|ψ⟩ = α|0⟩ + β|1⟩U(θ, φ, λ) = RZ(φ) RX(-π/2) RZ(θ) RX(π/2) RZ(λ)H = -∑ J_{ij} σ_i^z σ_j^z - h ∑ σ_i^xCNOT |ab⟩ = |a, a ⊕ b⟩⟨ψ|ψ⟩ = |α|² + |β|² = 1S(ρ) = -Tr(ρ log ρ)QFT |x⟩ = 1/√N ∑ e^{2πixk/N} |k⟩⟨φ|ψ⟩ = Σ φᵢ* ψᵢρ = |ψ⟩⟨ψ|P(|1⟩) = |β|²Grover: O(√N)E(|ψ⟩) = ⟨ψ|H|ψ⟩
|ψ⟩ = α|0⟩ + β|1⟩U(θ, φ, λ) = RZ(φ) RX(-π/2) RZ(θ) RX(π/2) RZ(λ)H = -∑ J_{ij} σ_i^z σ_j^z - h ∑ σ_i^xCNOT |ab⟩ = |a, a ⊕ b⟩⟨ψ|ψ⟩ = |α|² + |β|² = 1S(ρ) = -Tr(ρ log ρ)QFT |x⟩ = 1/√N ∑ e^{2πixk/N} |k⟩⟨φ|ψ⟩ = Σ φᵢ* ψᵢρ = |ψ⟩⟨ψ|P(|1⟩) = |β|²Grover: O(√N)E(|ψ⟩) = ⟨ψ|H|ψ⟩
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    Quantum Chemistry

    Electronic structure, reaction pathways, and materials simulation.

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    Second quantisation & encodings
    Introduce Fermionic encodings and Hamiltonian construction.