Quantum Annealing Minimum-Gap Explorer

Problem Instance (5-Spin Ring)

Anneal Schedule

Sweep time 8.0 s

Progress s = t / T 0.000

Live Readouts

Ground energy E₀(s)

0.000

Gap Δ(s) = E₁−E₀

0.000

Minimum gap Δ_min

0.000

P(diabatic jump)

0.0%

How it works

Five spins sit on a ring, each qubit i coupled to its two neighbours by J_ij. 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. Each animation frame reads off a pre-computed exact diagonalization (real-symmetric Jacobi eigenvalue algorithm run on H(s) in the computational σz basis) sampled across the sweep — the two lowest eigenvalues give the ground energy E₀(s) and the gap Δ(s), and the ground eigenvector gives every spin's expectation value ⟨σzᵢ⟩(s), which drives the arrows on the ring.

The adiabatic theorem says the system tracks the instantaneous ground state only if the sweep is slow compared to 1/Δ_min² 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⟩|) )

evaluated with the eigenvectors at the minimum-gap point and T = the sweep time you set above. Ferromagnetic coupling (J<0) has no frustration and a large, easy gap. An odd antiferromagnetic ring (J>0, 5 bonds) can't alternate up/down all the way around — it is geometrically frustrated — which collapses the minimum gap and forces a much slower, more careful sweep to stay adiabatic. This is exactly why frustrated problem graphs are the hard case for real quantum annealers like D-Wave.

Spin up ⟨σz⟩ > 0 Spin down ⟨σz⟩ < 0 Transverse field (1−s)

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.