Let me break this problem down 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) - 2 distance (can detect 1 error) - Stabilizers: \(XXXX\) and \(ZZZZ\) - Logical operators: - \(X_A = XIXI\) - \(X_B = XXII\) - \(Z_A = ZZII\) - \(Z_B = ZIZI\)

The goal is to prepare the logical GHZ state \((|00\rangle_{AB} + |11\rangle_{AB})/\sqrt{2}\).

The circuit is: \((CNOT_{03}) (H_0) (CNOT_{21}) (H_2)\)

Reading from right to left (as quantum circuits are executed):

1. Apply \(H_2\) (Hadamard on qubit 2)
2. Apply \(CNOT_{21}\) (control on 2, target on 1)
3. Apply \(H_0\) (Hadamard on qubit 0)
4. Apply \(CNOT_{03}\) (control on 0, target on 3)

Let me first figure out what the ideal state is without errors.

Starting state: \(|0000\rangle\)

After \(H_2\): \(|00\rangle_{01} \otimes \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)_2 \otimes |0\rangle_3 = \frac{1}{\sqrt{2}}(|0000\rangle + |0010\rangle)\)

After \(CNOT_{21}\) (control on 2, target on 1):

\(\frac{1}{\sqrt{2}}(|0000\rangle + |0110\rangle)\)

After \(H_0\), the state becomes more complex. I’ll split the superposition components carefully, applying the Hadamard gate to the first qubit. This creates a superposition that will ultimately lead to the desired logical GHZ state.

The CNOT operation between qubits 0 and 3 introduces entanglement, transforming the state. I’m tracking how the quantum gates modify the initial superposition, preparing to explore the logical basis states of this quantum error correction code.

I’ll investigate the stabilizer generators to understand the logical computational basis, focusing on the \(XXXX\) and \(ZZZZ\) operators that define the code’s logical subspace.

The logical basis states require careful examination of their eigenvalue relationships with key stabilizer operators. Each state must satisfy specific symmetry conditions across the four-qubit system, revealing the intricate structure of the quantum code’s logical encoding.

I’ll construct the logical states using a symmetric superposition approach, ensuring they simultaneously satisfy the \(XXXX\) and \(ZZZZ\) stabilizer constraints. The states are precisely balanced combinations of basis states that maintain the required symmetry properties.

The logical subspace spans four distinct basis states, representing two logical qubits embedded within the four-physical-qubit quantum error correction framework. Each state is meticulously designed to preserve quantum information through its stabilizer relationships.

The \(|00\rangle_{AB}\) state demonstrates this principle, comprising four equally weighted basis states that exhibit symmetric transformation under \(XXXX\) operations.

When applying \(XXXX\), each basis state undergoes a complete bit-flip, yet remarkably returns to its original configuration. Similarly, the \(ZZZZ\) operation reveals phase consistency, with each term maintaining its original phase regardless of the number of 1s present.

The \(Z_A\) stabilizer condition requires further investigation to confirm its preservation of the quantum state’s fundamental properties. I carefully examine the logical operators, checking their commutation properties with stabilizers. \(X_A\) and \(X_B\) both commute with the stabilizer generators, as do \(Z_A\) and \(Z_B\). I verify their anticommutation relationships, noting the critical interaction between logical X and Z operators on each qubit. The operators satisfy the key requirements for defining logical qubits in this quantum error correction code.

I’m now focusing on constructing the logical computational basis states. The CSS code approach seems promising, and I’ll explore the [[4,2,2]] code as a minimal example. I’m preparing to derive the logical state using the stabilizer group’s X-type generators.

The stabilizer group \(S_X = \{I, XXXX\}\) suggests a symmetric superposition state. This initial construction yields \(|0\rangle_L = \frac{1}{\sqrt{2}}(|0000\rangle + |1111\rangle)\), but I recognize this doesn’t fully capture the code’s potential.

I’ll shift to a more comprehensive representation that encodes two logical qubits. The standard construction reveals four distinct logical basis states, each carefully constructed to maintain the code’s quantum error correction properties.

The states systematically distribute superpositions across the four-qubit register, ensuring balanced representations of \(|00\rangle_{AB}\), \(|01\rangle_{AB}\), \(|10\rangle_{AB}\), and \(|11\rangle_{AB}\). Each state involves precise combinations of basis states, demonstrating intricate quantum encoding strategies.

I’ll verify the stabilizer transformations, focusing on how \(ZZZZ\) acts across different computational basis states. The phase-preserving nature suggests symmetrical quantum state properties. Checking the logical operators will help confirm the state’s quantum error correction characteristics.

The proposed final state represents a carefully constructed superposition, balancing different four-qubit configurations. By examining how \(Z_A\) transforms this state, I can validate its quantum error correction potential.

Specifically, the \(Z_A\) and \(Z_B\) operators reveal intriguing symmetries: some basis states gain a negative phase, while others remain positive. This phase behavior indicates a robust quantum encoding mechanism.

I notice the initial assumption of simple logical states doesn’t capture the full complexity. The code’s logical basis requires a more nuanced superposition approach, involving multiple computational basis states with equal amplitudes.

The online reference provides precise logical state definitions, suggesting a sophisticated quantum error correction strategy that distributes quantum information across multiple physical qubits.

For the \(|00\rangle_L\) logical state, I’m checking stabilizer properties by applying \(Z_A\) and \(Z_B\) operators, verifying the state’s consistency with quantum error correction principles.

The \(Z_A\) operator confirms even parity across the first two qubits, preserving the logical state. Similarly, the \(Z_B\) operator validates the state’s stability by maintaining even parity for specific qubit configurations.

For the \(|11\rangle_L\) logical state, I’m examining the stabilizer effects more critically. The \(Z_A\) operator reveals an interesting sign change, indicating odd parity in specific qubit arrangements. This sign flip is crucial for understanding the logical state’s quantum error correction behavior.

The GHZ state representation looks complex, involving intricate superpositions across multiple qubit configurations. However, something seems inconsistent with my previous calculations, prompting a careful recalculation of the circuit’s quantum operations.

I’m now focusing on the logical operators, particularly \(X_A\) and \(X_B\), which might reveal critical insights into the quantum error correction mechanism.

The state after the circuit appears to be a carefully constructed superposition, with specific amplitudes for different basis states. I’ll systematically verify its properties by examining the stabilizer group and checking its quantum error correction characteristics.

Applying \(ZZZZ\) to each basis state reveals a nuanced phase structure. I’ll track the number of ones in each computational basis state to determine the associated phase factors systematically.

For the \(|0000\rangle\) term, the phase calculation begins with counting the number of ones, which will determine the overall sign transformation under the \(ZZZZ\) operator.

The computational basis states \(|1001\rangle\), \(|0110\rangle\), and \(|1111\rangle\) each undergo similar phase calculations based on their specific one-count configurations. By meticulously tracking these phase factors, I can verify the operator’s action on the quantum state.

The result confirms that \(ZZZZ\) preserves the original quantum state’s structure, with each basis state experiencing a consistent phase transformation determined by its one-count configuration.

For \(Z_A\), the phase shifts reveal a symmetric pattern across different computational basis states, ultimately generating a balanced quantum superposition that maintains the state’s fundamental symmetry.

The logical operators \(Z_A\) and \(Z_B\) demonstrate intricate phase manipulations that preserve the overall quantum state’s essential characteristics while introducing subtle transformational nuances.

When considering the logical qubit state \(|\psi\rangle = (|00\rangle_L + |11\rangle_L)/\sqrt{2}\), the \(Z\) operators induce specific phase modifications that reflect the underlying quantum error correction code’s structural properties.

I notice a discrepancy between the expected and calculated \(Z\) operator transformations. This suggests the need to carefully examine the logical basis states and their interactions with stabilizer operators.

To resolve this, I’ll define logical basis states that satisfy specific \(Z\) operator constraints. The \(|00\rangle_L\) state must maintain positive eigenvalue under both \(Z_A\) and \(Z_B\), while the \(|11\rangle_L\) state must exhibit negative eigenvalues for these operators.

The computational approach requires precise alignment of the logical encoding with the desired quantum error correction code’s symmetries. I’ll decompose the state’s components, noting the \(Z_A\) and \(Z_B\) actions across different basis states. The state includes: - \(|0000\rangle\) with \(Z_A = +1, Z_B = +1\) (part of \(|00\rangle_L\)) - \(|1001\rangle\) with \(Z_A = -1, Z_B = -1\) (part of \(|11\rangle_L\)) - \(|0110\rangle\) with \(Z_A = +1, Z_B = -1\) (part of \(|01\rangle_L\)) - \(|1111\rangle\) with \(Z_A = -1, Z_B = +1\) (part of \(|10\rangle_L\))

This reveals the state is not a standard GHZ state of \((|00\rangle_L + |11\rangle_L)/\sqrt{2}\). I’ll need to verify the circuit calculation carefully. After \(CNOT_{03}\), the state transforms with specific changes to basis states. The final quantum state reflects these precise modifications, showing how control and target qubit interactions modify the superposition.

The calculation appears valid, but the interpretation requires deeper examination. A potential misalignment exists between the desired logical state and the current quantum representation, possibly stemming from different conventions in defining logical qubit states.

The problem statement’s goal of preparing a specific entangled state seems not fully achieved by the current quantum circuit operations. A careful reconsideration of logical state definitions might reveal the underlying discrepancy.

Logical operators are defined through multi-qubit interactions, creating complex eigenstates that depend on specific qubit combinations. These definitions establish how computational basis states transform under logical operations, revealing intricate quantum state relationships.

The eigenstates of logical \(Z\) operators demonstrate how qubit interactions determine the fundamental computational basis for these logical qubits.

For each 4-qubit state, the logical \(Z\) operators generate phase shifts based on specific qubit index combinations, creating a nuanced mapping between physical and logical quantum states.

I’ll carefully track how different physical states map to logical states, noting the phase relationships that emerge from these sophisticated quantum encodings.

Key observations include: - \(|0000\rangle\) maps to \(|00\rangle_{AB}\) with symmetric positive phases - \(|0110\rangle\) and \(|1001\rangle\) both map to \(|11\rangle_{AB}\) with symmetric negative phases - The phase calculations depend on precise index interactions across the quantum register

I’m tracking how different basis states transform under specific eigenvalue constraints, noting the systematic mapping between input and output quantum states.

The state \(|\psi\rangle\) reveals a nuanced quantum configuration where specific basis states are selectively represented. These states aren’t uniformly distributed but carefully chosen to reflect particular symmetry properties across the \(|00\rangle_{AB}\) and \(|11\rangle_{AB}\) subspaces.

The quantum representation suggests a structured approach where not all possible basis states are included, indicating a precise physical encoding mechanism beyond simple superposition.

I’m tracking how the circuit generates a specific physical state that, when projected into the code space, approximates the logical GHZ state through carefully constructed quantum interactions. The preparation involves unique superpositions of physical basis states representing logical quantum information.

The next step involves exploring how potential errors might modify this delicate quantum configuration, with depolarizing channels introducing probabilistic transformations across the two-qubit CNOT gates.

I’ll model the error-induced evolution by introducing probabilistic quantum channels that modify the initial quantum state, tracking how these stochastic interactions potentially deviate from the ideal quantum circuit’s intended transformation.

The mathematical representation will capture how these error channels probabilistically perturb the quantum state’s coherence and entanglement characteristics during each quantum gate operation.

By systematically tracing the quantum state’s progression through each noisy gate, I can analyze how depolarizing noise impacts the quantum computation’s fidelity and reliability.

The key focus involves understanding how small probabilistic errors can accumulate and potentially disrupt the quantum information processing.

I’m tracking the state transformation through multiple quantum operations, considering both ideal gate transformations and potential error channels that might introduce stochastic perturbations to the quantum system.

The mathematical representation allows me to model how different error probabilities and locations can modify the final quantum state’s characteristics, revealing the intricate dynamics of quantum noise propagation.

By systematically decomposing the circuit’s evolution, I can analyze how local errors on specific qubit pairs impact the overall quantum computation’s fidelity and output state.

The state transforms through a complex probabilistic mixture, where \((1-p)\) represents the ideal path and \(p\) represents error-induced variations across Pauli operators on qubits 2 and 1.

I’ll calculate the fidelity by comparing the output density matrix against the ideal quantum state, tracking how local noise introduces subtle deviations in the quantum information processing.

The circuit’s evolution involves sequential unitary transformations \(U_1\) through \(U_4\), with potential error channels \(\mathcal{E}_{03}\) and \(\mathcal{E}_{21}\) introducing stochastic perturbations that modify the quantum state’s coherence and entanglement structure.

After applying \(U_2\) and \(U_1\), the quantum state transforms to a superposition \(|\phi_1\rangle\) with equal amplitudes for \(|0000\rangle\) and \(|0110\rangle\). The subsequent error channel \(\mathcal{E}_{21}\) probabilistically introduces Pauli errors, modifying the state’s density matrix \(\rho_1\) with a parameter \(p\) representing the error probability.

I’ll trace the quantum circuit’s evolution, applying unitary transformations \(U_3\) and \(U_4\) sequentially. Each transformation modifies the state’s density matrix, with \(\rho_2\) resulting from \(U_3\) and \(\rho_3\) from \(U_4\). The final error channel \(\mathcal{E}_{03}\) introduces additional probabilistic Pauli errors, generating the output density matrix \(\rho_{out}\).

Calculating the fidelity reveals an interesting quantum state relationship. By examining the ideal state \(|\psi_{ideal}\rangle\) and the output state \(\rho_{out}\), I find the no-error term contributes \((1-p)^2\). This emerges from the specific unitary transformations \(U_3\) and \(U_4\) mapping \(|\phi_1\rangle\) precisely to \(|\psi_{ideal}\rangle\).

The key observation is the direct mapping between initial and final quantum states under these precise unitary operations, demonstrating a clean quantum circuit transformation with minimal error probability.

I’m now exploring the error term contributions, focusing on potential error pathways after the first CNOT gate. The complexity involves tracking probabilistic error states and their impact on overall quantum state fidelity.

The fidelity calculation involves summing probabilities of different error configurations, where each error has a specific occurrence probability. I’ll compute this by tracking how various error combinations modify the quantum state’s final representation.

By systematically mapping error probabilities and their transformative effects, I can quantify the deviation from an ideal quantum state. The key is understanding how different error sequences probabilistically alter the quantum circuit’s intended evolution.

The four distinct error scenarios range from no errors to multiple simultaneous errors, each with unique computational implications. Their probabilities decrease exponentially as more errors are introduced, reflecting the system’s inherent stability against quantum noise.

I’ll calculate the fidelity by integrating these probability-weighted overlap scenarios, systematically accounting for different error configurations across quantum gates and initial states.

The computation involves tracking how non-identity Pauli errors transform the initial quantum state \(|\phi_1\rangle\), which is a superposition of \(|0000\rangle\) and \(|0110\rangle\). I’ll methodically evaluate how each of the 15 non-identity Paulis acting on qubits 2 and 1 modify this initial state.

The key challenge is tracking how these Pauli errors propagate through the quantum circuit, preserving or disrupting the delicate quantum coherence of the initial superposition state.

I’m reconsidering my computational approach, recognizing the complexity of tracking every possible error pathway. The depolarizing channel’s probabilistic nature suggests a more strategic analysis might reveal deeper insights into the state’s fidelity under noise.

The fundamental question centers on how quantum errors impact the quantum information, particularly in a scenario with limited CNOT gates and a specific initial state configuration.

Mathematically, I’ll construct a comprehensive error probability framework, mapping out how different error combinations transform the initial quantum state through the circuit’s unitary operations.

The density matrix representation allows me to systematically capture the probabilistic evolution, tracking how various error combinations contribute to the final quantum state’s characteristics.

My approach will involve carefully computing the aggregate state transformation, considering the probabilistic nature of the depolarizing channel’s error mechanisms.

I’ll start by examining specific error scenarios, particularly focusing on the identity and non-identity error configurations, to understand their distinct contributions to the overall quantum state fidelity.

The key insight involves computing the overlap between the ideal quantum state and each potential error-induced state, systematically quantifying how errors probabilistically modify the quantum information.

For the identity case with no errors, the fidelity is straightforward: \((1-p)^2\). However, when introducing single-qubit errors like X or Z gates on specific qubits, the quantum state’s overlap becomes zero, indicating complete orthogonality between the ideal and error-transformed states.

By methodically analyzing these error scenarios, I can map out how different error types fundamentally disrupt the quantum state’s coherence and information preservation.

The systematic exploration reveals that single-qubit perturbations consistently eliminate state overlap, suggesting a robust mechanism for detecting quantum information corruption.

Specifically, bit-flipping operations fundamentally disrupt the quantum state’s coherence, rendering any modified state orthogonal to the original. Only phase-based transformations maintain potential quantum state continuity, with Z and Y rotations preserving critical quantum information characteristics.

Careful analysis of specific two-qubit projective measurements demonstrates the delicate nature of quantum state preservation, highlighting the precision required in quantum error detection protocols.

The computational pathway reveals nuanced interactions between quantum states under different error scenarios, suggesting complex constraints on quantum information stability.

Stabilizer groups define critical symmetry constraints. Specific Pauli combinations reveal fundamental state preservation mechanisms, particularly for qubits 0 and 3.

Local Pauli errors systematically disrupt quantum state coherence through two primary mechanisms: altering bit configurations or modifying phase relationships, ultimately rendering quantum states orthogonal.

These interactions demonstrate a profound symmetry breaking principle inherent in the quantum error correction framework, where local perturbations fundamentally transform quantum information representations.

Potential error contributions emerge through nuanced interaction pathways, with most error combinations likely producing zero fidelity contributions due to intrinsic state symmetries.

The quantum state’s delicate structural integrity suggests complex error propagation dynamics that critically depend on precise Pauli interaction patterns.

Careful circuit-level error analysis reveals subtle state transformation mechanisms where single-qubit perturbations fundamentally alter quantum coherence.

Systematic error mapping indicates non-trivial constraints on fidelity preservation, demanding granular examination of intermediate circuit stages and error occurrence probabilities.

Exploring error propagation pathways requires tracking quantum state modifications through sequential unitary transformations, highlighting the complexity of maintaining quantum information integrity.

Clifford gates transform Pauli errors in predictable ways. When an error occurs after a CNOT gate, subsequent Hadamard and CNOT operations can modify its representation, revealing intricate quantum error dynamics.

By examining how errors interact with different quantum gates, I can trace their evolution through the circuit’s computational landscape, uncovering subtle transformation mechanisms that preserve or alter quantum states.

The key observation emerges: certain errors can propagate unchanged through specific gate sequences, demonstrating the complex error transmission pathways in quantum computation.

I’ll explore the overlap scenarios systematically, considering different error combinations and their potential impacts on the quantum state’s fidelity. Each scenario represents a unique interaction between potential quantum errors, revealing nuanced quantum error dynamics.

The scenarios range from no errors to multiple non-trivial errors, each requiring careful computational analysis to determine the precise quantum state overlap.

For instance, when applying X or Y operators, the quantum state undergoes bit flips, fundamentally altering the original quantum configuration. These transformations generate orthogonal states, demonstrating the sensitivity of quantum systems to perturbative operations.

The Z operator presents intriguing behavior, potentially preserving state amplitudes while modifying phase relationships. This subtle interaction highlights the complex quantum mechanical principles governing error propagation and state preservation.

By systematically exploring different single-qubit error operators, we observe a critical property: each non-identity operation orthogonally transforms the ideal quantum state. This characteristic suggests a robust mechanism for detecting and characterizing quantum errors through state overlap measurements.

The mathematical framework reveals a profound insight: the quantum state’s resilience emerges from its intrinsic orthogonality under various error transformations.

Intriguingly, the analysis hints at potential complexities when multiple error operators interact simultaneously, suggesting a nuanced landscape of quantum error dynamics that requires careful probabilistic modeling.

For specific GHZ-like states, certain error combinations might yield non-trivial overlap effects. By examining a concrete example with Z operators on different qubits, I can explore how error interactions manifest in quantum state transformations.

Calculating the specific state transformations reveals subtle interference patterns between error operators, demonstrating that simple orthogonality assumptions may not always hold in quantum error scenarios.

The stabilizer state’s inherent symmetries suggest complex constraints on potential error mechanisms, potentially leading to unique fidelity characteristics that depend on precise error correlations.

I’ll systematically verify the state’s stabilization properties by applying key operators. The \(XXXX\) and \(ZZZZ\) operators both preserve the quantum state’s fundamental structure, indicating deep underlying symmetries that could impact error resilience.

These symmetry relationships hint at intricate quantum error dynamics that merit careful mathematical investigation.

I notice the state transforms predictably under certain tensor product operations, suggesting a rich stabilizer group with potentially complex interactions. The \(XIXZ\) operator’s action reveals interesting transformation patterns across different basis states.

The systematic exploration of potential stabilizers reveals nuanced quantum mechanical behaviors that could be critical for quantum information processing strategies.

The stabilizer group’s constraint on error propagation becomes evident through commutation relationships. Specifically, any error must align with the group’s fundamental symmetry principles to maintain state coherence.

When an error interacts with stabilizer elements, its contribution depends on commutation properties. If the error anti-commutes with a stabilizer, it generates zero expectation value, effectively canceling its potential impact.

For a general Pauli error, two key conditions emerge: - \(XXXX\) commutation requires an even count of \(Z\) and \(Y\) operators - \(ZZZZ\) commutation demands an even distribution of \(X\) and \(Y\) operators

This leads to specific error configurations: - Pure identity operator - Two-type mixed operators like \(XXYY\) or \(XXZZ\) - Single-type full operators such as \(XXXX\)

Stabilizer operators like \(XXXX\), \(YYYY\), and \(ZZZZ\) inherently produce complete overlap, which constrains potential error scenarios.

When attempting to construct \(XXYY\), I notice critical positioning constraints involving qubit interactions between \(E_1\) and \(E_2\). The two-qubit Pauli operators have restricted action across different qubit regions, preventing straightforward combination.

The stabilizer interactions require precise alignment, with each error operator needing specific commutation properties to maintain system coherence.

For \(E_1\) on qubits 2,1, valid configurations are limited to \(XX\), \(YY\), or \(ZZ\), excluding the identity operator. These configurations must satisfy commutation rules with both \(XXXX\) and \(ZZZZ\) stabilizers, ensuring balanced interactions across even and odd numbers of \(X\), \(Y\), and \(Z\) operators.

The complementary \(E_2\) on qubits 0,3 will similarly require carefully selected two-qubit Pauli operators to enable potential non-zero contributions.

Specifically examining \(E_1 = ZZ\), the operator transforms the ideal state \(|\psi_{ideal}\rangle\) with precise sign changes: - \(|0000\rangle\) remains unchanged - \(|1001\rangle\) and \(|0110\rangle\) acquire negative signs - \(|1111\rangle\) remains positive

The resulting transformed state reflects a balanced superposition with specific phase relationships, capturing the intricate quantum interactions.

Calculating the overlap reveals zero correlation, indicating the \(ZZ\) operator does not preserve the original quantum state’s structure. I’ll investigate the \(XX\) operator to understand its transformative properties.

The \(XX\) operation on qubits 2 and 1 appears to permute basis states while maintaining the superposition’s fundamental characteristics. This suggests a deeper symmetry in the quantum system where certain transformations preserve the overall state’s essential quantum information.

The logical operator \(X_B\) requires careful examination, as its action across all four qubits might reveal more complex quantum interactions that are not immediately apparent from simple basis state permutations.

Calculating the explicit transformations shows a precise mapping: - \(|0000\rangle\) transforms to \(|0110\rangle\) - \(|1001\rangle\) transforms to \(|1111\rangle\) - \(|0110\rangle\) transforms to \(|0000\rangle\) - \(|1111\rangle\) transforms to \(|1001\rangle\)

Critically, these transformations preserve the state’s overall structure, resulting in \(\langle\psi_{ideal}|E_1|\psi_{ideal}\rangle = 1\), which indicates an invariant quantum configuration.

I’ll now examine \(YY\) on qubits 2,1, noting its complex matrix representation involves sign changes and specific qubit interactions.

The state transformation reveals four key mappings: - \(|0000\rangle\) becomes \(-|0110\rangle\) - \(|1001\rangle\) becomes \(|1111\rangle\) - \(|0110\rangle\) becomes \(|0000\rangle\) - \(|1111\rangle\) becomes \(-|1001\rangle\)

Calculating the overlap yields zero, indicating \(E_1 = XX\) uniquely provides non-zero interaction.

I’m recognizing a pattern where only specific Pauli errors corresponding to logical operations or stabilizers generate meaningful state interactions for the GHZ state.

The state fundamentally remains invariant under two primary transformations: the identity operation and the \(XXXX\) operation.

Logical operators potentially introduce non-trivial phase shifts, particularly through qubit flips across specific register configurations.

The fidelity calculation involves summing probabilities of error combinations, with specific scenarios contributing to the overall quantum state preservation. Key contributions include:

1. No error scenario: \((1-p)^2\) probability
2. Partial error on specific qubits: \(p(1-p)/15\)

I’m investigating Pauli error transformations that maintain state invariance, focusing on how \(XX\) operations on qubits 0 and 3 modify quantum state configurations.

The \(XX\) operation cycles through specific basis states symmetrically, preserving the original quantum state’s structure. This symmetry suggests robustness against certain error types, demonstrating interesting quantum error dynamics.

Subsequent analysis will explore additional Pauli error modes like \(YY\) and \(ZZ\) to comprehensively understand state preservation mechanisms.

For \(ZZ\) interactions, the operation introduces phase changes, with some basis states remaining invariant while others acquire a negative sign. This reveals subtle quantum mechanical phase transformations.

The \(YY\) interaction exhibits more complex state transitions, mapping basis states to different configurations with sign variations. These intricate mappings highlight the nuanced quantum mechanical interactions between qubits.

The systematic exploration of these two-qubit error operations provides insights into quantum state resilience and transformation properties.

Focusing on non-zero contributions, the identity operation and specific two-qubit interactions emerge as significant, with the first-order probability scaling with \((1-p)^2\).

The subsequent terms involve single and double \(XX\) operations on different qubit pairs, each contributing \(p(1-p)/15\) or \(p^2/225\) to the total probability, maintaining a consistent overlap of 1.

Combining these contributions yields a compact expression tracking the error probability progression, revealing intricate interactions between quantum error correction mechanisms.

By factoring and rearranging terms, I can expose the underlying probabilistic structure, highlighting how different error scenarios contribute to the overall system reliability.

The algebraic manipulation suggests a nuanced relationship between single and double error probabilities, demonstrating the complexity of quantum error correction strategies.

I’m seeing an elegant transformation that reveals the intrinsic symmetry in the probability distribution, where the squared term captures both success and failure modes simultaneously.

The algebraic manipulation suggests a refined approach, exploring how \((1 - \frac{14p}{15})^2\) might represent a more compact representation of the underlying probabilistic structure. By expanding this expression, I can uncover potential insights into the relationship between different probability parameters.

The systematic deconstruction reveals nuanced interactions between the probability components, hinting at a deeper mathematical relationship that wasn’t immediately apparent in the initial formulation.

I’m carefully examining the error probabilities for two-qubit gates, recognizing that not all errors have equivalent impacts on the quantum state. The key observation centers on distinguishing between errors that alter the state and those that preserve its fundamental characteristics.

Specifically, I’m focusing on understanding which specific Pauli errors maintain the state’s integrity, with particular attention to the XX error as a unique preservation mechanism.

The fidelity calculation reveals a quadratic relationship between error probability and state preservation, where \((1 - \frac{14p}{15})^2\) captures the cumulative probability of maintaining quantum coherence across two-qubit interactions.

The mathematical structure suggests a systematic approach to quantifying quantum gate reliability, where each potential error contributes a precise probabilistic component to the overall system dynamics.

Expanding the expression confirms the intricate interplay between error rates and quantum state fidelity, demonstrating how small perturbations can significantly impact quantum information preservation.

I calculate the algebraic transformations, systematically simplifying the complex fraction by manipulating terms and consolidating like components. The result elegantly reduces to a squared term representing the quantum state’s resilience against probabilistic errors.

The final form reveals a compact representation of fidelity that directly relates to the probability of quantum state deviation, capturing the fundamental relationship between error probability and quantum information integrity.

Alright, now I need to calculate the logical state fidelity after post-selecting on detectable errors.

Given the correct answer to the previous problem, the physical state fidelity is:

Let me recalculate which errors give non-zero overlaps. I must have made an error in my calculation.

Starting with the ideal state:

For errors on qubits (2,1), let me check all 15 non-identity Paulis: - \(IX\): Acts as \(I \otimes X\) on qubits 2,1. This gives \(\frac{1}{2}(|0010\rangle + |1011\rangle + |0100\rangle + |1101\rangle)\). This is orthogonal to \(|\psi_{ideal}\rangle\). - \(IY\): Acts as \(I \otimes Y\) on qubits 2,1. This gives \(\frac{1}{2}(i|0010\rangle + i|1011\rangle - i|0100\rangle - i|1101\rangle)\). This is orthogonal to \(|\psi_{ideal}\rangle\). - \(IZ\): Acts as \(I \otimes Z\) on qubits 2,1. This gives \(\frac{1}{2}(|0000\rangle + |1001\rangle - |0110\rangle - |1111\rangle)\). This is orthogonal to \(|\psi_{ideal}\rangle\). - \(XI\): Acts as \(X \otimes I\) on qubits 2,1. This gives \(\frac{1}{2}(|0100\rangle + |1101\rangle + |0010\rangle + |1011\rangle)\).

I notice the \(XX\) error produces the exact same state as the ideal state. This means the \(XX\) error creates a perfect overlap, which is a critical finding for understanding error characteristics.

The \(XY\) error calculation reveals a complex phase relationship that requires careful analysis of the quantum state transformation.

The \(XZ\) error generates a different quantum state configuration, demonstrating how different two-qubit error patterns can produce distinct quantum state representations. Each error type introduces unique phase and amplitude modifications to the original quantum state.

The \(YI\) error introduces imaginary components across multiple basis states, highlighting the intricate quantum mechanical behavior during error propagation.

Continuing the quantum error analysis, the \(YX\) operation creates another orthogonal quantum state with specific phase and amplitude characteristics. The \(YY\) error reveals interesting sign variations across quantum basis states, while the \(YZ\) operation begins to show additional quantum state transformations.

The \(ZI\) operation demonstrates a symmetric state inversion pattern, redistributing quantum amplitudes with consistent sign changes. The \(ZX\) error further explores quantum state manipulation, revealing intricate relationships between different computational basis configurations.

The \(ZY\) operation introduces complex phase shifts, generating a symmetric quantum state with imaginary coefficients. Notably, the \(ZZ\) operation directly yields the ideal quantum state, indicating a significant non-zero overlap that distinguishes it from previous error scenarios.

The analysis suggests a nuanced exploration of quantum error characteristics, with specific attention to interactions between qubits 2 and 1, revealing subtle state transformations under different operator combinations.

Examining qubits (0,3), the initial investigations demonstrate orthogonality through various tensor product operations. The \(IX\) and \(IZ\) transformations generate quantum states that maintain perpendicularity to the ideal state, suggesting a systematic approach to understanding quantum error propagation.

Notably, the \(XX\) operation reveals a critical deviation: it produces a state identical to \(|\psi_{ideal}\rangle\), indicating a potential non-zero overlap that disrupts the previous orthogonality assumptions. This unexpected alignment warrants careful examination of quantum state transformations.

The subsequent \(XZ\) operation appears to be mid-calculation, suggesting further complexity in the quantum error analysis.

Examining the remaining operations, \(YY\) and \(ZI\) maintain orthogonality, while \(ZZ\) mirrors \(XX\) in generating a state with non-zero overlap to the ideal quantum state.

Ultimately, the analysis reveals two specific error scenarios on qubits 0 and 3 - \(XX\) and \(ZZ\) - that produce significant quantum state deviations, potentially compromising quantum computation integrity.

The physical fidelity calculation involves multiple error probability scenarios, each contributing distinct quantum state transformations with specific overlap characteristics.

Calculating the probabilities reveals five key non-zero contribution pathways, involving independent and combined single-qubit error operations across different quantum register locations.

The total fidelity emerges from summing these probabilistic quantum error scenarios, incorporating terms representing single and double error configurations with uniform overlap values.

The emerging fidelity expression suggests a complex interplay between error probabilities and quantum state preservation mechanisms, capturing the nuanced quantum error correction dynamics.

I’ll expand the squared term, revealing intricate interactions between error probability and quantum state stability, focusing on how different error configurations contribute to the overall system fidelity.

The mathematical structure indicates a systematic approach to quantifying quantum information preservation under probabilistic error conditions, with each term representing a specific error scenario’s contribution to total system reliability.

I’m tracking how the error probability \(p\) modulates the potential deviation from an ideal quantum state, carefully mapping out the probabilistic landscape of quantum gate performance.

The notation suggests a nuanced decomposition of error mechanisms across multiple quantum gates, where each potential error pathway introduces specific transformations to the quantum information.

My current analysis centers on understanding how these probabilistic error pathways interact and aggregate across different quantum operations, revealing the complex dynamics of quantum error propagation. I carefully trace the circuit’s error propagation. The error on qubits (2,1) commutes with \(H_0\) and \(CNOT_{03}\), meaning it remains unchanged through these gates.

I’m examining which Pauli errors preserve the stabilizer state’s structure. The key constraint is that any error must commute with the state’s fundamental stabilizers \(XXXX\) and \(ZZZZ\), maintaining the quantum information’s integrity.

For an error on qubits (2,1), I need to verify its compatibility with both stabilizer generators. The Pauli operators must align in a way that preserves the state’s fundamental symmetries, specifically ensuring the error doesn’t disrupt the delicate quantum correlations.

The potential error configurations include \(XX\), \(YY\), and \(ZZ\), each representing a potential quantum perturbation. However, my initial calculation of \(YY\) reveals unexpected state transformations that deviate from the ideal quantum state.

This suggests a more nuanced approach is required to understand the precise constraints on allowable quantum errors.

I’ll systematically evaluate each error type by computing its overlap with the ideal quantum state, focusing specifically on the interaction between the error matrix and the quantum state vector. The goal is to determine which errors genuinely preserve the quantum state’s fundamental characteristics.

For the \(IZ\) error, I calculate the state overlap by carefully tracking the sign changes and amplitudes across different basis states. The computation reveals a symmetric cancellation, resulting in zero net contribution to the state preservation.

Continuing with the \(XX\) error, I’ll compute its state preservation potential by examining how the error transforms the basis states and their corresponding amplitudes. The calculation involves tracking the transformation and maintaining precise phase relationships between different quantum states.

The \(XY\) and \(XZ\) errors similarly demonstrate zero impact on the ideal quantum state, with their transformations precisely canceling out when averaged across the representative basis states.

I methodically evaluate each error channel, noting how the quantum state’s symmetry and carefully constructed basis states lead to robust error suppression. The systematic zero results across multiple error channels suggest a fundamentally resilient quantum encoding strategy.

Intriguingly, only \(XX\) and \(ZZ\) operations on specific qubit pairs produce non-zero overlap, indicating a nuanced error sensitivity pattern. This observation prompts deeper investigation into the underlying error model’s precise mechanics.

My current calculation suggests a potential misalignment with the stated error probabilities, hinting at a possible misunderstanding of the depolarizing channel’s implementation following each controlled-NOT gate.

The problem specifies a two-qubit depolarizing error channel after each gate, with equal probability of applying one of 15 non-identity Pauli operators. This means each qubit pair involved in a CNOT gate experiences a probabilistic error transformation, carefully constrained to the gate’s immediate participants.

I’m tracking the error probabilities and their impact on quantum state overlap. The calculation involves tracking which specific Pauli errors maintain the quantum state’s essential characteristics, focusing on how \(XX\) and \(ZZ\) errors interact with the circuit’s underlying structure.

The mathematical expression captures the nuanced error probabilities, suggesting a complex interplay between error occurrence and state preservation. Something about the stabilizer structure seems critical to understanding the full error dynamics.

I need to carefully examine the stabilizers \(S_1 = XXXX\) and \(S_2 = ZZZZ\), and how specific two-qubit Pauli errors might commute with these fundamental quantum circuit constraints.

The logical operators \(X_A\), \(X_B\), \(Z_A\), and \(Z_B\) provide additional context for understanding potential undetectable error pathways, which requires precise analysis of their commutation relationships.

For an error \(E\) on specific qubits to be undetectable, it must satisfy two critical conditions: complete commutation with both stabilizer operators when extended across the entire quantum system. This means the error cannot change the fundamental quantum state’s symmetry properties.

The key constraint emerges when examining how the error interacts with the stabilizer group, specifically checking whether \(E|\psi\rangle\) remains proportional to the original state \(|\psi\rangle\).

I need to carefully analyze the commutation relationships between potential errors and the system’s fundamental stabilizer generators, ensuring precise alignment with the quantum code’s symmetry constraints.

The stabilizer group includes \(IIII\), \(XXXX\), \(ZZZZ\), and \(YYYY\), each acting across all four qubits. When examining potential error operators for qubits (2,1), I observe that no two-qubit non-identity Pauli operators exist within this stabilizer group.

This suggests logical operators become the critical pathway for preserving quantum state overlap, requiring deeper investigation into the code’s fundamental symmetry transformations.

An error \(E\) on specific qubits maintains state overlap if it can be decomposed as \(E = c' \cdot S \cdot L\), where \(S\) represents a stabilizer, \(L\) a logical operator preserving the encoded state, and \(c'\) a phase factor.

After careful verification, I confirm that only \(XX\) and \(ZZ\) errors on qubits (2,1) satisfy the state preservation condition \(E_{21}|\psi_{ideal}\rangle = c|\psi_{ideal}\rangle\).

I’m now systematically reviewing the 15 possible errors, comparing my calculations with the given solution that suggests 3 errors per location. The problem involves expanding the expression \((1-\frac{4}{5}p)^2\) to derive the fidelity calculation.

My detailed computation breaks down the \(p\) and \(p^2\) coefficients, carefully tracking each term. I’ve found the coefficient of \(p\) is \(-\frac{26}{15}\) and the coefficient of \(p^2\) is \(\frac{169}{225}\), which differs from the initial approach.

The key is meticulously tracking how each term transforms through algebraic manipulation, ensuring precision in the mathematical derivation.

I notice a discrepancy between the expanded form \((1-\frac{4}{5}p)^2\) and my previous calculation, suggesting potential computational errors that require careful re-examination.

The investigation shifts toward understanding potential systematic errors, particularly exploring the hypothesis of three preservation errors per location instead of two, which might explain the mathematical inconsistencies.

Examining \(YY\) operations on specific quantum states reveals complex transformation patterns that do not maintain the ideal quantum state’s structure. By systematically checking all possible two-qubit Pauli operators for the (2,1) configuration, I discover only \(XX\) and \(ZZ\) preserve the quantum state’s essential characteristics.

The problem involves understanding how depolarizing error channels impact quantum circuit fidelity. I’m focusing on identifying specific error mechanisms that could potentially preserve quantum state overlap, carefully analyzing the probability distribution of non-identity Pauli operators.

My current investigation centers on determining which error configurations might yield non-zero quantum state projection, requiring meticulous examination of error probabilities and their potential quantum state transformations.

The mathematical model suggests a nuanced approach to quantifying quantum noise, with careful consideration of how different Pauli error channels interact with the ideal quantum state.

I’m examining the coefficient distribution across non-identity two-qubit Paulis, noting the subtle distinctions in probability allocation. The specific problem constraints indicate a precise error mechanism where 15 non-identity Paulis emerge with equal probability.

My verification involves meticulously tracing quantum state transformations, specifically analyzing how operators like \(XI\) modify the initial quantum configuration across different basis states.

For the \(XI\) operator applied to the ideal state, I’m systematically computing the transformed state vector, tracking how each basis state transforms and confirming orthogonality through inner product calculations. The computational approach involves decomposing the transformation across multiple basis configurations to validate the quantum mechanical behavior.

The \(ZI\) operator calculation follows a similar methodical process, examining how the state transforms under this specific two-qubit interaction. By carefully tracking sign changes and amplitudes, I’m verifying the expected quantum mechanical properties.

I’m now preparing to explore an alternative computational strategy, focusing on calculating the final fidelity through a probabilistic approach that considers different error scenarios and their respective probabilities.

The fidelity calculation involves summing over all possible error configurations on two specific qubit regions, tracking how non-identity Pauli operators might impact quantum state overlap. I’ll count the number of Pauli operators that preserve quantum state characteristics, specifically identifying two preservation-capable operators for each qubit region.

My computational strategy involves tracking: - Preservation-capable Pauli operators on specific qubit regions - Probability interactions between these error configurations - Quantum state overlap under different transformation scenarios

The resulting fidelity equation captures these probabilistic interactions, showing how quantum state coherence degrades under different error probabilities. By systematically breaking down potential error pathways, I can model the quantum system’s resilience and information preservation potential.

I expand the squared term, revealing a nuanced relationship between probability parameters. Comparing coefficients reveals a precise algebraic constraint linking the original expression to this transformed representation.

Solving the algebraic manipulation shows a critical relationship: when \(p\) varies, the system’s fidelity transforms predictably. The algebraic steps methodically isolate the key variable, demonstrating how quantum error correction fundamentally depends on precise probability management.

The unexpected result suggests an underlying computational complexity. My algebraic exploration indicates potential systematic constraints that require deeper mathematical investigation.

I notice a discrepancy between the coefficient calculations, specifically around the \(p\) terms. The coefficient of \(p\) differs between two computational paths: \(-\frac{26}{15}\) versus \(-\frac{24}{15}\). This variance implies a need to adjust the error model, potentially expanding the error locations from 4 to 6.

The qubit state representation hints at a more intricate quantum error correction strategy, where preserving specific quantum states demands precise error management across multiple locations.

I’m systematically exploring two-qubit Pauli transformations, tracking how each operation modifies the quantum state’s amplitude and phase. The transformations reveal subtle symmetries and state transitions that could be critical for understanding quantum error correction mechanisms.

The next sequence of Pauli operations continues to demonstrate intricate phase and amplitude manipulations across different basis states, with imaginary coefficients introducing complex quantum interactions.

Specifically, the transformations from \(XY\) through \(YX\) showcase how quantum gates can induce phase rotations and state superpositions through precise mathematical mappings between computational basis states.

The detailed \(YY\) calculation reveals nuanced sign and phase changes when applying the Y gate twice to different qubit configurations, highlighting the non-trivial nature of quantum gate compositions.

I carefully track the transformations, noting how \(Y\) introduces complex phase factors and sign changes when applied to basis states. The explicit calculations show how tensor product interactions generate intricate quantum state mappings.

The key observation emerges that \(Y \otimes Y\) produces non-trivial transformations across different input states, with specific sign and phase variations that depend on the initial qubit configuration.

I systematically map out these transformations, revealing the complex quantum mechanical behavior under \(Y\) operations across multiple qubits.

The state transformations show that only \(XX\) and \(ZZ\) preserve the ideal quantum state \(|\psi_{ideal}\rangle\), indicating specific symmetry constraints in this quantum system. These two operations uniquely maintain the state’s structural integrity without introducing significant alterations.

By meticulously tracking sign changes and qubit interactions, I’ve identified the precise conditions under which quantum state preservation occurs, highlighting the nuanced quantum mechanical principles governing these transformations.

I’m now shifting focus to a different analytical approach, examining the logical state fidelity through post-selection of detectable errors in the [[4,2,2]] quantum error correction code. The key challenge involves characterizing undetectable errors by understanding their relationship to stabilizer and logical operator groups.

Specifically, I’m investigating how errors on specific qubit pairs can remain undetected when extended across the entire quantum system, with particular attention to the stabilizer group’s structural properties.

My analysis centers on the code’s 4-dimensional state space, carefully tracing how potential errors might map between valid code states without disrupting the fundamental quantum information encoding.

Two-qubit errors present an intriguing challenge. They can only remain undetectable by either perfectly preserving the code space or acting as a logical transformation. However, the constraints are tight - stabilizers and logical operators both engage all 4 qubits, which complicates potential undetectable pathways.

I’m focusing on precise conditions: an error must map every code state to another valid code state without triggering detection mechanisms. This requires a nuanced understanding of how quantum errors propagate through the specific encoded quantum information.

The key is examining how errors interact with stabilizer generators like \(XXXX\) and \(ZZZZ\). By tracking whether an error commutes with these fundamental symmetry operations, we can determine its detectability within the quantum error correction framework.

Specifically, undetectable errors must preserve the code space’s essential properties. This means maintaining the state’s stabilizer eigenvalues while potentially mapping the quantum state to an equivalent representation.

For our specific scenario, \(XX\) and \(ZZ\) errors on qubits (2,1) represent known undetectable transformations. These operations leave the ideal state unchanged, effectively becoming invisible to the error detection mechanism.

However, there might exist additional subtle logical operations that can transform the GHZ state in ways not immediately apparent, potentially revealing more complex undetectable error pathways.

By systematically examining logical operations, I find that only the identity transformation truly preserves the precise quantum state. Other operations like \(X_A X_B\) or single-qubit \(Z\) rotations fundamentally alter the quantum configuration.

This analysis suggests that for the logical fidelity calculation, we must carefully post-select scenarios where no detectable errors have occurred, which will significantly impact our error probability estimation.

The probability calculation reveals a nuanced error landscape, where the probability of no detectable error involves a complex interplay between error rates. When no detectable error emerges, the logical state appears to perfectly reproduce the original quantum state, suggesting an intriguing quantum error correction mechanism.

However, my initial confidence seems premature. Something in this reasoning requires deeper scrutiny, prompting me to re-examine the underlying assumptions and carefully re-read the original problem statement.

The key challenge lies in distinguishing between detectable and undetectable errors within the quantum error correction framework. Specifically, I need to analyze how different error types interact with the stabilizer group and potentially modify the logical state’s fidelity.

For the [[4,2,2]] code, an error’s detectability hinges on its commutation relationships with the stabilizer generators \(XXXX\) and \(ZZZZ\). By systematically examining two-qubit Pauli errors, I can map out which transformations remain undetected by the code’s measurement protocol.

Starting with the \(IX\) error on qubits (2,1), I observe it commutes with \(XXXX\) but anticommutes with \(ZZZZ\), suggesting this particular error would be detectable through stabilizer measurements.

Continuing through the list, I find most two-qubit Paulis are detectable. The notable exception emerges with \(XX\) and \(YY\) errors, which commute with both \(XXXX\) and \(ZZZZ\), rendering them undetectable by the code’s standard verification mechanism.

These undetectable errors occur at three specific locations, suggesting a potential vulnerability in the quantum error correction strategy. The pattern reveals a systematic weakness where certain tensor product operations fail to distinguish these particular error types.

I’ll need to investigate how \(YY\) specifically interacts with the ideal state to understand its preservation properties.

Comparing the initial state transformation shows \(YY\) introduces a non-trivial state modification, indicating a subtle logical operation that isn’t immediately apparent through direct algebraic manipulation.

The post-selection process involves measuring stabilizers and conditioning on +1 eigenvalue outcomes, which occurs with probability \((1 - \frac{4p}{5})^2\). Interestingly, this probability precisely matches the physical fidelity from earlier analysis.

When no detectable error occurs, the final state emerges from a mixture of the ideal state and potential undetectable error states. For specific errors on the (2,1) qubit, \(XX\) preserves the quantum state, while \(YY\) induces a more complex transformation.

I’ll carefully construct the density matrix by summing over undetectable error combinations, normalized by the no-error probability. This reveals the post-selected state isn’t pure, but a nuanced mixture reflecting potential quantum error pathways.

I’ll define basis states representing different error configurations relative to the ideal quantum state, tracking how \(YY\) errors can transform the initial quantum system. These basis states capture the probabilistic landscape of quantum error propagation, allowing precise characterization of the post-selection process.

The normalization factor \((1-\frac{4p}{5})^2\) scales the density matrix, accounting for the probability of avoiding detectable errors during the quantum computation.

I’m focusing on understanding how specific errors like \(YY\) on different qubit locations modify the quantum state’s fundamental structure, particularly examining how these transformations relate to the logical GHZ state’s preservation.

The key insight involves tracing how undetectable errors impact the logical quantum information, recognizing that stabilizer code properties uniquely determine the logical state after post-selection.

My calculation reveals \(YY\) on specific qubits produces non-trivial state transformations that don’t obviously preserve the original quantum configuration. This suggests potential complexities in error characterization that require careful mathematical verification.

The physical fidelity’s probabilistic nature implies most undetectable errors maintain state integrity, yet my specific \(YY\) calculation challenges this initial assumption, prompting a deeper investigation into potential calculation errors or alternative interpretive frameworks.

I’m meticulously tracking qubit ordering and transformation rules, systematically examining how \(YY\) operates across different quantum states to understand its precise transformation mechanisms.

The \(YY\) operation on qubits (2,1) generates predictable sign changes and state transformations, revealing a consistent pattern of quantum state manipulation where specific input states map to specific output states with consistent negative signs.

By applying \(YY\) across various basis states like \(|1001\rangle\), \(|0000\rangle\), \(|0110\rangle\), and \(|1111\rangle\), I observe a remarkable symmetry: each transformation results in a negated state, ultimately preserving the overall quantum state structure while introducing phase inversions.

The final result demonstrates that \(YY^{(2,1)}\) acts as a precise quantum operation that transforms \(|\psi_{ideal}\rangle\) to its own negation, highlighting the intricate quantum mechanical principles governing these transformations. I realize the GHZ state’s physical encoding involves a superposition across four qubits. The ideal state spans multiple computational basis states, representing a logical encoding that preserves quantum information through carefully constructed error syndromes.

The logical state maintains the original two-qubit entangled structure, despite being physically represented across four qubits. Post-selection ensures that only specific error patterns survive, which fundamentally preserve the quantum state’s essential characteristics.

Undetectable errors transform the state in ways that do not disrupt its core quantum information, effectively maintaining the logical quantum state’s integrity through sophisticated error correction mechanisms. I’ll carefully complete the partial thought from the previous thinking and then summarize the new thinking about error scenarios.

Continuing the thought about \(YY\) errors on different locations, I calculate the effect of applying \(YY\) operators on specific qubit positions. When \(YY\) errors occur simultaneously on (2,1) and (0,3), the quantum state transforms in a specific way. The calculation reveals a sign change and state transformation that depends on the initial quantum state and the precise error locations.

The detailed computation shows how \(Y\) operators acting on specific qubits can invert and modify the quantum state’s sign and configuration, demonstrating the complex dynamics of quantum error interactions.

Ultimately, this analysis reveals that certain combinations of \(YY\) errors can preserve the original quantum state’s fundamental structure, highlighting the intricate symmetries in quantum error correction mechanisms.

The investigation explores how undetectable errors interact, demonstrating that specific error combinations maintain the quantum state’s essential characteristics, potentially up to a phase factor.

The key insight emerges from examining how different error combinations - specifically \(YY^{(0,3)}\) and \(XX^{(2,1)}\) - interact with the ideal quantum state, revealing preservation of quantum information under certain error scenarios.

However, a deeper complexity arises when considering the post-selected state’s density matrix, suggesting potential nuances in the state’s fidelity that require further mathematical scrutiny.

The critical observation centers on proportionality: each error combination can be represented as a scaled version of the original state, with scaling factors maintaining unit magnitude.

The post-selected state emerges as a pure \(|\psi\rangle\), achieving perfect logical fidelity of 1. However, the nuanced aspect involves understanding the post-selection success probability, which depends on the parameter \(p\).

I’m wrestling with potential interpretations - whether the fidelity calculation should incorporate the post-selection probability or strictly consider the state’s purity upon successful selection. The definition of logical fidelity for mixed states introduces intriguing complexity.

My current intuition suggests carefully examining the \(YY\) operation across different qubits, meticulously tracking sign conventions and transformation rules to validate the initial assessment.

Systematically computing \(Y \otimes Y\) for each basis state reveals intricate phase and sign interactions. The transformations show how \(Y\) gates modify basis states, introducing complex phase factors and sign changes across different two-qubit configurations.

The detailed calculations demonstrate non-trivial quantum state manipulations, highlighting the subtle interplay between quantum gates and basis states. \(Y\) transforms basis states with specific phase and sign changes. Applying \(YY\) to different input states reveals consistent transformation patterns.

For \(|0000\rangle\), the operation yields \(-|0110\rangle\), introducing complex phase factors through qubit rotations. Similarly, \(|1001\rangle\) transforms to \(-|1111\rangle\) under the same quantum gate sequence.

The systematic application of \(Y\) gates demonstrates how quantum operations modify quantum states through precise phase manipulations. I’ll complete the partial thought from the previous thinking and then continue with the new calculations:

$$

I verify the quantum gate transformations across different basis states. The Y gate operations systematically transform the quantum states, showing consistent sign and phase changes. Each calculation demonstrates how the Y gates modify the quantum state’s configuration, revealing intricate quantum mechanical interactions.

The transformations map specific input states to their corresponding output states, preserving the underlying quantum mechanical principles of superposition and phase manipulation.

I’m now exploring the logical state fidelity for the [[4,2,2]] quantum error correction code. The logical states represent encoded quantum information across multiple physical qubits, providing a robust mechanism for quantum information preservation.

The logical states encode different two-qubit combinations, mapping binary representations across a four-qubit system with specific transformation rules.

I’m carefully examining the logical operators and computational basis states, focusing on how \(Z_A\) and \(Z_B\) eigenvalues define these logical states. The eigenstates reveal intricate relationships between quantum states and their transformations under specific logical operations.

The computational basis states show complex phase and eigenvalue interactions that determine the logical state representations, suggesting nuanced quantum information encoding strategies.

I recognize the logical GHZ state encoding as \((|00\rangle_L + |11\rangle_L)/\sqrt{2}\), which implies a symmetric quantum superposition across the logical subspace.

Undetectable errors preserve the logical state up to phase, ensuring complete logical state fidelity. This means the post-selection process maintains the quantum information perfectly.

I’m carefully examining the error probabilities, noting the undetectable error configurations across different qubit locations. The probability calculation seems complex, requiring precise tracking of error types and their interactions.

The current analysis suggests the logical state fidelity remains 1, independent of the error probability parameter \(p\). However, the detailed error counting mechanism needs careful verification. I’m reconsidering the logical state fidelity calculation. The post-selection process introduces a renormalization factor that could impact the state’s characteristics. My previous conclusion about perfect fidelity might need more careful examination. The density matrix calculation suggests a more nuanced analysis of how undetectable errors affect the quantum state’s preservation during post-selection.

The unnormalized state reveals potential complexities in the fidelity computation that weren’t immediately apparent. I need to carefully trace how the different error probabilities and post-selection mechanism interact to truly understand the logical state’s final configuration. The fidelity reaches perfect unity, unaffected by the error probability. This confirms the post-selection process guarantees an ideal logical state reconstruction with absolute certainty.

The user wants me to populate the code template with the final answer from my derivation. From my analysis in Step 6, I found that the logical state fidelity after post-selection is 1, independent of p.

So I need to set F_logical = 1 in the code template.