Five spins sit on a ring, each qubit i coupled to its two neighbours by Jij. The annealing Hamiltonian interpolates linearly between a transverse driver and the classical cost (problem) Hamiltonian as s sweeps 0→1:
H(s) = −(1−s) Σᵢ σxᵢ + s · [ Σ<i,j> Jᵢⱼ σzᵢσzⱼ + Σᵢ hᵢ σzᵢ ]
s=0: ground state = |+⟩^⊗5, a uniform quantum superposition
s=1: ground state = the classical spin config minimizing the Ising cost
The 5-spin ring lives in a 2⁵ = 32-dimensional Hilbert space. This 2D view exactly diagonalizes H(s) at 41 points across the sweep (real-symmetric Jacobi eigenvalue algorithm) and plots every one of the 32 eigenvalues, not just the ground state and gap — the top panel is a genuine spectral "fan diagram" that reveals every avoided level crossing the ring passes through, not only the narrowest one. The two lowest branches (bold) give E₀(s) and the gap Δ(s); the ground eigenvector's ⟨σzᵢ⟩(s) drives the 2D ring arrows below.
The adiabatic theorem says the system tracks the instantaneous ground state only if the sweep is slow compared to 1/Δmin² at the narrowest point of the gap. Sweep too fast and a Landau–Zener transition can kick the system into the excited state instead of the true optimum:
P(diabatic jump) ≈ exp( −π Δmin² T / (2·|⟨1|dH/ds|0⟩|) )
The bottom-right panel plots this probability as a continuous curve against sweep time T (log scale), so you can see the whole exponential falloff at once instead of a single number. An odd antiferromagnetic ring (J>0, 5 bonds) can't alternate up/down all the way around — it is geometrically frustrated — which collapses its minimum gap and pushes the curve's safe region out to longer T. This is why frustrated problem graphs are generally the hard case for real quantum annealers like D-Wave. But the full 32-level fan diagram reveals a subtlety a single Δmin readout hides: with the small symmetry-breaking field used here, the ferromagnetic ring's two nearly-degenerate classical minima (all spins up vs. all spins down) are connected only by a collective, all-5-spins-flip tunneling channel — a narrow interior avoided crossing near s≈0.68, not at the endpoint, and numerically it comes out even narrower than the frustrated ring's minimum gap for this instance. "No frustration" does not automatically mean "easy gap" once a global symmetry is only weakly broken; watch the fan diagram's interior crossings, not just the two endpoints, to see why.