristics with minimal phase alterations.

For error scenarios following the second controlled-NOT operation, the fidelity calculations demonstrate probabilistic outcomes. Different Pauli error channels contribute distinctly to the overall quantum state probability distribution, with identity errors having the highest likelihood at \((1-p)\) and specific tensor product errors contributing smaller fractional probabilities around \(p/15\).

The quantum state evolution involves complex transformations across multiple qubits, tracking error propagation through controlled operations and Hadamard gates. These intricate quantum mechanical interactions require precise probabilistic modeling to capture the nuanced state transitions.

I define intermediate states \(|\psi_0\rangle\) and \(|\phi_0\rangle\) for the scenario with no initial error, showing how quantum gates modify the initial state. The calculations reveal subtle changes in quantum amplitudes, reflecting the probabilistic nature of quantum error correction processes.

The fidelity calculation sums over potential error configurations, systematically examining how different error paths impact the final quantum state’s integrity.

For the zero-error case, I observe the quantum state transforms with specific amplitude modifications, maintaining a normalized superposition across multiple basis states. The quantum error correction protocol tracks these precise transformations to detect and potentially correct quantum decoherence.

The key calculation reveals a nuanced probability distribution across different Pauli error channels, with three specific error modes contributing \(\sqrt{p(1-p)/15}\) to the overall quantum state evolution. These error modes correspond to specific tensor product operators acting on the quantum system.

The total contribution from the zero-error channel combines squared probability terms, capturing the intricate interplay between error suppression and quantum state preservation. This suggests a sophisticated error mitigation strategy embedded within the quantum error correction framework.

I’m preparing to analyze scenarios where the initial quantum state index \(k_1\) differs from zero, which will likely introduce additional complexity to the error correction calculation. I’m exploring how errors propagate through the quantum circuit, tracking the state transformations. The complexity increases with each gate and potential error, suggesting a systematic approach to analyzing fidelity degradation.

The initial state preparation and gate operations create intricate quantum state interactions. By explicitly computing the output density matrix for small error probabilities, I can extract the leading-order error terms that impact the quantum state’s integrity.

The goal is to understand how Pauli errors accumulate and modify the intended logical GHZ state, revealing the circuit’s noise sensitivity through precise mathematical analysis. The fidelity can be expressed as the trace of the logical GHZ state with the output state, considering a depolarizing channel followed by a unitary transformation. For a two-qubit system, this involves a probabilistic mixture of the unitary transformation and random Pauli errors, with the trace calculation leveraging the cyclic property to analyze state preservation.

I’ll explore the error patterns systematically. Two CNOT gates, each followed by a depolarizing channel, create a complex error landscape. By tracking the probabilities of different two-qubit Pauli errors after each gate, I can systematically compute the overall state fidelity. The key is understanding how these probabilistic error patterns accumulate and impact the quantum state’s integrity.

The error probabilities follow a structured distribution: a high probability of no error, and a uniform low probability across 15 possible non-identity Pauli errors for each two-qubit gate. This probabilistic framework allows a comprehensive analysis of potential quantum state deviations.

When no errors occur, the state perfectly matches the logical GHZ state, yielding a contribution that depends on the depolarizing channel’s error probability. The mathematical formulation captures the intricate relationship between gate operations, error channels, and quantum state preservation.

For scenarios with a single error after specific quantum gates, the contribution involves carefully tracking the error propagation. The calculation considers different error types and their impact on the quantum state’s fidelity, revealing how local errors modify the overall quantum information.

The analysis systematically explores error scenarios, computing their probabilities and effects on the quantum state. By examining each potential error configuration, we can quantify the robustness of the quantum error correction protocol under a depolarizing noise model.

The computation involves tracking state transformations after applying Hadamard gates and controlled-NOT operations, with careful attention to how Pauli errors on different qubits modify the quantum state’s evolution.

I’ll systematically apply each Pauli operator combination to the current quantum state, observing how the amplitudes and basis states change. The first few transformations reveal interesting state modifications, like \(I_2X_1\) shifting basis states through precise quantum manipulations.

The process requires meticulously tracking each potential error scenario across multiple qubits, demonstrating the complex quantum state evolution under different Pauli perturbations.

Notably, the transformations preserve the normalized \(\frac{1}{\sqrt{2}}\) amplitude while strategically rearranging quantum basis states, highlighting the fundamental symmetries inherent in quantum operator interactions.

The quantum gate operations systematically modify the quantum state, revealing intricate relationships between different computational basis configurations through precise unitary transformations.

Each operation demonstrates how single-qubit gates can induce complex phase and state reconfigurations within a two-qubit quantum system, showcasing the nuanced quantum mechanical principles governing such interactions.

The transformations highlight the probabilistic nature of quantum states, where gate applications generate superposition states with carefully calibrated amplitudes and relative phases.

Quantum gates like \(Y_2\), \(Z_2\), and \(I_1\) systematically manipulate the initial state \(\frac{1}{\sqrt{2}}(|0000\rangle + |0110\rangle)\), producing distinct quantum configurations through precise rotational and phase-shifting operations.

Each matrix transformation reveals intricate quantum state evolution, demonstrating how single-qubit operations can generate complex entangled quantum representations with nuanced amplitude and phase relationships.

The state transformations reveal symmetrical patterns across different Pauli error scenarios, suggesting underlying quantum computational principles that preserve state information while introducing controlled variations.

After applying Hadamard and CNOT operations, the states will undergo systematic transformations. For states with specific bit configurations, the Hadamard gate introduces superposition across computational basis states, while the CNOT gate implements controlled bit flipping based on the first qubit’s value.

The quantum circuit’s operations demonstrate intricate state manipulation techniques that maintain probabilistic relationships between initial and transformed quantum states.

I notice an interesting symmetry in the target GHZ state. Its invariance under certain error operations suggests robust quantum information encoding. By exploring error expansion techniques for small error probabilities, I can potentially characterize the state’s resilience more systematically.

The fidelity expansion provides a mathematical framework to quantify how small perturbations impact quantum state preservation, offering insights into the circuit’s error tolerance.

I’m now focusing on computing the first-order infidelity contributions by carefully analyzing error propagation through different circuit stages. The computational approach will help me systematically verify how various Pauli errors affect the quantum state’s integrity.

I’ll examine the state transformation when introducing specific errors at different points in the circuit preparation, tracking how these perturbations modify the initial GHZ state. By propagating Pauli errors through \(CNOT\) and Hadamard gates, I can determine which errors lead to orthogonal states.

My strategy involves methodically exploring error scenarios, starting with a specific case where a Pauli error \(P^{(21)}\) is applied before the second CNOT gate, and tracking its impact on the quantum state’s fidelity.

The key observation is how Hadamard gates transform different Pauli operators, creating interesting mappings between \(X\), \(Y\), and \(Z\) errors that reveal subtle symmetries in quantum error propagation.

For the CNOT gate between qubits 0 and 3, specific Pauli transformations emerge, showing how controlled operations modify error representations across entangled qubits. These transformations demonstrate non-trivial interactions between quantum gates and error operators.

When considering the Hadamard gate’s interaction with a Pauli error on qubits 2 and 1, the critical insight is that \(H_0\) leaves the error unchanged since it does not directly act on those qubits. This preservation of error structure highlights the selective nature of single-qubit gates in quantum error correction protocols.

The commutation of \(CNOT_{03}\) and \(P^{(21)}\) follows from their disjoint qubit interactions, enabling straightforward gate reordering. This allows me to manipulate the quantum state by rearranging gate applications without altering the underlying quantum information.

I can now explore which specific Pauli errors maintain the logical encoded state’s overlap, probing the robustness of the quantum error correction code.

The state \(|\text{GHZ}_L\rangle\) contains four basis states, and I’ll systematically examine how different Pauli operators transform this logical quantum state. By investigating the 15 non-identity Paulis acting on qubits 2 and 1, I can determine which preserve the state’s quantum coherence.

Some Paulis like \(X_2I_1\) and \(I_2X_1\) change the basis states, rendering them orthogonal to the original state. I’m looking for Paulis that map the computational basis states within the superposition to each other, maintaining the state’s fundamental structure.

My goal is to identify which specific Pauli operators can act on this logical qubit without fundamentally altering its quantum information encoding.

\(X_2X_1\) looks promising - it cycles through the four basis states \(|0000\rangle, |0110\rangle, |1001\rangle, |1111\rangle\) in a symmetric way, effectively preserving the state’s quantum configuration while permuting its components.

\(Y_2Y_1\) behaves similarly to \(X_2X_1\), maintaining the fundamental structure of the quantum state through a phase-equivalent transformation.

\(Z_2Z_1\) introduces problematic phase shifts. When applied to the GHZ state, it alters the relative signs between basis states, disrupting the delicate quantum interference pattern that defines the original state’s coherence.

Checking the detailed action of \(Z_2Z_1\) reveals a non-trivial phase transformation that breaks the state’s symmetry, preventing it from being a valid preservation operation.

In contrast, \(X_2X_1\) demonstrates a perfect state preservation, mapping the GHZ state exactly onto itself without introducing unwanted phase variations.

The algebraic manipulation reveals \(Y_2Y_1\) as a complex combination of \(X_2\) and \(Z_2\) operations, suggesting a more intricate quantum transformation.

Calculating \(Y_2Y_1\) requires careful tracking of sign changes and phase factors, indicating potential subtleties in the quantum operator’s behavior.

The computation shows unexpected sign and phase shifts when applying \(Y_2Y_1\) to the GHZ state, hinting at deeper quantum mechanical complexities in this specific operator interaction.

I’ll methodically break down the \(Y\) matrix operations, focusing on precise state transformations and sign conventions to ensure accurate quantum state manipulation.

\(Y_2Y_1\) generates complex phase shifts across different computational basis states, revealing intricate quantum interactions. The transformations show systematic sign changes and phase rotations between \(|0110\rangle\), \(|0000\rangle\), \(|1001\rangle\), and \(|1111\rangle\) states.

The final superposition state emerges as a balanced quantum combination, with carefully balanced amplitudes reflecting the underlying symmetries of the quantum system.

Calculating the overlap reveals zero interaction, indicating \(Y_2Y_1\) does not preserve the original quantum state. Only \(X_2X_1\) maintains state coherence among non-identity Pauli operators.

For errors after the first CNOT, the fidelity contribution depends on specific operator interactions, with identity and \(X_2X_1\) preserving state overlap, while other Pauli operators introduce destructive quantum interference.

The mathematical expression captures this nuanced quantum error dynamics, showing how \((1-p) + \frac{p}{15}\) represents the probabilistic preservation of quantum information under specific error conditions.

For the second scenario with \(E_1 = I\) and \(E_2 = P^{(03)}\), three specific Pauli operators (\(X_0X_3\), \(Y_0Y_3\), \(Z_0Z_3\)) maintain state integrity, resulting in a fidelity contribution of \((1-p) + \frac{p}{5}\) that reflects quantum error resilience.

Now, examining \(E_1 = P^{(21)}, E_2 = P^{(03)}\), the quantum state transformation involves commuting Pauli operators acting on disjoint qubits. The analysis reveals eight potential combinations where both \(P^{(21)}\) and \(P^{(03)}\) preserve the quantum state, indicating complex error correction mechanisms.

The investigation suggests intricate quantum error dynamics where operator interactions determine state preservation probabilities.

I’ll carefully track the systematic calculation of overlap probabilities, focusing on the nuanced interactions between different Pauli operators across specific qubit subspaces.

The formula \(F = \sum_{E_1 \in \mathcal{P}_{21}} \sum_{E_2 \in \mathcal{P}_{03}} P(E_1) P(E_2) |\langle \text{GHZ}_L | E_2 E_1 | \text{GHZ}_L \rangle|^2\) captures the probabilistic quantum error landscape, with \(F = (1-p)^2 \cdot 1\) representing the baseline error scenario.

I’m now examining the more complex error terms: first the single-CNOT error cases with probability \(\frac{p}{15}\), then investigating the intricate double-CNOT error interactions where the overlap calculations become substantially more nuanced.

The specific Pauli operators \(X_0X_3, Y_0Y_3, Z_0Z_3\) and \(X_2X_1\) provide critical insights into how quantum errors propagate through the entangled GHZ state’s logical subspace.

By analyzing the commutation properties of these operators, I can identify three distinct non-zero contribution scenarios, each representing a unique error configuration that preserves the underlying quantum information structure.

The final probability calculation reveals a precise quantification of these error interactions, scaling quadratically with the error probability \(p\) and capturing the subtle quantum error correction dynamics.

The algebraic manipulation systematically transforms the initial expression, consolidating similar terms and revealing an elegant representation of the quantum error probability landscape.

I simplify the equation by redistributing terms, combining fractional components and isolating polynomial terms in \(p\). The process involves carefully rearranging coefficients to expose the underlying mathematical structure.

The final form emerges as a compact polynomial representation, highlighting the intricate relationship between error probability and system dynamics through precise algebraic reduction.

I calculate the simplified expression, revealing \(1 - \frac{26p}{15} + \frac{56p^2}{75}\) as the condensed algebraic result.

I carefully track each algebraic transformation, noting potential computational nuances. The key steps involve systematically breaking down the polynomial expansion, tracking coefficient interactions, and maintaining precision through each algebraic manipulation.

My focus remains on verifying each algebraic step, ensuring no computational errors emerge during the complex polynomial reduction process.

For the Pauli operator contributions, I’m methodically examining each non-identity operator’s potential overlap and computational significance. Specifically, I’m tracking:

• Unique Pauli operator interactions
• Precise overlap calculations
• Contribution mechanisms for different operator configurations

The current analysis centers on distinguishing between zero and non-zero overlap scenarios, with particular attention to specific operator combinations like \(X_2X_1\) and \(X_0X_3, Y_0Y_3, Z_0Z_3\).

I notice the intricate probability scaling, where \(p\) and \((1-p)\) modulate the computational landscape, revealing nuanced interaction probabilities across different operator sets.

The fidelity calculation progressively transforms, systematically reducing complex fractional terms through algebraic manipulation. Each line simplifies the expression, tracking how probability components interrelate.

The final steps consolidate the probability terms, revealing a compact representation that balances linear and quadratic probability interactions, ultimately converging toward a refined probabilistic characterization.

The emerging equation suggests a sophisticated mapping between probabilistic states, where small parameter shifts dramatically influence overall system fidelity.

Initial verification reveals nuanced boundary behaviors: as probability approaches zero or one, the fidelity demonstrates non-trivial transformative properties requiring careful mathematical scrutiny.

The unexpected low fidelity at maximum probability triggers deeper investigation, signaling potential complexity in the underlying quantum mechanical interactions.

Algebraic manipulation exposes an elegant factorization revealing intricate dependencies between probability parameter and state transformation, suggesting sophisticated quantum noise dynamics.

The derived expression captures probabilistic quantum state degradation through a compact, revealing formula that maps non-linear distortion mechanisms across different probability regimes.

This is asking for the logical state fidelity after post-selecting on no detectable errors.

In the [[4,2,2]] code, errors are detectable if they anti-commute with at least one stabilizer (XXXX or ZZZZ).

Let me analyze what happens:

1. We start with the circuit that prepares the logical GHZ state
2. Errors occur after each CNOT gate
3. We post-select on cases where no detectable error occurred
4. We calculate the logical fidelity of the resulting state

From the previous problem, I know: - The circuit correctly prepares \(|\text{GHZ}_L\rangle = \frac{1}{2}(|0000\rangle + |1111\rangle + |0110\rangle + |1001\rangle)\) with no errors - Errors \(E_1\) occur after CNOT_{21} (on qubits 2,1) - Errors \(E_2\) occur after CNOT_{03} (on qubits 0,3)

For an error to be undetectable, it must commute with both stabilizers XXXX and ZZZZ.

Let me check which two-qubit Paulis on qubits (2,1) commute with both stabilizers: - \(I_2I_1\): commutes with both (identity) - \(X_2X_1\): commutes with XXXX, commutes with ZZZZ - \(Y_2Y_1\): commutes with XXXX, commutes with ZZZZ
- \(Z_2Z_1\): commutes with XXXX, commutes with ZZZZ

For qubits (0,3): - \(I_0I_3\): commutes with both (identity) - \(X_0X_3\): commutes with XXXX, commutes with ZZZZ

• \(Y_0Y_3\): commutes with XXXX, commutes with ZZZZ
• \(Z_0Z_3\): commutes with XXXX, commutes with ZZZZ

I’ve identified the complete set of undetectable errors for both qubit pairs. These errors can occur without being detected by the stabilizer measurements, which means they won’t trigger a post-selection rejection. This is crucial for understanding the potential state modifications that can happen silently during the quantum error correction process.

The next step involves examining how these specific error combinations interact with the logical GHZ state. I’ll investigate the effects of each error type, focusing on how they transform the encoded quantum information.

For \(Y_2Y_1\), I’m starting to analyze its impact on the logical state, carefully tracking the potential phase and amplitude changes that might occur during this transformation.

The calculation reveals that \(Y_2Y_1\) introduces a negative sign, effectively inverting the entire logical GHZ state. This suggests a specific symmetry in how certain Pauli operators interact with the encoded quantum information.

For \(Z_2Z_1\), the computation shows a more nuanced effect, where some basis states maintain their sign while others change, indicating a complex interaction between the error operator and the logical state’s structure.

I’ll carefully examine the logical basis states to understand their precise composition and how these transformations impact the quantum code’s fundamental properties.

The logical states represent a carefully constructed superposition across different four-qubit configurations, each encoding a specific two-qubit logical state through intricate quantum correlations.

When applying \(Z_2Z_1\) to the logical GHZ state, I observe an interesting orthogonality emerging between different logical state representations, revealing subtle quantum mechanical interactions within the error-correcting code.

The density matrix computation will require tracking the probabilities of undetected errors across specific qubit configurations, suggesting a complex probabilistic landscape of quantum state evolution.

I’ll systematically explore error scenarios, noting how different Pauli error combinations transform the initial logical state while maintaining specific probabilistic constraints.

The first few cases demonstrate how identity and small perturbative errors preserve core state characteristics, with probabilities scaling proportionally to the error rate \(p\).

For the specific case of \(E_1 = Y_2Y_1, E_2 = I\), the probability remains \(\frac{p}{15}(1-p)\), interestingly producing the same logical state with only a global phase shift. This suggests robustness against certain correlated two-qubit errors.

The \(Z_2Z_1\) error introduces more substantial state modification, transitioning to an equal superposition between computational basis states, highlighting the nuanced error dynamics within the logical encoding.

Continuing to track these probabilistic transformations reveals intricate error correction mechanisms inherent in the logical qubit representation.

The next error configuration involving \(I\) and \(Y_0Y_3\) presents another symmetric probability of \((1-p)\frac{p}{15}\), maintaining the global phase preservation characteristic of these quantum error scenarios.

For double errors, the quantum state transformations reveal interesting symmetries. Some configurations like \(X_2X_1\) with \(Y_0Y_3\) or \(Z_0Z_3\) maintain the \(|\text{GHZ}_L\rangle\) state, while others introduce a global phase inversion.

Specific double error patterns demonstrate consistent quantum state behavior, with some transformations resulting in an unchanged state and others inducing a sign change, reflecting the intricate quantum error correction dynamics. \(X_0X_3\) acts on logical states by considering its relationship to logical operators. The definition involves physical qubits, but the key insight is understanding logical operators up to stabilizer multiplication. The physical operator \(X_0X_3\) might relate to logical operations in ways not immediately apparent from direct computation.

Examining the physical state transformation reveals a symmetric pattern of bit flips across the four-qubit system. The operator induces a specific sign change and state permutation that suggests deeper structural properties of the logical encoding.

Exploring the interaction between \(X_0X_3\) and the logical GHZ state shows a precise relationship between physical and logical operators. The computation demonstrates how certain tensor products transform the logical state with predictable sign and configuration changes.

Investigating the subsequent operator composition \(Y_0Y_3Z_2Z_1\) indicates a systematic approach to understanding quantum state manipulations within this logical subspace. The emerging pattern hints at intricate connections between different operator combinations and their effects on the logical quantum state.

The matrix representation of the \(Y\) gate reveals critical phase and sign transformations during quantum gate operations, suggesting a nuanced approach to tracking quantum state evolution through precise computational steps.

Applying \(Y_0Y_3\) to different basis states yields specific phase and sign changes: \(|0000\rangle\) transforms to \(|1001\rangle\), \(|1111\rangle\) becomes \(|0110\rangle\), \(|0110\rangle\) shifts to \(-|1111\rangle\), and \(|1001\rangle\) converts to \(-|0000\rangle\). These transformations demonstrate the complex interactions between quantum gates and computational basis states.

The resulting state represents a symmetric superposition, which when further manipulated by \(Z_2Z_1\) and \(Z_0Z_3\) gates, preserves its fundamental logical qubit structure. Specifically, the logical GHZ state remains invariant under these specific gate operations, highlighting its robust quantum encoding.

I compute the gate transformations systematically, noting how \(Z_0Z_3\) interacts with basis states. For \(|0000\rangle\), the gate leaves the state unchanged, while for \(|1111\rangle\), the gate introduces a phase factor that effectively preserves the state’s overall configuration.

When applying \(Z_0Z_3\) to \(|0110\rangle\) and \(|1001\rangle\), I observe a sign change, with each state acquiring a negative phase. This reveals a consistent pattern of gate interactions across different computational basis states.

Combining these observations leads to a simplified representation of the transformed logical qubit state, showing how local Pauli operations modify the quantum information encoded in the logical subspace.

The computation demonstrates that \(Z_0Z_3\) transforms the logical basis states in a predictable manner, preserving the overall quantum state structure while introducing phase variations.

By applying \(Z_0Z_3\) to the logical GHZ state, I’ve shown that \(|00\rangle_L\) remains unchanged and \(|11\rangle_L\) acquires a negative sign. This results in the original GHZ state, confirming the stabilizer property.

The calculation reveals a key symmetry in the logical quantum error correction code, where certain combinations of Pauli operators leave the encoded quantum state invariant.

The probabilities span multiple error scenarios, ranging from single-qubit errors to correlated two-qubit errors, with probabilities scaling as \((1-p)^2\), \((1-p)p/15\), and \((p/15)^2\). These probabilities capture the intricate error dynamics of the quantum code.

The error configurations include interactions across different qubit pairs, with specific probability distributions reflecting the underlying quantum noise model. Each configuration represents a potential error pathway in the quantum error correction scheme.

The subsequent error scenarios introduce additional complexity, particularly focusing on \(Z_2Z_1\) interactions with identity operations, suggesting nuanced error propagation mechanisms within the quantum system.

The total probability calculation for generating the logical GHZ state involves intricate combinations of error probabilities, capturing the probabilistic nature of quantum state preparation under noisy conditions.

By combining the probabilities of different error scenarios, we derive a comprehensive expression that accounts for successful state generation and potential error modes. The probability includes squared and linear terms of the error parameter, reflecting the complex quantum error correction landscape.

The post-selection success probability integrates multiple quantum state configurations, systematically accounting for different error pathways and their respective likelihoods.

The resulting probability expression simplifies to a quadratic form in the error parameter, capturing the nuanced probability distribution across different quantum states.

The unnormalized density matrix incorporates two distinct quantum states: the GHZ state and an orthogonal perpendicular state, defined as an equal superposition of logical zero and logical one states.

The normalization process involves dividing each state’s probability by the total probability, creating a weighted representation that preserves the quantum state’s essential characteristics.

The logical fidelity calculation begins by examining the overlap between the GHZ state and the normalized density matrix, suggesting a quantitative measure of quantum state preservation.

I’ll compute the fidelity by simplifying the complex probability expressions, focusing on how different error probabilities contribute to the quantum state’s integrity.

The numerator’s algebraic manipulation reveals intricate relationships between error probabilities, showing how quantum states degrade and interact under different conditions.

By carefully rearranging terms and consolidating \(p^2\) coefficients, I’m tracking how quantum error correction mechanisms might influence the overall system’s performance.

The denominator’s transformation suggests similar complexity in tracking quantum state fidelity, with multiple error probability terms interacting in nuanced ways.

The emerging equation reveals a sophisticated relationship between error probabilities and system resilience, where \(p^2\) terms play a critical role in characterizing quantum information preservation.

I’m carefully tracking the probability contributions from different quantum state configurations. These include scenarios involving identity operations and specific two-qubit gate interactions, each contributing a nuanced probability component to the overall system dynamics.

The current analysis suggests multiple pathways through which quantum states can transition, with probabilities ranging from \((1-p)^2\) for perfect alignment to fractional probabilities like \(\frac{p}{15}(1-p)\) representing more complex quantum interactions.

I notice the probabilities systematically explore combinations of \(X\), \(Y\), and \(Z\) gates across different qubit positions, methodically mapping out potential quantum state transformations with precise probability calculations.

The pattern reveals symmetric probability distributions around \(\frac{p^2}{225}\) for various gate combinations, suggesting a structured quantum computational approach where each configuration contributes a specific probabilistic contribution.

The total probability calculation aggregates these individual probabilities, combining squared and mixed terms to represent the comprehensive quantum state evolution probability.

Orthogonal states emerge through specific gate interactions, generating complementary quantum configurations with precise probability amplitudes, like \(\frac{1}{\sqrt{2}}(|00\rangle_L - |11\rangle_L)\) under different gate combinations.

The total probability calculation involves intricate summation of quantum state probabilities, consolidating terms to simplify the complex expression across different quantum configurations.

Probability terms progressively combine, reducing fractional components and revealing underlying quantum state interaction patterns through systematic algebraic manipulation.

The logical fidelity calculation requires careful expansion of both numerator and denominator, methodically transforming polynomial expressions to expose fundamental quantum probability relationships.

Initial algebraic steps involve distributing terms, combining like coefficients, and preparing for further simplification of the quantum probability representation.

I’ll consolidate the \(p^2\) terms in the numerator, revealing a compact representation that reduces multiple fractional components to a single, more elegant expression. The result transforms the complex initial form into \(1 - \frac{5p}{3} + \frac{158p^2}{225}\), which captures the essential probabilistic interactions more concisely.

The denominator expansion follows a similar algebraic strategy, systematically breaking down compound terms to reveal underlying structural relationships between quantum probability parameters.

The final fractional representation \(F_L\) emerges from this careful algebraic manipulation, presenting a nuanced relationship between the numerator and denominator terms that encapsulates the probabilistic dynamics of the system.

By converting each term to a common denominator, I’ve transformed the complex fractions into a more structured multiplicative form. The numerator now reflects \(25(225 - 375p + 158p^2)\), while the denominator represents \(225(25 - 40p + 16p^2)\), revealing a systematic approach to simplifying the mathematical expression.

This reformulation allows for a clearer understanding of how the polynomial terms interact, providing insight into the underlying mathematical structure without losing the essential computational complexity.

I notice potential discrepancies in my previous calculation and decide to carefully re-examine the algebraic manipulation, sensing an opportunity to uncover a more precise representation of the function.

At zero, the function clearly evaluates to 1. For small values of \(p\), I expand both numerator and denominator to first-order terms, revealing a subtle relationship between their coefficients.

By multiplying the expanded terms, I derive an approximation that converges to \(1 - \frac{p}{15}\), which suggests an intriguing pattern in the function’s local behavior near the origin.

I’m now carefully examining the state transformations, particularly focusing on how \(Z_2Z_1\) interacts with the GHZ state’s complex structure. The specific action of this operator on the encoded quantum state requires precise analysis.

The result reveals an orthogonal state emerges after this transformation, which fundamentally changes the quantum information landscape. When post-selection occurs, we’re left with a mixed quantum state containing two distinct components.

The key insight involves calculating the logical fidelity, which depends on the probability of successfully preserving the original quantum information after error processes. This requires tracking the probabilities of different error scenarios and their impacts on quantum coherence.

By multiplying the numerator and denominator by 225, I can simplify the complex fidelity expression, revealing the intricate relationship between error probability and quantum state preservation.

The emerging formula captures the nuanced interplay between quantum error correction mechanisms, showing how different error probabilities fundamentally alter the quantum state’s integrity.

The final logical fidelity expression elegantly represents the system’s resilience, demonstrating how quantum information can be maintained under probabilistic error conditions.

Let me work through this step-by-step.

First, I need to understand the circuit. The circuit is:

This is in matrix multiplication order, so the operations occur in reverse:

1. \(H_1\) on qubit 1
2. \(CNOT_{12}\) with control 1, target 2
3. \(CNOT_{10}\) with control 1, target 0
4. \(CNOT_{23}\) with control 2, target 3
5. \(CNOT_{34}\) with control 3, target 4
6. \(CNOT_{04}\) with control 0, target 4
7. \(M_4\) measure qubit 4

Let’s trace through this without errors first. We start with \(|00000\rangle\) (qubits 0,1,2,3,4).

1. \(H_1|00000\rangle = \frac{1}{\sqrt{2}}(|00000\rangle + |01000\rangle)\)
2. \(CNOT_{12}\): \(\frac{1}{\sqrt{2}}(|00000\rangle + |01100\rangle)\)
3. \(CNOT_{10}\): \(\frac{1}{\sqrt{2}}(|00000\rangle + |11100\rangle)\)
4. \(CNOT_{23}\): \(\frac{1}{\sqrt{2}}(|00000\rangle + |11110\rangle)\)
5. \(CNOT_{34}\): \(\frac{1}{\sqrt{2}}(|00000\rangle + |11111\rangle)\)

6. \(CNOT_{04}\): \(\frac{1}{\sqrt{2}}(|00000\rangle + |11111\rangle)\)

7. When measuring qubit 4, there are two possible outcomes: \(|0\rangle\) with 50% probability resulting in \(|00000\rangle\), or \(|1\rangle\) with 50% probability resulting in \(|11111\rangle\).

I’m skeptical about my previous calculation and need to verify the steps more rigorously.

Tracing the quantum circuit from the initial state \(|00000\rangle\), I carefully apply each gate:

1. Hadamard on first qubit creates superposition
2. CNOT gates progressively entangle qubits
3. Final state shows equal probability of \(|00000\rangle\) and \(|11111\rangle\)

Measurement of qubit 4 will collapse the state to one of these two possibilities.

When measuring qubit 4, the post-selection process reveals two distinct scenarios: - Zero outcome means qubits 0-3 collapse to \(|0000\rangle\) - One outcome means qubits 0-3 collapse to \(|1111\rangle\)

This suggests a deeper exploration of the [[4,2,2]] quantum error correction code’s logical states might provide additional insights.

The projection of \(|0000\rangle\) involves complex quantum operations, specifically applying stabilizer generators to transform the initial state into a valid code space representation. This projection introduces multiple quantum state modifications through different stabilizer combinations.

After normalization, the process yields the logical qubit state \(|00\rangle_L\), which emerges when the ancilla measurement results in \(|0\rangle\). This represents a successful quantum error correction code preparation.

The subsequent analysis focuses on potential quantum errors introduced by two-qubit depolarizing channels following each CNOT gate. These channels probabilistically apply various two-qubit Pauli operators, creating a complex error landscape with 15 potential non-identity transformations.

I’ll track specific error locations: - \(E_1\) occurs after \(CNOT_{12}\) - \(E_2\) follows \(CNOT_{10}\) - \(E_3\) emerges after \(CNOT_{23}\) - \(E_4\) appears after \(CNOT_{34}\) - \(E_5\) happens after \(CNOT_{04}\)

The final quantum state becomes intricate, representing a sequence of CNOT gates and potential error transformations applied to the initial \(|00000\rangle\) state.

The ancilla qubit serves a critical error detection mechanism. Its measurement reveals system integrity: a \(|0\rangle\) indicates no errors, while a \(|1\rangle\) signals specific error occurrences.

The circuit’s quantum state evolves through strategic CNOT operations. After initial Hadamard and CNOT gates, the state transforms to \(\frac{1}{\sqrt{2}}(|00000\rangle + |11110\rangle)\), demonstrating quantum superposition.

Subsequent CNOT gates like \(CNOT_{34}\) and \(CNOT_{04}\) propagate and entangle qubit states, with qubit 4 capturing the XOR of qubits 0 and 3.

In error-free scenarios, both quantum branches yield zero ancilla measurements. However, single-qubit errors can trigger distinct measurement outcomes, enabling selective error detection within the quantum system.

Undetectable errors must satisfy two critical constraints: they must commute with the code’s stabilizers and preserve the ancilla’s measurement state. Specifically, errors flipping both qubits 0 and 3, or neither, won’t alter the ancilla’s measurement result.

I’m developing a systematic approach to analyze potential error sequences. The circuit preparation involves post-selection under two key conditions: no detectable errors and a zero ancilla measurement. My goal is to comprehensively map error probabilities and their corresponding logical quantum states.

To achieve this, I’ll first establish the circuit’s baseline behavior by carefully tracing its state transformations from the initial \(|00000\rangle\) configuration, examining how Hadamard and CNOT gates progressively modify the quantum representation.

The measurement of qubit 4 introduces a probabilistic collapse, yielding either \(|0000\rangle\) or \(|1111\rangle\) with equal likelihood. Post-selection on the zero state provides a critical starting point for understanding potential error dynamics.

The projection onto the code space reveals a key transformation. By applying specific quantum operators, the state evolves to an unnormalized logical qubit state \(|00\rangle_L\), highlighting the circuit’s intricate quantum error correction mechanism.

I’m analyzing the circuit’s fault-tolerant design, focusing on how single errors might propagate through the quantum gates. The ancilla qubit plays a crucial role in error detection, suggesting a sophisticated approach to maintaining quantum coherence.

The complexity of tracking error propagation requires a systematic breakdown of potential quantum state modifications.

Examining the logical operators and stabilizers provides insight into the code’s error detection capabilities. Single-qubit errors that commute with stabilizers remain undetectable, with the identity operation being the only such transformation.

For two-qubit interactions, specific undetectable error patterns emerge. These include the identity operation on both qubits, coordinated X and Z operations, and specific combinations of Y rotations between qubits.

I’ll need to systematically analyze how these potential errors propagate through the quantum circuit, tracing the state evolution and tracking potential error mechanisms.

The key insight is that fault-tolerant circuits should detect most single-qubit errors through ancilla measurements. This means I’m looking for sophisticated error patterns that could potentially bypass detection.

My focus will be on identifying error sequences that: - Maintain the overall quantum state’s integrity - Avoid triggering ancilla qubit state changes - Represent coordinated, subtle perturbations across multiple qubits

Specifically, I’m analyzing error probabilities:

1. No error: \((1-p)^5\)
2. Single errors: \(5p(1-p)^4\)
3. Double errors: \(10p^2(1-p)^3\)

These probabilities reveal potential scenarios where quantum error correction might fail, particularly in complex multi-qubit interactions where error patterns could strategically “cancel out” or remain undetected.

The circuit prepares a specific entangled state \(\frac{1}{\sqrt{2}}(|0000\rangle + |1111\rangle)\). When no errors occur and the ancilla measurement is zero, the state collapses to \(|0000\rangle\), which maps to the logical \(|00\rangle_L\).

I’m investigating potential undetectable errors that could modify the logical state while maintaining ancilla = 0. These include: - Logical \(\bar{X}_A\) transforming to \(|10\rangle_L\) - Logical \(\bar{X}_B\) transforming to \(|01\rangle_L\) - Logical \(\bar{X}_A\bar{X}_B\) transforming to \(|11\rangle_L\)

The fault-tolerant design means single errors should either trigger the ancilla or preserve the logical state. Double errors present a more complex scenario, potentially implementing logical operations while evading detection.

The critical insight is the circuit’s robust error-handling mechanism: single errors are either caught by the ancilla measurement or fundamentally unable to corrupt the quantum information.

I’m breaking down the circuit’s structure to understand potential error propagation. The preparation and entanglement stages create a sophisticated error detection framework, with the ancilla playing a crucial role in identifying specific error patterns.

The probabilistic nature of error occurrence suggests a nuanced approach to analyzing potential state corruptions, focusing on how different error combinations might interact with the quantum circuit’s inherent protections.

By examining the parity of specific qubit sets and tracking how errors might commute with stabilizer measurements, I can map out scenarios where errors remain undetected or potentially compromise the logical state.

The undetectable error patterns reveal fascinating symmetries in the quantum error correction mechanism, particularly around how certain error transformations preserve the initial quantum information without triggering the ancilla’s detection mechanism.

I’m focusing on understanding which error combinations maintain the logical zero state while evading the code’s error detection capabilities, considering both single and double error scenarios across different qubit interactions.

The key challenge involves identifying precise error patterns that simultaneously satisfy three critical conditions: preserving ancilla measurement, remaining undetectable, and maintaining the intended logical quantum state.

My analysis centers on probabilistic error contributions, particularly examining how specific error sequences might slip past the quantum error correction mechanism without fundamentally altering the encoded quantum information.

I’m systematically exploring error propagation through the stabilizer circuit, recognizing that fault-tolerant design introduces nuanced constraints on potential error transformations.

The dominant fidelity contribution emerges from error-free preparation, with single and double error scenarios presenting increasingly complex computational challenges for precise characterization.

My current focus shifts to granular examination of individual CNOT gate interactions and their potential error susceptibilities, tracing how specific error interventions might subtly modify the quantum state’s fundamental properties. I’m tracking the complexity of error propagation in a quantum circuit, focusing on fault-tolerant state preparation. The circuit involves 5 CNOTs with an error probability \(p\), and I’m approximating the logical fidelity.

To first order, the no-error probability is close to 1, with undetectable errors being very low probability. I recognize the need for a systematic approach to identifying specific error patterns that might corrupt the logical state while evading detection tests.

The key challenge is understanding how errors might slip through the ancilla measurements and potentially compromise the quantum information. I’ll need to carefully map out potential error scenarios in the [[4,2,2]] code structure. In a well-designed quantum error correction circuit, single errors that bypass detection mechanisms are unlikely to significantly disrupt the logical qubit state.

To first-order approximation, the logical fidelity remains essentially unchanged, with error contributions scaling quadratically with the error probability. I’m interested in identifying specific double error patterns that could introduce subtle state modifications.

The circuit preparation involves carefully constructing a quantum state that requires precise tracking of potential error mechanisms across multiple qubits. By systematically analyzing error propagation, I can map out how different error combinations might impact the final logical state.

Starting from the initial state, I’ll trace the quantum gates and their potential error transformations, focusing on how CNOT operations might introduce or propagate quantum errors across the multi-qubit system.

When measuring the ancilla qubit as 0, the state collapses to \(\frac{1}{\sqrt{2}}(|0000\rangle + |1111\rangle)\), representing a logical encoded state. However, subtle error mechanisms could potentially compromise the quantum information.

Specific error scenarios might involve single-qubit X errors that preserve the ancilla’s state while subtly altering the underlying quantum information. An X error on qubit 1, for instance, could silently flip the logical state without triggering ancilla detection, presenting a nuanced challenge in quantum error correction.

The key concern involves identifying error patterns that maintain the ancilla’s zero state while introducing undetected quantum state modifications across the computational qubits.

Probabilistically, the logical fidelity depends on: - Perfect state preservation: \((1-p)^5\) - Single error detection mechanisms - Double error corruption probability: \(O(p^2)\)

Consequently, the logical fidelity approximates \(F_L \approx \frac{(1-p)^5}{(1-p)^5 + c \cdot p^2}\), capturing the delicate balance between error introduction and quantum information preservation.

To first and second order approximations, the fidelity trends toward unity, with subtle \(p^2\) corrections emerging from complex error interactions.

Investigating specific error patterns reveals nuanced detection mechanisms. Two-qubit errors exhibit intriguing commutation properties with stabilizer operators, determining their detectability through intricate quantum mechanical interactions.

For an undetectable two-qubit error in the [[4,2,2]] code, only specific error forms are permissible: identity, XX, YY, or ZZ on identical qubits.

The logical fidelity calculation involves a probabilistic ratio, capturing the system’s quantum state preservation under noisy conditions. Key contributions arise from error-free scenarios and specific error pattern probabilities, tracking the quantum information’s resilience across computational processes.

I’ll focus on understanding how error patterns might subtly corrupt the logical quantum state while evading detection mechanisms. The goal is to characterize the system’s fault tolerance by examining how paired errors could potentially implement undetected logical transformations.

The mathematical structure suggests a quadratic error dependence, with the logical fidelity potentially maintaining near-perfect preservation under low-noise conditions. Specific error pair interactions become critical in understanding the quantum error correction’s fundamental limits.

Exploring these intricate error pathways reveals how seemingly independent quantum operations could collectively induce subtle logical state modifications, challenging the initial assumption of complete error protection.

The logical fidelity appears bounded by quadratic error probabilities, with specific error pair configurations potentially compromising quantum information integrity without triggering standard detection mechanisms.

Single-qubit errors are systematically detectable in this quantum error correction framework, but two-qubit correlated errors present a more nuanced challenge to maintaining quantum state fidelity.

Investigating undetectable two-qubit Pauli error patterns becomes crucial for understanding the precise boundaries of fault-tolerance in this quantum circuit design.

The key insight involves tracking how error pairs might subtly modify logical states without triggering immediate detection, suggesting a complex interplay between error mechanisms and quantum information preservation.

Analyzing potential error transformations reveals nuanced state modifications: single errors might leave the logical state intact, while paired errors could introduce subtle phase or bit flip variations within the logical subspace. The critical challenge involves precisely characterizing these minute state perturbations at the \(O(p^2)\) error scale.

Computational basis states demand exact representation, meaning any deviation - even phase shifts - constitutes a meaningful logical error. This necessitates careful enumeration of potential error pathways that could imperceptibly alter the quantum information.

The fundamental goal remains quantifying how probabilistic errors might incrementally degrade the logical state’s fidelity, requiring meticulous tracking of potential error transformation mechanisms. I’ll complete the partial thought from the previous thinking first, then summarize:

CNOT_04 introduces additional complexity to the error pattern. The two undetectable errors \(E_1 = X_1X_2\) and \(E_5 = X_0X_3\) create an interesting scenario where the errors commute with the stabilizer generators.

The key observation is that these specific error combinations don’t necessarily corrupt the logical quantum information. They implement \(XXXX\), which is a stabilizer, meaning the logical state remains fundamentally unchanged. This reveals a nuanced behavior in quantum error correction where not all error combinations lead to logical state degradation.

To implement a true logical error, more sophisticated error combinations are required, specifically targeting logical operators like \(\bar{X}_A\) or \(\bar{X}_B\). However, individual two-qubit errors from the noise model cannot directly achieve this transformation.

The analysis suggests a complex error suppression mechanism where certain error patterns preserve the logical quantum state, hinting at the robust nature of the quantum error correction approach.

The fault-tolerant circuit demonstrates remarkable error resilience:

1. Clean state preparation has high probability \((1-p)^5\)
2. Most error patterns are detectable
3. Logical fidelity remains extremely high, with corrections at \(O(p^2)\)

The logical fidelity equation \(F_L = 1 - \alpha p^2 + O(p^3)\) captures this error suppression, showing how quantum error correction can maintain quantum information integrity.

Undetectable errors emerge from specific two-qubit error patterns: - Correlated X, Y, Z errors on identical qubit pairs - Each undetectable error occurs with \(p/15\) probability

The challenge involves meticulously tracking error combinations that:

1. Preserve ancilla state
2. Potentially alter logical qubit state

A generalized fidelity expression will capture these nuanced error propagation mechanisms, revealing the delicate balance between error introduction and correction in quantum circuits.

The expected fidelity follows a polynomial expansion in error probability, with second-order terms revealing critical error dynamics. These terms capture both independent and correlated error scenarios across circuit elements.

Specifically, second-order error contributions arise from two distinct scenarios: independent single errors across different circuit locations and simultaneous double errors from identical CNOT operations.

For independent single errors: - 10 possible two-CNOT error configurations exist - Each configuration involves 225 unique error combinations - Probability scales with \((p/15)^2 (1-p)^3\)

For double errors within a single CNOT: - 5 potential CNOT locations - 210 unique two-qubit error combinations per location - Probability characterized by \((p/15)^2 (1-p)^4\)

The complexity suggests nuanced error propagation mechanisms requiring careful probabilistic analysis. At \(p^2\), two CNOTs experience errors out of five total, with 10 possible error placement configurations. Each configuration involves 225 potential error combinations, occurring at \((p/15)^2(1-p)^3\) probability.

The goal is to systematically analyze these 2250 two-error scenarios, determining which pass detection tests and map to specific logical states. A fault-tolerant circuit’s fidelity depends on carefully characterizing these error patterns.

The expected fidelity structure balances error-free performance against specific two-error correction mechanisms, represented by a ratio comparing ideal and error-influenced circuit behaviors.

The key insight emerges in the coefficient comparing successful \(|00\rangle_L\) preservation versus alternative logical state transitions. This coefficient quantifies the circuit’s resilience against specific error configurations, suggesting a linear degradation mechanism at \(p^2\) scale.

While the exact calculation remains complex, the mathematical framework reveals how quantum error correction fundamentally tracks state preservation probabilities across different error scenarios.

The potential exists for a perfectly designed fault-tolerant circuit where all error patterns maintain the original logical state, though practical implementations likely involve some state corruption probability.

The emerging model suggests a quadratic error suppression mechanism, with a constant \(c\) capturing the nuanced error dynamics specific to the quantum circuit’s architectural design.

I’m contemplating the intricate error detection landscape, where single errors might be fully detectable, but subtle two-error combinations could potentially compromise the logical qubit’s integrity without triggering standard detection mechanisms.

Ultimately, a comprehensive error pattern analysis would require an exhaustive computational approach beyond quick analytical estimation, revealing the complex probabilistic nature of quantum error correction.

The logical fidelity emerges as a nuanced function: \(F_L = 1 - \alpha p^2 + O(p^3)\), where the coefficient \(\alpha\) quantifies specific two-error corruption pathways. Perfect fault tolerance implies \(\alpha = 0\), while realistic scenarios suggest a second-order fidelity reduction.

The critical insight lies in post-selection criteria: we retain only outcomes where detected errors commute with the [[4,2,2]] code’s stabilizer generators, fundamentally constraining potential state transformations.

Potential error patterns must simultaneously satisfy two stringent conditions: triggering zero ancilla measurements while preserving the logical \(|00\rangle_L\) state. This suggests errors must be equivalent to stabilizer products, carefully avoiding logical operator interventions that would disrupt the quantum state’s integrity.

Logical Z errors pose no threat since \(|00\rangle_L\) remains stable under such transformations. However, logical X errors would fundamentally alter the quantum information, potentially mapping the state to alternative logical configurations.

The critical investigation centers on identifying whether noise-induced error patterns can simultaneously evade detection and implement a logical X error - a determination that directly impacts the logical fidelity \(F_L\).

While initially suspecting perfect fault tolerance, deeper analysis suggests a more nuanced scenario. Two-error interactions might strategically circumvent detection mechanisms, introducing a second-order fidelity reduction characterized by \(1 - \alpha p^2 + O(p^3)\). Such subtle error correlations could compromise quantum information integrity, even within seemingly robust fault-tolerant architectures.

Determining the precise \(\alpha\) coefficient requires exhaustive examination of potential two-error configurations, potentially involving systematic enumeration across a substantial pattern space of 2250 distinct error scenarios.

The logical X error pathways reveal complex transformation possibilities from the initial \(|00\rangle_L\) state, with specific error combinations potentially inducing state transitions to \(|10\rangle_L\), \(|01\rangle_L\), or \(|11\rangle_L\).

I’m investigating whether the two-qubit error model can generate these logical X error patterns, focusing on the specific qubit interactions that might trigger state changes.

The logical fidelity appears to follow a nuanced error probability model, potentially capturing imperfect fault-tolerance through a quadratic error term with possible higher-order corrections.

I want to explore a concrete scenario involving specific error patterns. Consider an undetectable error sequence where \(X_1X_0\) occurs after a CNOT gate, which might not immediately disrupt the logical state. This specific error configuration seems particularly intriguing because it could potentially bypass standard error detection mechanisms.

The circuit’s fault-tolerant design aims to prepare a logical zero state, with post-selection on ancilla measurement playing a critical role in error suppression. However, the complexity of tracing error propagation through quantum gates makes precise analysis challenging.

My current understanding suggests that while single errors are likely manageable, two-error patterns present a more nuanced challenge. These patterns might potentially implement logical errors while remaining undetected, leading to a logical fidelity that scales quadratically with the error rate.

The mathematical form \(F_L = 1 - \alpha p^2 + O(p^3)\) captures this quadratic error scaling, with \(\alpha\) representing a key parameter that quantifies the resilience of the quantum error correction scheme.

The post-selection criteria involve two critical conditions: the ancilla measuring |0⟩ and ensuring no detectable errors that disrupt the stabilizer code’s logical state. This means carefully examining error patterns that can pass both verification tests.

The logical fidelity calculation becomes a ratio of probabilities, comparing successful logical state preservation against total acceptable error scenarios. The no-error case dominates the numerator, while the denominator accounts for all permissible error patterns.

The fundamental question remains whether all error patterns that pass the tests preserve the intended \(|00\rangle_L\) logical state, which would maintain perfect fidelity.

Preliminary analysis suggests a fidelity expression of \(F_L = 1 - \alpha p^2 + O(p^3)\), indicating potential second-order error corrections with a non-negative constant \(\alpha\). The goal is determining whether perfect fault tolerance (\(\alpha = 0\)) is achievable.

I’m exploring circuit structural insights that might reveal error suppression mechanisms. Specifically, I’m investigating whether undetectable errors preserving the \(|00\rangle_L\) logical state can be systematically characterized through stabilizer and logical Z operator interactions.

The complexity demands careful error pattern enumeration, which isn’t feasible within current computational constraints. My current approach focuses on understanding how two-error sequences might implement logical X errors while circumventing detection syndromes.

The ancilla extraction circuit becomes crucial, with specific attention to how CNOT operations might propagate or isolate potential error mechanisms.

By tracking parity relationships between qubits 0 and 3, I can potentially map out error detection boundaries and understand syndrome measurement limitations.

Given the fault-tolerant circuit’s constraints, I’m developing an approximate logical fidelity model. The key parameters suggest a quadratic error suppression mechanism, where single errors are likely neutralized but two-error configurations might introduce logical errors.

My preliminary expectation is a logical fidelity expression of \(F_L = 1 - c p^2\), where \(c\) represents the number of two-error patterns capable of bypassing detection tests. An ideal scenario would eliminate these patterns, potentially achieving perfect fidelity.

However, the precise value of \(c\) requires comprehensive circuit analysis to definitively characterize potential error propagation pathways. I’ll carefully complete the partial thought from the previous rewritten thinking and then summarize the new thinking.

c$ exactly. So my answer is: \(F_L = 1 - c p^2 + O(p^3)\) where \(c \geq 0\) is a constant depending on error patterns and circuit structure.

The analysis reveals a nuanced scenario where two-qubit Pauli errors could potentially corrupt the quantum state in unexpected ways. Some two-qubit Paulis might slip through undetected, passing the ancilla test while introducing subtle logical errors. This means the fidelity calculation needs careful consideration of these undetectable error mechanisms, potentially introducing first-order corrections to the error model. Single errors in a fault-tolerant circuit should not compromise the logical state after passing verification checks. This fundamentally defines fault tolerance, ensuring the system maintains integrity against initial error patterns.

The logical fidelity emerges as a quadratic correction, with potential small deviations represented by a constant \(c\). Specific error pair interactions might introduce subtle logical errors that circumvent initial detection mechanisms.

I anticipate the logical fidelity will follow the form \(F_L = 1 - c p^2 + O(p^3)\), where perfect fault tolerance implies \(c = 0\). The ancilla-based error detection strategy suggests robustness against single and double error scenarios.

Exploring potential error symmetries could reveal deeper insights into the system’s error-correction capabilities. Undetectable errors that preserve stabilizer relationships might not fundamentally alter the logical state, potentially simplifying the fidelity calculation.

Detailed error counting reveals complex interactions: ten possible two-CNOT error locations, with 225 potential error combinations per location. Not all combinations will pass ancilla tests or implement meaningful logical errors.

The proposed fidelity model suggests a quadratic error suppression mechanism, with a constant \(\alpha\) capturing two-error pattern contributions. Determining whether \(\alpha\) equals zero requires careful examination of error propagation through the quantum circuit.

Ancilla-checked qubits (0 and 3) provide partial error detection, while qubits 1 and 2 remain less directly monitored. This asymmetry hints at potential nuanced error dynamics within the fault-tolerant state preparation circuit.

An \(X_1X_2\) error reveals intriguing commutation properties, potentially passing undetected through syndrome measurements. Its transformation involves complex logical operator relationships that challenge straightforward error classification.

The logical operators form rich equivalence classes, not simple single-term representations. Exploring how \(X_1X_2\) relates to these classes requires careful algebraic manipulation and understanding of stabilizer group interactions.

I’m investigating whether this error can be decomposed into fundamental logical operators and stabilizers, which could illuminate its impact on quantum information preservation.

By systematically checking commutativity and algebraic relationships, I can trace its potential connections to established logical operator structures like \(\bar{X}_A\) and \(\bar{X}_B\).

The current algebraic exploration suggests complex interrelationships between quantum error correction components, revealing intricate transformation pathways.

I’ve discovered that \(X_1X_2\) implements a specific logical operation mapping \(|00\rangle_L\) to \(|11\rangle_L\) through the logical operator \(\bar{X}_A \bar{X}_B\), which emerges from precise algebraic manipulations involving symmetry properties like \(S_1^2 = I\).

This single error is deeply problematic. It passes both code tests undetected - slipping through the ancilla verification and remaining invisible within the code’s error detection mechanism. The consequence is critical: the logical qubit state becomes corrupted with a first-order fidelity correction of \(1 - \beta p + O(p^2)\).

I’m now investigating whether additional single errors might exhibit similar destructive characteristics after the CNOT operation, probing the circuit’s fundamental vulnerability.

Preliminary analysis reveals multiple undetectable error combinations: \(X_1X_2\), \(Y_1Y_2\), and \(Z_1Z_2\) each present unique transformation potentials that could potentially compromise the quantum information integrity.

After careful examination of \(Y_1Y_2\), I’ve determined it introduces a phase-based state corruption, transforming \(|00\rangle_L\) to \(-|11\rangle_L\). This suggests significant state manipulation beyond simple error detection.

The CNOT_12 operation exposes two critical undetectable error pathways, each occurring with a probability of \(p/15\): \(X_1X_2\) and \(Y_1Y_2\). These errors fundamentally challenge the quantum error correction strategy.

I need a systematic approach to comprehensively map these error interactions across different CNOT configurations.

Analyzing XX errors across qubit pairs reveals no straightforward logical X implementations. Each pair’s XX combination (\(X_0X_1\), \(X_0X_2\), etc.) fails to produce a clean logical X operation, indicating complex error propagation dynamics.

Investigating ancilla test constraints, I find specific error patterns that preserve or disrupt the XOR measurement. Errors on qubit pairs (0,3) and (1,2) maintain the ancilla’s zero state, suggesting these are potentially stable error configurations. Other qubit pair errors introduce detectable ancilla state changes, revealing intricate error detection mechanisms.

YY errors introduce additional complexity, potentially implementing logical X with concurrent logical Z transformations. ZZ errors remain fundamentally benign, only inducing logical Z modifications without disturbing the logical ground state.

Analyzing CNOT interactions, CNOT_12 emerges as uniquely vulnerable, representing the sole circuit operation where XX or YY errors can simultaneously pass the ancilla test and compromise the logical quantum state.

Single error probabilities contribute \(\frac{2p}{15}\) to state corruption, with the logical fidelity approximated as \(1 - \frac{2p}{15} + O(p^2)\).

I recognize potential nuances in error detection mechanisms that warrant deeper scrutiny, particularly around undetectable \(X_1X_2\) errors passing ancilla verification.

Specific error scenarios emerge across different CNOT operations: - \(X_1X_2\) and \(Y_1Y_2\) remain undetectable, producing distinct logical states - \(Z_1Z_2\) preserves original logical state - Certain qubit interactions like \(X_1X_0\) and \(Y_1Y_0\) trigger ancilla failures

These intricate error pathways suggest more complex fidelity calculations beyond initial linear approximations. Ancilla interactions reveal critical error detection challenges. Certain two-qubit errors can unexpectedly trigger ancilla measurement failures, compromising quantum error correction. Specifically, \(X_3X_4\) and \(X_0X_4\) type errors directly flip the ancilla, invalidating post-selection. The most significant error modes emerge after specific CNOT operations, particularly involving \(X_1X_2\) and \(Y_1Y_2\) interactions, which collectively manifest with a \(\frac{2p}{15}\) probability of occurrence. I need to account for errors that pass both tests and result in \(|11\rangle_L\), which include specific two-qubit errors after CNOT operations. These errors have probabilities proportional to \(\frac{p}{15}(1-p)^4\), specifically \(X_1X_2\) and \(Y_1Y_2\) after CNOT_12.

Calculating the probabilities to first order in \(p\), I find the probability of \(|00\rangle_L\) is \((1-p)^5 + \frac{3p}{15}(1-p)^4\), while \(|11\rangle_L\) has a probability of \(\frac{2p}{15}(1-p)^4\). These calculations help me understand the error dynamics in the quantum error correction process.

By carefully analyzing the fidelity \(F_L\), I derive an approximation that shows a slight degradation with error probability \(p\), specifically \(F_L = 1 - \frac{2p}{15} + O(p^2)\). This suggests a small but systematic reduction in quantum state preservation.

I’m now examining how CNOT operations involving the ancilla create entanglement and potentially introduce measurement complications. The key observation is that errors on the ancilla can fundamentally alter the post-measurement data state, making error analysis intricate.

My strategy shifts to isolating data qubit errors occurring after specific CNOT gates, recognizing that ancilla-related errors present a distinct challenge from pure data qubit perturbations. I’ll methodically trace potential error pathways to understand their impact on quantum information preservation.

Focusing on CNOT_12, CNOT_10, and CNOT_23, I’m mapping out which error types survive the quantum error correction tests. The critical insight emerges that only certain error combinations - specifically those preserving specific qubit correlations - can pass the stringent measurement criteria.

The mathematical representation suggests a first-order fidelity correction, with error probability scaling proportionally to the system’s noise characteristics. My next step involves rigorously validating the ancilla measurement conditions for these error scenarios.

By systematically checking error propagation through each CNOT interaction, I’m confirming the consistency of the error preservation mechanisms across different qubit interactions.

The preliminary fidelity estimate remains \(F_L = 1 - \frac{2p}{15} + O(p^2)\), pending further detailed verification of error pathways, particularly around the ancilla-involving CNOT operations.

Investigating the \(Z_3Z_4\) error reveals nuanced state transformations. When applying this error, the quantum state maintains its fundamental structure but experiences a subtle phase modification, critically impacting ancilla measurement outcomes and subsequent state evolution.

Specifically, after CNOT_04, the state transformation demonstrates precise error propagation characteristics, with measurement of the ancilla yielding a definitive \(|00000\rangle\) state under certain error conditions.

Analyzing potential error scenarios, I identify multiple error pathways that preserve the logical qubit state while maintaining quantum coherence. These include specific two-qubit errors like \(Z_1Z_2\), \(Z_1Z_0\), \(Z_2Z_3\), \(Z_3Z_4\), and \(Z_0Z_4\), each occurring with a probability of \(\frac{p}{15}(1-p)^4\).

Combining these error probabilities yields a comprehensive error model: \((1-p)^5 + \frac{5p}{15}(1-p)^4\), capturing both no-error and specific two-qubit error scenarios that preserve quantum information integrity.

For \(|11\rangle_L\) states, I observe two primary error mechanisms: \(X_1X_2\) and \(Y_1Y_2\), each contributing \(\frac{p}{15}(1-p)^4\) to the total error probability.

Calculating the logical fidelity reveals a nuanced error correction landscape, with \(F_L\) approximating \(1 - \frac{2p}{15}\) to first order, indicating a slight degradation proportional to the error probability.

Seeking precision, I’ll expand the terms to second order, carefully tracking the \((1-p)^5\) and \((1-p)^4\) expansions to capture more subtle error dynamics.

The numerator and denominator expansions will help me derive a more accurate representation of the logical fidelity, accounting for quadratic error contributions.

By applying the approximation \(\frac{1}{1+x} \approx 1 - x + x^2\) for small \(x\), I can systematically break down the complex fraction into manageable components, revealing underlying mathematical structure.

I recognize the complexity is increasing, so I’ll pivot to a more streamlined approach by defining \(F_L\) as the ratio of numerator \(N\) to denominator \(D\), which will simplify subsequent calculations.

To first order, I calculate the linear terms of both numerator and denominator. By carefully extracting the first-order coefficients, I derive \(N' = -\frac{14}{3}\) and \(D' = -\frac{68}{15}\).

Combining these terms yields \(F_L \approx 1 - \frac{2}{15}\), a concise representation that captures the essential first-order behavior of the complex fraction.

I’m now cross-checking the calculation using an alternative approach involving probability terms for quantum error correction. By defining specific error probabilities \(a\), \(b\), and \(c\), I aim to validate the previous result through a different computational pathway.

My new method involves analyzing the probabilities of no error and single error scenarios, which will help confirm the earlier calculation’s accuracy.

The algebraic manipulation reveals a systematic reduction of the fidelity expression, progressively simplifying the fraction by factoring out common terms and consolidating the error probability components.

The key transformation involves strategically canceling \((1-p)^4\) terms and reorganizing the numerator and denominator to expose the underlying error probability structure.

The process culminates in expressing the logical qubit fidelity as a ratio that captures the nuanced relationship between error probabilities at different stages of quantum error correction.

By applying the first-order approximation technique, the expression converges to a linear function of the error probability, highlighting the initial error mitigation characteristics of the quantum protocol.

Carefully manipulating the algebraic terms reveals a systematic reduction in logical fidelity, with the leading correction term being \(-\frac{2p}{15}\). This indicates a slight degradation of the quantum state’s reliability under single-error scenarios.

The calculation suggests the quantum circuit experiences imperfect fault tolerance, specifically noting potential state corruption following the CNOT_12 operation. The linear dependence on error probability provides insight into the circuit’s error propagation mechanism.

The analysis leaves an open question about investigating second-order error terms, which could offer deeper understanding of the quantum error dynamics. I’m carefully tracking error propagation through the quantum circuit, examining how specific error patterns affect the quantum state. By systematically analyzing each branch of the superposition, I can trace how \(X_1X_2\) errors transform the initial state through successive CNOT gates. The detailed step-by-step calculation reveals critical insights into error dynamics, showing how each gate interaction modifies the quantum state’s configuration.

The key observation emerges that error transformations aren’t straightforward - they depend intrinsically on the sequence of quantum operations. My previous assumption about error commutativity was incorrect, highlighting the complex quantum mechanical interactions within the error correction circuit.

The analysis suggests a nuanced approach is necessary to understand how errors propagate and potentially compromise quantum information preservation.

I’m recognizing that fault-tolerance requires careful examination of error dynamics. Single errors must either be definitively detected or fundamentally unable to disrupt the logical quantum state.

The mathematical representation \(F_L = 1 + O(p^2)\) implies a probabilistic threshold for error tolerance, but my detailed circuit analysis reveals potential inconsistencies that demand deeper investigation.

The critical insight involves understanding how errors interact with the quantum state preparation and ancilla coupling, which might fundamentally alter error propagation mechanisms.

My current calculations suggest potential discrepancies between theoretical fault-tolerance expectations and actual error behaviors, indicating a need to re-examine the underlying quantum circuit model and error assumptions.

The core challenge lies in understanding how single errors propagate and potentially corrupt quantum states, revealing nuanced interactions that deviate from simple logical operation models. My analysis points to a logical fidelity that differs from ideal fault-tolerant expectations, specifically showing \(F_L = 1 - \frac{2p}{15} + O(p^2)\) instead of the expected \(F_L = 1 + O(p^2)\).

I recognize multiple potential explanations: analytical errors, circuit-specific fault-tolerance limitations, or fundamental misunderstandings of the error propagation mechanism. The complexity demands careful, iterative investigation to resolve these uncertainties.

The user wants me to fill in the code template with the final answer from Subproblem 3.

From my analysis, the logical fidelity is:

I need to express this as a SymPy expression.

In SymPy, this would be:

F_logical = (1 - 2*p/3) / (1 - 8*p/15)

Let me double-check this expression: - Numerator: \(1 - \frac{2p}{3}\) - Denominator: \(1 - \frac{8p}{15}\)

Yes, this matches my derivation.