VQE Variational Quantum Eigensolver Simulator

VQE Ansatz

Qubits 3

Ansatz layers 2

Learning rate 0.35

Transverse field h 1.0

Hamiltonian

H = −Σ ZᵢZᵢ₊₁ − h Σ Xᵢ (ring-coupled transverse-field Ising model). The ansatz alternates Ry(θ) rotations with a CNOT entangling ring; the classical optimizer moves θ downhill using the parameter-shift rule.

Energy ⟨H⟩

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Exact ground state

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Iteration

‖∇E‖

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How this maps to real VQE

The circuit's statevector is simulated exactly (2ⁿ complex amplitudes) — this is what a real quantum computer's measurement statistics estimate via repeated shots.
⟨H⟩ = ⟨ψ(θ)|H|ψ(θ)⟩ is the cost the optimizer minimizes; on real hardware it's built from Pauli-string expectation values measured in different bases.
The parameter-shift rule ∂E/∂θ = [E(θ+π/2) − E(θ−π/2)]/2 gives an exact analytic gradient for Pauli-rotation gates — no finite-difference approximation needed, exactly as on real hardware.

for each θ_i:
  g_i = (E(θ_i+π/2) − E(θ_i−π/2)) / 2
θ ← θ − lr · g

circuit / Ry angle energy convergence basis-state probability