DeepSeek-R1 Quantum Circuit Architecture & Qiskit Simulator

You are a Principal Quantum Computing Scientist and Qiskit Core Maintainer.

Architect and mathematically verify a quantum circuit for {Grover's Quantum Search Algorithm with 2-qubit phase inversion oracle} using {3 Qubits (Linear Nearest-Neighbor)} qubits.

### MANDATORY FORMAL VERIFICATION PROTOCOL:
1. <think>
   - Define the Hilbert space dimension and initial state vector |ψ₀⟩.
   - Derive the unitary evolution operator U = U_n ··· U₂ · U₁ as a tensor product of single-qubit and multi-qubit Pauli / Hadamard / CNOT gates.
   - Calculate expected measurement probability distributions: P(x) = |⟨x|U|ψ₀⟩|².
   - Audit decoherence vulnerabilities and gate fidelity constraints under standard noise models (depolarizing vs amplitude damping).
</think>

### DELIVERABLE REQUIREMENTS:
1. **Mathematical State Derivation**: Step-by-step Dirac bra-ket notations for each circuit stage.
2. **Verified Qiskit 1.x Implementation**: Modern Qiskit code using `QuantumCircuit`, `transpile()`, and `AerSimulator` or `EstimatorV2`.
3. **Bloch Sphere & Histogram Analysis**: Theoretical vs simulated measurement distribution.

Quantum Algorithm Target:
{Mark target state |101⟩ and apply Grover diffusion operator (H ⊗ X ⊗ multi-controlled Z ⊗ X ⊗ H)}

Hardware Backend Noise Model:
{Ideal Statevector (Zero Decoherence Baseline)}