NQS Parameters
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RBM Wavefunction
Carleo-Troyer (2017): ψ_θ(σ) encodes quantum state amplitudes as an RBM. Parameters θ={aᵢ,bⱼ,Wᵢⱼ} are complex-valued for sign structure.
log ψ(σ) = Σᵢ aᵢσᵢ + Σⱼ log cosh(Λⱼ(σ))
where Λⱼ = bⱼ + Σᵢ Wᵢⱼσᵢ
SR optimization: Stochastic reconfiguration minimizes ⟨H⟩ using the quantum geometric tensor Sₖₗ = ⟨O_k* O_l⟩ - ⟨O_k*⟩⟨O_l⟩ as metric.
Expressibility: α hidden units per spin: RBM captures area-law states exactly; with α→∞ can represent any state. Critical systems need α ~ poly(N).
TFI exact: Critical at h=1 (J=1); E₀/N = -1.0 at h→∞ (product state).
log ψ(σ) = Σᵢ aᵢσᵢ + Σⱼ log cosh(Λⱼ(σ))
where Λⱼ = bⱼ + Σᵢ Wᵢⱼσᵢ
SR optimization: Stochastic reconfiguration minimizes ⟨H⟩ using the quantum geometric tensor Sₖₗ = ⟨O_k* O_l⟩ - ⟨O_k*⟩⟨O_l⟩ as metric.
Expressibility: α hidden units per spin: RBM captures area-law states exactly; with α→∞ can represent any state. Critical systems need α ~ poly(N).
TFI exact: Critical at h=1 (J=1); E₀/N = -1.0 at h→∞ (product state).