This is the ring-diagram view of the same VQE ansatz as the 3D Bloch-sphere simulator: an identical hardware-efficient circuit — alternating RY/RZ rotation layers with a ring of CNOT entanglers — runs on a genuine complex statevector of 2N amplitudes, and a parameter-shift gradient descent minimizes the same antiferromagnetic Ising-ring Hamiltonian. Instead of watching 3D Bloch arrows, you watch three 2D-native readouts of that exact math.
Per layer l = 1..p, per qubit i:
RY(θ_ry[l,i]) RZ(θ_rz[l,i])
then one ring of CNOTs: (0→1)(1→2)…(N−1→0)
Target Hamiltonian (antiferromagnetic Ising ring):
H = Σᵢ Zᵢ Zᵢ₊₁ (mod N)
Parameter-shift gradient (exact, no finite-difference error):
∂⟨H⟩/∂θ = [⟨H⟩(θ+π/2) − ⟨H⟩(θ−π/2)] / 2
Coupling ring (top): each node is one qubit, colored by its own single-qubit expectation ⟨Zᵢ⟩ (blue = spin down, red = spin up), computed by summing the measurement probability of every basis state weighted by that qubit's bit. Each ring edge is colored and thickened by the two-qubit correlator ⟨ZᵢZᵢ₊₁⟩ — green when the pair is anti-aligned (lowering the Ising energy), orange when aligned. This is a genuine joint expectation value read directly off the statevector, not a product of the two single-qubit values, so a green edge with both nodes pale is the graph's own entanglement signature.
Energy vs iteration (bottom-left): the cost C(θ) plotted every optimizer step, with the brute-force classical ground energy drawn as a dashed reference line.
Gradient heatmap (bottom-right): one cell per rotation parameter (rows = ansatz layers, columns = qubit×{RY,RZ}), brightness = |∂C/∂θ| from the same parameter-shift rule shown above. As N and p grow, watch the grid dim almost uniformly — a direct, hands-on view of the barren plateau phenomenon (McClean et al. 2018).
- Qubits / layers — grows the ansatz; more layers and qubits generally reach lower energy, but the gradient heatmap dims faster too.
- Entangler ring — toggling CNOTs off severs every edge's genuine correlation (⟨ZᵢZᵢ₊₁⟩ then always equals ⟨Zᵢ⟩⟨Zᵢ₊₁⟩); the optimizer can then only align each spin individually.
- Run Optimization — repeated parameter-shift gradient descent steps, driving the energy curve down and the ring colors toward the antiferromagnetic ground pattern.
The reference "Classical ground E_min" is found by brute-force enumeration over all 2N classical spin configurations.