Let me work through this step by step.

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

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

The circuit operations happen in reverse order, so:

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

Let’s trace through the circuit starting from \(|0000\rangle\):

1. After \(H_2\): \(|0000\rangle \rightarrow |00\rangle \otimes \frac{|0\rangle + |1\rangle}{\sqrt{2}} \otimes |0\rangle = \frac{|0000\rangle + |0010\rangle}{\sqrt{2}}\)

2. After \(CNOT_{21}\): \(\frac{|0000\rangle + |0010\rangle}{\sqrt{2}} \rightarrow \frac{|0000\rangle + |0110\rangle}{\sqrt{2}}\)

3. After \(H_0\), the state becomes a superposition with increased complexity. The Hadamard gate splits the first qubit’s state, creating four possible combinations. This results in a normalized state that combines different bit configurations, effectively spreading the quantum information across multiple basis states.

4. The \(CNOT_{03}\) operation further transforms the quantum state. By applying this controlled-NOT gate between specific qubits, the entanglement is modified, producing a new state where certain bit flips occur based on the control qubit’s value. The resulting state represents a carefully constructed quantum configuration that maintains specific correlations between qubits.

The ideal final state emerges as a precisely balanced superposition, representing a key quantum error correction configuration that encodes logical information across multiple physical qubits.

Exploring the logical computational basis requires understanding how stabilizer operators interact with quantum states. The logical states must satisfy specific symmetry conditions, particularly maintaining +1 eigenstates for critical stabilizer operators. This involves examining how different basis states transform under specific logical operations, revealing the intricate structure of quantum error correction codes.

The standard approach to defining logical states centers on eigenvalue properties, specifically how states relate to the logical \(Z_L\) operator and the code space’s fundamental symmetries.

By carefully analyzing the \(|00\rangle_{AB}\), \(|01\rangle_{AB}\), \(|10\rangle_{AB}\), and \(|11\rangle_{AB}\) states, I can map out their precise eigenvalue characteristics under \(Z_A\) and \(Z_B\) transformations, which provides insight into the code’s logical computational structure.

The code space fundamentally requires states satisfying two key stabilizer conditions: XXXX and ZZZZ must both return the same state. Focusing on the ZZZZ stabilizer reveals a critical constraint - only computational basis states with an even number of 1s are permitted.

This means the valid computational basis states include: \(|0000\rangle\), \(|0011\rangle\), \(|0101\rangle\), \(|0110\rangle\), \(|1001\rangle\), \(|1010\rangle\), \(|1100\rangle\), and \(|1111\rangle\). These states represent the +1 eigenspace of the ZZZZ transformation, ensuring parity-based consistency.

Examining the XXXX stabilizer introduces additional complexity, as it systematically flips all qubits. The resulting +1 eigenstates emerge through specific superposition configurations that remain invariant under this bit-flip operation.

I’ll construct a code space basis by normalizing these symmetric states, creating four distinct quantum states that represent the fundamental encoding:

• |C₁⟩ represents the all-zero and all-one state superposition
• |C₂⟩ captures the first two qubits’ symmetric configuration
• |C₃⟩ reflects symmetric behavior in the middle qubits
• |C₄⟩ demonstrates symmetry in the last two qubits

These normalized basis states will help me investigate the logical computational properties under specific stabilizer transformations.

Examining \(Z_A = ZZII\), I observe: - |C₁⟩ remains unchanged - |C₂⟩ and |C₃⟩ acquire a negative phase - |C₄⟩ maintains its original configuration

This suggests certain computational basis states are eigenvalues of the \(Z_A\) stabilizer.

For \(Z_B = ZIZI\), the pattern shifts: - |C₁⟩ preserves its initial state - |C₂⟩ receives a negative sign - |C₃⟩ now appears positive - |C₄⟩ remains to be determined

, likely with a negative sign

Continuing the eigenvalue mapping: - \(|00⟩_{AB}\) shows symmetric entanglement - \(|01⟩_{AB}\) demonstrates asymmetric phase relationships - \(|10⟩_{AB}\) suggests a similar transformation pattern - \(|11⟩_{AB}\) indicates complete phase inversion

I’ll carefully analyze the logical qubit states to verify the eigenvalue calculations, focusing on the Z-basis transformations and their implications for quantum state representation.

The eigenvalue matrix reveals systematic phase relationships across different computational basis states, showing how Z-operators interact with each logical qubit’s state. Each two-qubit configuration demonstrates predictable sign changes that encode fundamental quantum information properties.

By mapping these eigenvalue patterns, I can trace the precise quantum state transformations and understand the underlying symmetries in this quantum system.

The logical Bell states emerge through carefully constructed superpositions, revealing intricate entanglement structures where specific basis states combine with precise amplitude relationships. These states represent fundamental quantum information building blocks with well-defined computational characteristics.

The logical GHZ state demonstrates a remarkable symmetry, combining computational basis states with balanced amplitudes that highlight quantum coherence and entanglement principles.

Critically, the actual circuit preparation diverges from the intended GHZ state, instead generating a distinct entangled configuration with different eigenvalue characteristics and state composition.

I’ll carefully trace the quantum gate transformations, noting how each operation modifies the initial quantum state. The Hadamard gate on qubit 2 introduces superposition, while the controlled-NOT gate creates entanglement between specific qubits, fundamentally altering the quantum circuit’s output.

The intermediate state reveals a nuanced quantum transformation that deviates from the original expectation, suggesting a need for precise gate sequence verification.

Continuing the calculation, the next CNOT operation will further manipulate the quantum state, potentially introducing additional complexity to the circuit’s final configuration.

The state transforms, with specific qubits flipping based on control qubit conditions. The result shows a symmetric pattern of quantum state modifications, where certain basis states change while maintaining overall quantum coherence.

The quantum circuit’s progression reveals intricate interactions between entangled qubits, demonstrating how controlled operations can systematically alter quantum information representations.

I’m questioning whether this circuit genuinely produces the intended GHZ state, recognizing potential discrepancies between the designed outcome and actual quantum state generation.

The analysis will now focus on error mechanisms, specifically examining the two-qubit depolarizing channel’s probabilistic Pauli transformations that introduce noise into the quantum system.

The circuit’s error model involves CNOT gates followed by stochastic quantum noise, where each gate experiences potential random Pauli perturbations across two qubits. Single-qubit Hadamard gates remain unaffected by these error processes.

I’ll track how these probabilistic errors might deviate the quantum state from the ideal GHZ configuration, systematically mapping the circuit’s quantum transformations under noisy conditions.

The initial state transforms through Hadamard and controlled-NOT operations, with each CNOT gate potentially introducing stochastic quantum noise that could disrupt the delicate quantum coherence.

After the first Hadamard gate on qubit 2, the state becomes a superposition of \(|0000\rangle\) and \(|0010\rangle\). The subsequent CNOT gate between qubits 2 and 1 further entangles these basis states, creating a new superposition of \(|0000\rangle\) and \(|0110\rangle\).

The error channel introduces probabilistic Pauli noise, where each non-identity Pauli operation on qubits 2 and 1 contributes a small probability of state perturbation, represented by a weighted sum across the Pauli group.

I’ll apply the Hadamard gate on qubit 0, which transforms the state’s basis representation. The CNOT gate between qubits 0 and 3 then propagates potential entanglement or error correlations.

The final error channel again introduces probabilistic Pauli noise, this time across qubits 0 and 3, completing the quantum circuit’s error propagation analysis.

I can leverage the predictable transformation of Pauli errors through Clifford gates, which allows me to systematically track error evolution across different quantum operations.

By defining specific unitary transformations for each gate sequence, I can map how errors propagate through the quantum circuit’s computational steps.

When \(H_0\) acts on qubit 0, Pauli errors on qubits 2 and 1 remain unchanged, demonstrating the commutative nature of these quantum gates with specific error types.

The \(CNOT_{03}\) gate introduces more complex error propagation dynamics across qubits 0 and 3, requiring careful analysis of how Pauli errors transform through this entangling operation.

Critically, Pauli errors on qubits 2 and 1 continue to commute with the \(CNOT_{03}\) gate, preserving their original error configuration throughout the quantum circuit’s evolution.

Consequently, an initial error \(P_{21}\) persists unaltered through the quantum gate sequence, maintaining its fundamental quantum error characteristics.

The comprehensive density matrix representation captures the probabilistic nature of these quantum errors, systematically accounting for all possible two-qubit Pauli error combinations across different qubit configurations.

The expansion reveals multiple error contribution terms, ranging from the ideal state preservation to increasingly complex error interactions, weighted by precise probability coefficients that reflect the likelihood of each quantum operation.

I’ll calculate the fidelity by comparing this error-prone state against the original quantum state, tracking how these probabilistic transformations impact quantum information integrity.

The first term directly represents the probability of no error occurring, while subsequent terms capture increasingly intricate error scenarios across different Pauli transformation sets.

I’ll focus on computing matrix elements for the quantum state \(|\psi\rangle\), which is a superposition of four specific four-qubit basis states. Each term involves calculating overlap integrals between this state and various Pauli operators.

The quantum state is symmetrically constructed: \(|\psi\rangle = \frac{1}{2}(|0000\rangle + |1001\rangle + |0110\rangle + |1111\rangle)\), suggesting interesting symmetry properties for the matrix element calculations.

I’ll systematically evaluate \(\langle\psi|P|\psi\rangle\) across different Pauli operators to understand how these transformations interact with the specific quantum state.

The key observation is that matrix elements will be non-zero only when the Pauli operator maps basis states to themselves or their scalar multiples, which requires careful tracking of phase and sign transformations.

I’ll systematically list the Pauli operators on qubits 2 and 1, continuing the previous enumeration with additional operators involving X, Y, and Z interactions. These operators represent specific quantum transformations within the computational basis, each introducing unique phase relationships between quantum states.

The sequence includes combinations like \(I \otimes X \otimes Y \otimes I\) and \(I \otimes Y \otimes Z \otimes I\), which demonstrate the rich structure of two-qubit Pauli interactions. Each operator represents a potential contribution to quantum computational processes, revealing intricate quantum mechanical symmetries and transformation rules.

I’m noticing a pattern emerging in these operator combinations, suggesting a systematic exploration of quantum state manipulations across different qubit configurations.

The two-qubit depolarizing channel formula seems complex, involving a weighted mixture of the original state and a maximally mixed state. The parameter \(p\) controls the noise level, with the sum over Pauli operators indicating a comprehensive noise model that considers multiple potential quantum perturbations.

I’ll need to carefully analyze how these Pauli operators interact within the depolarizing channel’s mathematical framework. The channel introduces a probabilistic transformation where each non-identity two-qubit Pauli occurs with equal probability \(p/15\). The transformation can be expressed as a weighted sum of the original state and Pauli perturbations, with the identity term having a different weight.

I’ll compute the fidelity for a pure state by examining how the state transforms under this channel, focusing on the dominant term and potential error contributions.

The circuit involves sequential CNOT gates with independent error channels, where the unitary operation \(\mathcal{U}\) transforms the initial state \(|0000\rangle\). Crucially, errors on specific qubits can commute through later gates, simplifying the error analysis. This means the effective error propagation can be modeled as a composition of independent error channels acting on different qubit subsets.

By leveraging the commutativity of error channels on disjoint qubits, I can rearrange the error operations. The resulting density matrix reflects the accumulated errors, with each error channel introducing a probabilistic mixture of the original state and potential error projections.

The mathematical representation captures how small error probabilities compound, showing a quadratic decay of the ideal state fidelity as errors propagate through the quantum circuit.

I’ll calculate the expected fidelity by tracking the contributions from different error scenarios, considering both single and combined error projections across distinct qubit subspaces.

The quantum state involves symmetric superpositions across four computational basis states. I’ll systematically evaluate the overlap between this state and various Pauli error projections, focusing on specific qubit interactions and their probabilistic contributions to the overall quantum process.

By carefully decomposing the error probabilities and projection operators, I can systematically compute the intricate quantum fidelity components, tracking how different error configurations modify the initial quantum state’s coherence and probability amplitudes.

For the specific Pauli projection \(I \otimes I \otimes X \otimes I\), I observe how this operator transforms each basis state, revealing precise state transitions and phase modifications across the quantum system’s computational subspace.

Calculating the expectation value reveals an intriguing symmetry where the inner product vanishes, suggesting orthogonal interactions between the quantum state and non-identity Pauli operators.

I’m investigating whether this quantum state exhibits eigenstate properties under different Pauli transformations, systematically checking each potential operator’s effect on the state’s structure.

For the \(I \otimes Z \otimes Z \otimes I\) Pauli operator, I note its specific action on the \(|0000\rangle\) basis state, which provides initial insights into potential eigenstate relationships.

The computational exploration shows the operator does not simply map the state to a scalar multiple of itself, indicating complex transformation dynamics within the quantum system.

This suggests a nuanced approach is needed to understand the precise interaction between the quantum state and various Pauli operators, potentially revealing deeper structural constraints.

I’ll systematically examine single-qubit Pauli transformations, tracking how they interact with the state’s specific support configuration. Initial investigations reveal most elementary Pauli operators fail to preserve the state’s intrinsic structure, requiring more sophisticated analysis.

The computational basis states form a delicate network where simple transformations often disrupt the precise quantum configuration, demanding careful mapping of potential permutations and phase relationships.

Exploring tensor product combinations of Pauli matrices emerges as a promising strategy. The \(X_0 X_3\) product demonstrates remarkable preservation properties, perfectly cycling through the support states while maintaining their fundamental relationships.

Subsequent investigations will explore additional tensor product combinations to uncover potential symmetry-preserving transformations within this quantum system.

The computational approach involves systematically mapping state transitions, verifying support preservation, and analyzing overlap characteristics with the initial quantum state.

I examine \(Z\) operators, noting they maintain computational basis states while introducing phase variations. Checking \(Z_0\) reveals zero overlap, indicating no significant interaction with the original quantum state.

For \(Z_1\), I begin calculating the overlap, but the computation appears incomplete. The quantum state transformation suggests potential phase cancellations, though the final overlap calculation remains unresolved.

Investigating \(Z_2\) and \(Z_3\) confirms similar zero-overlap behavior. However, when combining specific \(Z\) operators like \(Z_0 Z_3\) and \(Z_1 Z_2\), the quantum state remains unchanged, resulting in a complete overlap of 1.

These observations hint at intriguing quantum operator interactions, suggesting complex phase relationships within the quantum system.

The list of operators producing unit overlap includes fundamental transformations like the identity and specific cross-operator combinations of \(X\) and \(Z\) operators across different qubit indices.

Exploring \(Y\) operators reveals additional complexity, with their composition involving intricate sign and phase interactions through the combination of \(X\) and \(Z\) transformations.

Specifically, \(Y_0 Y_3\) and \(Y_1 Y_2\) both generate a phase inversion, resulting in an expectation value of -1 when applied to the quantum state \(|\psi\rangle\). These mixed product interactions demonstrate subtle quantum mechanical symmetries within the system.

The analysis suggests intricate relationships between different operator combinations and their effects on the quantum state’s phase and overlap characteristics.

Examining \(X_0 Z_3\) reveals zero expectation value, indicating orthogonality with the original state’s support. This systematic investigation uncovers precise interaction patterns across quantum operators.

The quantum state can be elegantly represented as a symmetric superposition, highlighting its balanced probabilistic nature across specific basis configurations.

The state fundamentally emerges as a tensor product of two maximally entangled Bell states connecting specific qubit pairs, demonstrating intricate quantum correlational structures.

Preservation of Bell states under specific Pauli transformations reveals fundamental symmetry operations, with identity and cross-X interactions maintaining core quantum state characteristics.

Certain Pauli operators induce phase transformations while preserving the essential entanglement topology, suggesting nuanced quantum manipulation strategies.

For the Bell state pairs, I’ll track the non-zero overlap Pauli products across qubits 0-3 and 1-2, revealing intricate interaction patterns. The computational basis reveals symmetric transformations where specific tensor products generate interesting quantum correlations.

The overlap calculations demonstrate how local Pauli operators interact with entangled quantum states, showing systematic phase and symmetry preservation across different qubit combinations.

Exploring error channel projections, I’m identifying non-trivial Pauli subsets on specific qubit pairs, excluding identity operations to understand potential quantum error dynamics.

For the \(\mathcal{P}_{21}\) subset, key interactions emerge through \(X_1 X_2\), \(Y_1 Y_2\), and \(Z_1 Z_2\) operators. Each interaction yields a magnitude of 1, indicating significant quantum correlations between these qubits under different Pauli transformations.

These measurements reveal consistent overlap patterns, suggesting robust quantum state interactions across different Pauli bases. I’ll complete the partial thought from the previous rewritten thinking by copying over the next thinking to finish the sentence, and then summarize:

The cross-product terms reveal interesting interactions between different Pauli operator sets. When considering products of Pauli operators from \(\mathcal{P}_{21}\) and \(\mathcal{P}_{03}\), we find nine potential non-zero overlap scenarios. These arise from the specific combinations of operators acting on disjoint qubits, creating a systematic pattern of quantum correlations.

The calculation shows that when \(P = X_1 X_2\) and \(Q = X_0 X_3\), the resulting operator \(QP\) transforms the state in a way that preserves the code’s stabilizer structure, highlighting the intricate symmetries of the quantum error-correcting code.

I notice the product \(X_0 X_3 \cdot X_1 X_2\) yields a stabilizer operator \(XXXX\), which remains invariant when applied to the code’s basis state. This suggests a deep connection between the operator composition and the code’s fundamental symmetry properties.

The systematic exploration of different operator combinations reveals interesting algebraic relationships within the quantum code’s structure, particularly how specific Pauli operator products interact with the code space.

By carefully computing \(X_0 Y_1 Y_2 X_3 |\psi\rangle\), I find the transformation leads to a sign change, indicating non-trivial phase dynamics in the quantum state manipulation.

The comprehensive analysis shows that all nine possible product combinations exhibit consistent behavior, with squared expectation values uniformly equal to 1, which points to a robust quantum error correction mechanism.

I’m tracking the fidelity calculation through algebraic transformations, systematically expanding and simplifying the expression to reveal its underlying mathematical structure. The goal is to precisely characterize how quantum state fidelity evolves under specific transformations.

The current algebraic manipulation involves carefully tracking each term’s contribution, methodically reducing the complex expression to its most fundamental representation.

The final result emerges as a quadratic function of p, with coefficients that reflect intricate probabilistic interactions. The equation

captures the nuanced relationship between the quantum states, distilling multiple computational steps into an elegant, compact form.

In this subproblem, we need to compute the logical state fidelity after post-selecting on all detectable errors in the [[4,2,2]] code. This is different from the physical state fidelity.

First, let me understand what this means: - The [[4,2,2]] code encodes 2 logical qubits into 4 physical qubits - The stabilizers are \(XXXX\) and \(ZZZZ\) - The logical operators are: - \(X_A = XIXI\), \(X_B = XXII\) - \(Z_A = ZZII\), \(Z_B = ZIZI\) - The code can detect (but not correct) single-qubit errors

The target logical state is \((|00\rangle_{AB} + |11\rangle_{AB})/\sqrt{2}\).

From subproblem 1, we found that the ideal physical state is:

Let me verify this encodes the correct logical state: - \(|00\rangle_{AB}\) should be in the +1 eigenspace of both \(Z_A = ZZII\) and \(Z_B = ZIZI\) - \(|11\rangle_{AB}\) should be in the -1 eigenspace of both \(Z_A\) and \(Z_B\)

For \(|0000\rangle\): - \(Z_A|0000\rangle = ZZII|0000\rangle = |0000\rangle\) ✓ - \(Z_B|0000\rangle = ZIZI|0000\rangle = |0000\rangle\) ✓

For \(|1111\rangle\):

• \(Z_A|1111\rangle = ZZII|1111\rangle = -|1111\rangle\) ✓
• \(Z_B|1111\rangle = ZIZI|1111\rangle = -|1111\rangle\) ✓

I’ll carefully examine the other state components to verify their logical qubit properties. The eigenvalue checks for \(|1001\rangle\) and \(|0110\rangle\) will reveal their logical state characteristics.

For \(|1001\rangle\): - \(Z_A\) and \(Z_B\) both yield \(|1001\rangle\), suggesting this state maintains its sign under logical Z operations.

For \(|0110\rangle\): - Inconsistent \(Z_A\) and \(Z_B\) eigenvalues indicate potential encoding complexities.

The discrepancy signals I need to re-evaluate the logical basis state definitions in this [[4,2,2]] quantum error correction code, focusing on how simultaneous Z operations interact with the encoded states.

I’ll carefully map the logical computational basis states, paying close attention to their eigenvalue relationships across different physical qubit configurations.

The stabilizers \(XXXX\) and \(ZZZZ\) define the code’s fundamental symmetry constraints. By examining how these stabilizers act on the codeword state, I can verify the state’s consistency with the code’s encoding principles.

The specific codeword \(|\psi\rangle\) exhibits symmetric transformation properties under these stabilizer operations, suggesting a carefully constructed quantum error correction mechanism.

I’ll carefully track the stabilizer actions, confirming each basis state transforms predictably under \(ZZZZ\), ultimately validating the quantum state’s structural integrity.

For \(Z_A\) and \(Z_B\) logical operators, I’ll systematically compute their effects across the four basis states. The first calculation shows \(Z_A\) producing a symmetric superposition, with \(|1111\rangle\) acquiring a negative phase.

I’ll continue analyzing \(Z_B\) to understand its precise transformation properties, tracking how each basis state responds to this logical operator.

The state is clearly not a pure eigenstate, revealing complex quantum superposition dynamics. I’m interested in mapping these physical basis states to their corresponding logical computational basis representations.

My next step involves decomposing the quantum state by identifying the joint eigenstates of \(Z_A\) and \(Z_B\) across the four-qubit system.

For each computational basis state, I’ll track the parity conditions that determine eigenvalue signs for \(Z_A\) and \(Z_B\), specifically examining how the sum of qubit indices affects the overall eigenstate characteristics.

The key observations involve tracking the parity relationships between different qubit indices, which will help me understand how the logical computational basis states emerge from the physical qubit configuration.

I need to systematically explore the code space dimensionality, recognizing that the [[4,2,2]] quantum error correcting code requires being in the +1 eigenspace of specific stabilizer generators. The code’s structure implies a 4-dimensional space with two independent logical qubits, which means carefully constructing a basis that satisfies stabilizer constraints.

My next step involves methodically generating basis states by applying logical operators to the initial \(|0000\rangle\) state, tracking how each transformation maintains the stabilizer group’s +1 eigenvalue requirements.

I’m realizing the logical operators have significant flexibility - they can be modified by multiplying with stabilizer elements, which fundamentally changes how I conceptualize the code space’s structure.

The code space is defined by two key constraints: states where the total number of 1s is even. This means the basis includes states with specific parity configurations across the four qubits. Specifically, states with zero or two 1s will be valid code space elements, ensuring the total parity remains even.

The initial basis states include \(|0000\rangle\) and specific two-1 configurations like \(|0011\rangle\) and \(|0101\rangle\), representing the fundamental states that satisfy the even parity requirement.

However, something seems incomplete about this initial characterization. The stabilizer conditions involving \(XXXX\) and \(ZZZZ\) suggest more complex symmetry requirements beyond simple parity. The states must not only have even parity but also exhibit a specific symmetry under bit flips.

I need to carefully reconsider how the stabilizer transformations constrain the possible quantum states in this code space.

The key observations involve the eigenvalue conditions for \(XXXX\) and \(ZZZZ\), which imply specific constraints on the quantum states. These stabilizer operators define a subspace with precise symmetry properties.

The logical Pauli operators \(X_A\), \(X_B\), \(Z_A\), and \(Z_B\) have critical anticommutation relationships that further characterize the code’s structure. Their specific interactions reveal fundamental properties of the quantum error correction scheme.

The proposed state \(|\psi\rangle\) appears to represent a carefully constructed superposition that encodes quantum information across multiple qubits, suggesting a sophisticated encoding strategy for protecting quantum information.

I’ll systematically verify the eigenvalue properties for each basis state, checking how they relate to the \(Z\) operators for qubits \(A\) and \(B\). The computational basis states reveal interesting symmetries in their transformation under these operators.

For \(|0000\rangle\) and \(|1111\rangle\), the eigenvalue signs align perfectly with the expected \(|00\rangle_{AB}\) and \(|11\rangle_{AB}\) states, indicating a precise quantum state construction.

The \(|0110\rangle\) state introduces complexity, showing different eigenvalue behaviors for \(Z_A\) and \(Z_B\) that suggest mixed state components across \(|00\rangle_{AB}\), \(|01\rangle_{AB}\), and \(|11\rangle_{AB}\) subspaces.

Calculating systematically reveals nuanced interactions: \(|1001\rangle\) yields \(-1\) for \(Z_A\) but \(+1\) for \(Z_B\), mapping to \(|10\rangle_{AB}\). Similarly, \(|0110\rangle\) produces \(-1\) for both \(Z_A\) and \(Z_B\), indicating \(|11\rangle_{AB}\) contribution. The \(|1111\rangle\) state interestingly returns to \(|00\rangle_{AB}\) with symmetric \(+1\) eigenvalues.

These calculations suggest a complex quantum state superposition emerging from the intricate interactions of local \(Z\) operators across different basis states. I’ll carefully map out the eigenvalue calculations for different basis states, showing how the quantum state transforms under specific operations. The key observations involve tracking parity across different qubit combinations, revealing the state’s complex transformation.

For \(Z_A\) and \(Z_B\) operations, the eigenvalues depend on specific bit combinations. Each basis state gets mapped differently based on the sum of its bits, creating intricate quantum state transitions. The current state doesn’t match the ideal GHZ configuration, suggesting a need for precise circuit recalculation.

The quantum state encoding reveals subtle parity relationships that don’t align with the desired quantum entanglement pattern, indicating potential errors in the initial circuit design or measurement approach. I’ll complete the partial thought from the previous rewritten thinking by copying over the next thinking up to the newline:

Now I’ll summarize the quantum circuit transformation concisely:

The quantum state evolves through Hadamard and CNOT gates, creating a superposition of four basis states. The probability amplitudes are distributed equally across |0000⟩, |1000⟩, |0110⟩, and |1110⟩, each with an amplitude of 1/2. The next gate will further modify this entangled quantum state.

The logical state mapping reveals an interesting pattern of qubit transformations across different basis states, showing how quantum gates can create complex entangled quantum information representations.

The current state differs from the ideal GHZ state, suggesting potential imperfections in the quantum circuit’s error correction mechanism. The nuanced mapping between physical and logical states requires careful analysis to understand the quantum information preservation.

The post-selection condition introduces an additional layer of complexity, hinting at a sophisticated error detection and correction strategy that will require precise examination of the quantum state’s error characteristics.

In the [[4,2,2]] quantum error correcting code, single-qubit errors become detectable through their interaction with stabilizer generators. This detection mechanism allows for filtering out specific error types during state preparation and measurement.

The depolarizing channel introduces probabilistic error scenarios across the quantum circuit’s CNOT gates, creating a multifaceted error landscape with distinct probability distributions and potential two-qubit Pauli error configurations.

Stabilizer interactions reveal critical constraints for error detectability. An error becomes undetectable only when it simultaneously maintains specific symmetry conditions across both stabilizer generators, requiring precise alignments of \(X\), \(Y\), and \(Z\) operators that preserve commutation relationships.

The undetectable error set includes fundamental symmetry elements like the identity, stabilizer products, and logical operators, forming a complex algebraic structure that preserves quantum information encoding.

These undetectable errors emerge from intricate group-theoretic relationships between stabilizer and logical operator groups, representing fundamental symmetry transformations within the quantum error correction framework.

I’ll systematically investigate the error propagation mechanisms, focusing on two-qubit Pauli channel interactions and their potential to generate undetectable quantum state modifications.

The analysis requires carefully mapping out the 15 non-trivial two-qubit Pauli transformations, examining their interactions with the quantum error correction code’s underlying symmetry structure.

I’ll translate these Pauli operators into 4-qubit representations, tracking how each transformation potentially impacts the logical quantum state’s integrity and fidelity.

The error channels \(\mathcal{E}_{21}\) and \(\mathcal{E}_{03}\) generate complex sets of single and multi-qubit Pauli errors. My focus is identifying which specific error configurations remain undetectable by checking their commutation relationships with stabilizer generators \(XXXX\) and \(ZZZZ\).

I’m considering how these error propagation mechanisms interact with the quantum circuit’s underlying stabilizer structure, recognizing that undetectable errors could potentially introduce logical errors without immediate syndrome detection.

The computational challenge involves systematically evaluating each error’s commutation properties to determine which might compromise quantum error correction protocols without triggering immediate warning signals. I’ll push the error channels through the gates, noting they commute with gates not acting on their qubits. Both \(\mathcal{E}_{21}\) and \(\mathcal{E}_{03}\) can effectively be moved to the end of the circuit. This means the final density matrix will be a sum over all possible error configurations, with probabilities determined by the error channels.

The key next step involves identifying undetectable errors - those that map the code space to itself without being detected. For the [[4,2,2]] code, these errors are specifically those in the normalizer group, which includes stabilizers and logical operators. I’ll systematically catalog these undetectable error combinations to understand how they impact the quantum state’s logical fidelity.

The undetectable errors span multiple categories: stabilizer group elements, logical X and Z operators, and their various products. These include transformations like \(XXXX\), \(ZZZZ\), \(YYYY\), and specific logical operators such as \(X_A = XIXI\) and \(Z_B = ZIZI\).

Initially, I thought the group would have a simple 64-element structure, but the stabilizers’ central nature complicates this assumption. The group’s precise composition requires more careful analysis to determine its exact mathematical structure and implications for error correction.

The key insight involves understanding how specific Pauli error combinations interact, with most two-qubit depolarizing channel errors being detectable. Only very precise, strategically constructed error combinations will remain undetectable in this quantum error correction scenario.

When combining Pauli errors across different qubits, the total error becomes a multi-qubit Pauli operation. Undetectability requires the error to match specific elements within the logical Pauli group modulo stabilizers.

The fundamental observation centers on post-selection, where remaining states represent different undetectable error transformations of the initial quantum state.

After post-selection, the density matrix emerges through a weighted sum of error-transformed states, normalized by the success probability. Each undetectable error maps the original state to a specific code space configuration, with its logical state determined by how the error acts on logical qubits.

Logical operators can transform the state in distinct ways: some preserve the original logical state, while others induce specific logical transformations like applying \(X\) or \(Z\) operations to different qubits. To calculate logical fidelity, I’ll systematically:

1. Detect undetectable error combinations
2. Analyze their logical state modifications
3. Assess the resulting density matrix’s fidelity to the target GHZ state

I’m questioning my previous understanding of the ideal state’s encoding. The circuit aims to prepare a GHZ state, but my initial calculations suggest a different encoded state. I need to carefully re-examine the circuit and verify the logical state representation, particularly for the [[4,2,2]] or \(C_4\) code.

I’m now focusing on the eigenvalue calculations for different bit combinations, tracking how \(Z_A\) and \(Z_B\) operators affect the state’s phase. The computational basis states seem to have specific constraints based on the parity of certain bit positions.

For the \(|01\rangle_{AB}\) subspace, the conditions require \(a+b\) to be even, \(a+c\) to be odd, with specific bit configurations emerging from these constraints. The resulting states include \(|0001\rangle, |0111\rangle, |1000\rangle, |1110\rangle\), each satisfying the parity requirements.

The pattern continues for \(|10\rangle_{AB}\) and \(|11\rangle_{AB}\) states, where different combinations of bit parities determine the allowed computational basis states.

I need to refine this by intersecting with the code space’s stabilizer eigenspace, which requires symmetry under X flips and even total parity. The stabilizer group \(\langle XXXX, ZZZZ \rangle\) constrains the possible states to symmetric superpositions with even weight.

The key code space basis states emerge from symmetric pairs like \((|0000\rangle + |1111\rangle)/\sqrt{2}\), ensuring both \(XXXX\) and \(ZZZZ\) stabilizer conditions are met.

I’ll now investigate how logical operators \(Z_A\) and \(Z_B\) transform these basis states, which will reveal the code’s logical qubit structure. The \(Z_A\) operator’s action looks complex, systematically flipping signs across different symmetric states.

I notice the sign changes aren’t straightforward. The eigenstates aren’t simple superpositions, suggesting a more nuanced quantum error correction mechanism. The operator’s behavior depends on specific bit configurations, hinting at intricate stabilizer relationships.

I’ll need to carefully trace how \(Z_A\) transforms each symmetric state, paying close attention to sign changes and preservation of the code space’s fundamental symmetries.

The logical operators reveal critical constraints: squared operations return identity, cross-operators introduce sign inversions, and inter-qubit interactions follow strict commutation rules. These properties define how computational basis states transform under logical Pauli operations.

Specifically, the joint eigenstates of \(Z_A\) and \(Z_B\) exhibit precise phase behaviors, where each logical qubit contributes a distinct sign modification based on its computational state.

I recognize the ideal state can be strategically decomposed, revealing an elegant two-part structure. By carefully factoring the quantum state, I can expose underlying symmetries and computational relationships between the basis states.

The decomposition allows me to separate the state into two orthogonal components, each representing a distinct quantum configuration with balanced probability amplitudes.

When examining the logical operators’ actions, I discover these components do not maintain eigenstate properties under \(Z_A\) transformations. This reveals a subtle complexity in the quantum state’s behavior, where simple linear combinations do not preserve expected spectral characteristics.

The systematic investigation suggests deeper quantum mechanical constraints are emerging, requiring further precise analysis of the state’s fundamental symmetries and transformation properties.

I’ll explore alternative decomposition strategies, focusing on eigenstates of \(Z_A\) and \(Z_B\) within the code space’s constraints. By carefully examining parity conditions and stabilizer relationships, I aim to uncover more nuanced quantum state configurations that satisfy the underlying algebraic requirements.

The initial approach involves identifying basis states where specific parity conditions are met, potentially revealing intricate quantum correlations across different computational subspaces.

I notice an interesting eigenstate behavior when analyzing the symmetric superposition states. The first state maintains stabilizer symmetry, while the second exhibits more complex eigenvalue characteristics, specifically demonstrating a negative \(Z_A\) eigenvalue.

The computational investigation reveals subtle quantum mechanical interactions that challenge straightforward linear decomposition strategies.

I’m recognizing the need for a more nuanced approach to understanding code space representations, particularly focusing on stabilizer group constraints and logical basis state properties.

The key insight emerges from carefully examining how stabilizer operators interact with quantum states, suggesting that logical computational basis states require careful verification across multiple symmetry conditions.

The [[4,2,2]] code reveals a structured encoding strategy, mapping two-qubit logical states to specific four-qubit superposition states through precise amplitude distributions that maintain critical quantum error correction properties.

I notice a critical discrepancy when applying the \(Z_A\) operator to the initial state. The eigenvalue relationship doesn’t hold as expected, indicating a fundamental flaw in the proposed encoding mechanism. This suggests the need for a careful re-examination of the code’s quantum state construction.

I’ll systematically derive a new encoding by exploring the simultaneous eigenvector structure within the code space. By constructing basis states that transform predictably under \(Z_A\) and \(Z_B\), I can develop a more robust quantum error correction approach.

The code space basis will be constructed to ensure consistent eigenvalue behavior across stabilizer operators, providing a more reliable quantum information representation.

My current basis reveals inconsistent transformation properties, indicating I need to refine my approach and select eigenvectors that maintain coherent symmetry across quantum operations.

The fundamental challenge lies in constructing eigenstates that naturally inhabit the code space while respecting entanglement constraints. Product state assumptions fail, necessitating a more nuanced representation that captures intrinsic quantum correlations.

I’ll explore matrix representations that explicitly map the transformation behavior, seeking a basis where \(Z_A\) and \(Z_B\) exhibit predictable eigenstate interactions within the quantum code subspace.

The proposed alternative basis reveals subtle symmetries: two-qubit entangled states emerge, with specific sign relationships determining code space membership. These states demonstrate complex quantum interference patterns that challenge naive computational intuitions.

Critically, the \(XXXX\) stabilizer selectively preserves certain superposition states, revealing a nuanced quantum selection mechanism. Only states with symmetric amplitude distributions qualify for inclusion, excluding antisymmetric configurations.

The code space’s dimensionality hinges on these precise quantum constraints, suggesting a structured approach to quantum information encoding that transcends classical binary representations.

The \(Z_A\) operator’s action demonstrates intriguing symmetry-breaking properties, selectively inverting specific basis states while potentially disrupting the code space’s integrity by mapping some states outside its defined boundaries.

The key insight emerges from understanding logical operators as transformations that can differ from codewords by stabilizer elements. This nuanced perspective suggests the potential need to redefine how logical operators are conceptualized within this quantum error correction framework.

My investigation reveals a critical discrepancy: \(Z_A\) does not consistently preserve the code space’s structure, particularly when applied to \(|c_1\rangle\), which suggests a fundamental constraint or limitation in the current operator definition.

The resolution likely involves carefully examining how logical operators interact with stabilizers, recognizing that their definition allows multiplication by stabilizer elements while maintaining specific group-theoretic properties. This means exploring the precise conditions under which an operator can be considered a valid logical transformation.

The core requirements for a logical operator emerge as twofold: commuting with all stabilizers and existing outside the stabilizer group itself, forming a sophisticated quotient group structure within the Pauli group’s representation.

For the [[4,2,2]] code, specific Pauli operators like \(Z_A\) and \(X_A\) demonstrate these properties by carefully balancing their interaction with the code’s fundamental stabilizer generators. Their construction ensures they preserve the code space’s essential symmetries while introducing non-trivial logical transformations.

The critical insight centers on understanding how these operators map the entire code space to itself, rather than focusing on individual basis state transformations.

By examining \(Z_A\)’s action, I see it maintains the stabilizer’s +1 eigenspace structure, confirming its role as a valid logical operator. The algebraic manipulation reveals its preservation of the code’s fundamental quantum information encoding.

Carefully tracing the tensor product operations, I confirm \(Z_A\) introduces subtle phase changes across different computational basis states, highlighting its nuanced quantum mechanical behavior.

The operator \(Z_A\) systematically applies Pauli-Z gates selectively, revealing intriguing sign transformations for specific quantum states. By examining \(|0000\rangle\) and \(|1111\rangle\), I observe distinct eigenvalue characteristics that illuminate the operator’s fundamental quantum interactions.

Computational basis states like \(|0110\rangle\) and \(|1001\rangle\) demonstrate how \(Z_A\) modifies quantum amplitudes through targeted phase inversions, revealing the operator’s intricate quantum mechanical properties.

I’m tracking the systematic action of \(Z_A\) across different computational basis states, noting how the operator preserves or inverts quantum state amplitudes with precise quantum mechanical precision.

For \(Z_A\), the eigenvalue depends on the parity of specific qubit combinations. States \(c_3\) and \(c_4\) get phase-inverted because their constituent basis states have odd \(a+b\) parity. Conversely, states \(c_1\) and \(c_2\) remain unchanged due to their even parity configurations.

The \(Z_B\) operator follows a similar pattern, with eigenvalues determined by the \(a+c\) parity across different code space basis states. I’ll continue analyzing its specific transformations systematically.

The computational basis states emerge from carefully constructed superpositions, with each state having specific symmetry properties and eigenvalue characteristics under \(Z_A\) and \(Z_B\) transformations.

The logical states include: - \(|00\rangle_{AB}\) with symmetric \(+1\) eigenvalues - \(|01\rangle_{AB}\) with mixed \(Z_A\) and \(Z_B\) eigenvalues - \(|10\rangle_{AB}\) showing opposite \(Z_A\) and \(Z_B\) signs - \(|11\rangle_{AB}\) exhibiting symmetric \(-1\) eigenvalues

These states represent normalized superpositions that encode intricate quantum correlations across the two-qubit system.

I’m noting a discrepancy between the prepared state and the target GHZ state. The circuit’s output differs from the desired quantum configuration, which requires careful fidelity assessment.

The prepared state \(|\psi_0\rangle_{logical}\) shows a specific two-qubit entanglement pattern that doesn’t precisely match the canonical GHZ state, suggesting potential circuit design challenges or measurement constraints.

Calculating the fidelity reveals a low overlap of \(1/4\) between the actual and target quantum states, indicating significant deviation from the intended quantum configuration.

The mathematical analysis shows the inner product between states is \(1/2\), which when squared yields the fidelity value. This suggests the quantum circuit preparation might not be correctly mapping the desired quantum state.

The circuit notation requires careful interpretation, with operations applied right-to-left in matrix multiplication order. This means the sequence of quantum gates needs precise examination to understand the state preparation mechanism.

Tracing through each gate transformation reveals a systematic state evolution: Hadamard gates and controlled-NOT operations progressively modify the initial \(|0000\rangle\) state. The final state encodes a different quantum superposition than the intended target state.

The discrepancy indicates potential ambiguity in the problem statement or circuit design, potentially requiring further investigation into the precise quantum circuit implementation.

I’ll need to methodically analyze error scenarios, focusing on depolarizing channel effects across the two CNOT gates. The error analysis involves calculating probabilities for various error combinations, considering the 15 possible Pauli error configurations for each gate interaction.

The key steps involve post-selection error handling and subsequently computing fidelity against the target GHZ state, which requires careful probabilistic tracking of quantum state transformations.

I’m examining undetectable error combinations by systematically exploring Pauli error spaces on specific qubit interactions, particularly tracking errors that preserve the code space’s fundamental quantum properties.

My current investigation centers on identifying error patterns from the \(CNOT_{21}\) gate, specifically mapping out the Pauli error space affecting qubits 2 and 1.

The 4-qubit operator representation reveals a comprehensive set of potential error configurations, systematically exploring interaction possibilities between different Pauli operators across these qubits.

I’ll continue analyzing the error spaces for \(\mathcal{E}_{03}\) to understand how different qubit interactions might generate undetectable quantum errors.

The total error \(E\) emerges from combining \(E_{03}\) and \(E_{21}\), with specific conditions determining whether this composite error remains undetectable. Key undetectable scenarios include stabilizers, logical \(X\) and \(Z\) operators, and their intricate product combinations.

I’ll analyze specific error scenarios systematically. When no errors occur, the result is trivially undetectable. However, certain error combinations like a single \(X\) on a specific qubit become detectable through anticommutation with stabilizer operators.

Some error combinations, such as \(XIII \cdot IXII = X_B\), preserve the logical operator structure, rendering them fundamentally undetectable within the quantum error correction framework.

To determine undetectability, I need strict criteria: an error must commute with both \(XXXX\) and \(ZZZZ\) stabilizers. This requires an even distribution of Pauli operators across the error’s representation.

The key constraint involves balancing \(X\), \(Y\), and \(Z\) Paulis to maintain commutation properties with critical stabilizer operators.

When combining errors on disjoint qubits, the total Pauli content becomes a straightforward summation. Each error’s individual Pauli structure contributes to the overall error’s composition, creating complex interaction patterns that determine detectability.

I’ll systematically classify single-gate errors by their Pauli type, tracking the precise combinations that emerge across different qubits. The goal is understanding how these error configurations interact and potentially remain undetected.

The categorization reveals intricate error landscapes, mapping out potential single-gate transformations ranging from individual Pauli operators to mixed two-Pauli interactions. This granular breakdown helps predict error propagation and potential masking mechanisms within quantum systems.

For undetectable scenarios, I’m focusing on errors with even Pauli content, specifically examining configurations where total error remains imperceptible. The key constraint involves ensuring total Pauli operators balance out across gate interactions.

Probability calculations become crucial - like the \((1-p)^2\) chance of no error occurring, which represents a baseline quantum noise scenario. I’m now investigating specific undetectable error sequences, particularly those involving complex multi-qubit transformations.

The emerging pattern suggests tracking total Pauli operator counts becomes essential for understanding quantum error dynamics. After post-selection, I calculate the logical density matrix by averaging over all undetectable errors, weighted by their probabilities and normalized by the total success probability. This mapping transforms physical error states into logical quantum states, capturing how different errors affect the underlying quantum information. Each undetectable error \(E\) potentially encodes a distinct logical state, with the initial state being a superposition of \(|00\rangle\) and \(|10\rangle\).

For specific error scenarios: - A logical identity preserves the original state - \(X_A\) leaves the superposition unchanged - \(X_B\) transforms the superposition to a different basis - \(Z_A\) will introduce a phase shift or state modification - \(Z_B\) maintains the original state configuration

I need to compute the fidelity by comparing the logical state density matrix with the target state. The calculation involves tracking how different error channels affect the quantum state’s evolution.

I’ll systematically map out undetectable error transformations, focusing on identifying patterns that could simplify the complex quantum state analysis.

The goal is to understand how various single-qubit operations modify the initial quantum superposition and track potential error propagation mechanisms. I’ll investigate the undetectable error scenario, focusing on the \(E_{21} = ZZ\) case. By checking commutativity with \(XXXX\) and \(ZZZZ\), I confirm that \(IZZI\) indeed commutes with both stabilizers. This means \(IZZI = Z_A Z_B\) represents a genuine logical operator, making this specific error configuration undetectable.

When examining combined gate errors, I start exploring scenarios like \(E_{21} = IX\) and \(E_{03} = XI\), which produce \(XXII = X_B\), indicating another potential logical error pathway. The partial exploration suggests complex error propagation mechanisms in this quantum error correction context.

I’ll systematically analyze different error combinations, tracking their detectability and potential logical operator generation. The goal is to map out which two-gate error scenarios might compromise quantum information integrity.

Categorizing errors reveals interesting patterns across Pauli operators. Single-type errors like \(X\), \(Y\), and \(Z\) appear in different gate configurations. Mixed-type errors introduce complexity, especially when considering how they might combine undetectably.

The key constraint emerges: for errors to merge invisibly, their Pauli composition must balance out, creating an even distribution across operator types. This means carefully tracking not just individual error occurrences, but their collective mathematical behavior.

Undetectable error combinations follow specific rules, where paired errors of the same type or strategic mixed-type pairings can create imperceptible quantum disturbances. The goal is maintaining a symmetry that masks computational perturbations.

Key patterns include: - Matching 2 or 4 identical Pauli errors across gates - Specific cross-gate error combinations like \(X\) and \(Z\) that preserve logical quantum states - Strategic error pairings that result in equivalent identity or logical operator transformations

These undetectable error scenarios demonstrate the nuanced quantum error correction challenges, where certain error configurations can propagate without immediate detection.

The probabilities suggest increasingly rare multi-gate error sequences, with single-gate errors having higher likelihood of generating logical operator transformations.

Two-gate error combinations produce specific stabilizer and logical operator outcomes, each with a consistent \(p^2/225\) probability. These include combinations like \(XX\), \(YY\), and \(ZZ\) generating symmetric error patterns.

An interesting edge case emerges with mixed Pauli gate errors like \(XY\), where the interaction becomes more complex and initial assumptions about error propagation might require careful verification.

Checking undetectability involves analyzing commutation relationships. For \(XXYY\), this means verifying zero commutators with \(XXXX\) and \(ZZZZ\) by counting gate types, which confirms its potential as an undetectable error configuration.

I’m investigating whether \(XXYY\) represents a logical operator or stabilizer. Decomposing it reveals interesting algebraic properties, with \(IYIY\) presenting an intriguing commutation pattern that suggests membership in the normalizer group.

The systematic approach involves carefully examining commutation relations and exploring the operator’s behavior within the stabilizer framework, potentially uncovering deeper insights into its quantum computational significance.

Focusing on undetectable errors, I’ve identified key probabilistic scenarios:

1. No error occurs with \((1-p)^2\) probability
2. Specific two-qubit errors like \(Z_AZ_B\) emerge with \(p(1-p)/15\) likelihood
3. Logical operators from two-gate errors appear at \(p^2/225\) probability
4. Stabilizer errors like \(XXXX\) manifest with \(p^2\) probability

The success probability calculation involves combining these error modes, accounting for 4 logical operators and 3 non-identity stabilizers. The total probability integrates these distinct error pathways.

Examining potential error combinations reveals nuanced interactions between gate errors, particularly how specific error sequences can generate undetectable logical operations across different qubit configurations.

I’m systematically mapping out error combinations, recognizing they form a structured group with specific transformation properties. The undetectable error group for this quantum code contains 16 elements, representing the logical Pauli group’s fundamental structure.

The stabilizers and logical operators reveal intricate algebraic relationships. I’ll methodically compute their interactions, tracking how different logical operators combine through precise matrix multiplication. Each product provides insights into the code’s error correction capabilities.

Specifically, I’m tracking logical X and Z operators, calculating their cross-products to understand potential error propagation and detection mechanisms within this quantum error correction framework.

Key computations include: - \(X_AX_B\) generating \(ZIIX\) - \(Z_AZ_B\) producing \(IZZI\) - \(X_AZ_A\) yielding \(YIYI\) - \(X_BZ_B\) resulting in \(YYII\) - \(X_AZ_B\) transforming to \(YI\)

These transformations map complex quantum logical operator interactions, revealing subtle computational dynamics.

The user wants me to populate the code template with the final answer from subproblem 2. From my analysis, I found that the logical state fidelity after post-selection is F = 1/4, which is a constant independent of p.

So I need to return the SymPy expression 1/4 or sp.Rational(1, 4).