Adiabatic Quantum Optimization: Energy Gap Simulator

Problem Graph

Graph Triangle (frustrated)

Adiabatic Sweep

Sweep rate (1/T) 0.25×

Live State

Sweep parameter s

0.00

Instantaneous gap Δ

—

P(ground state)

100.0%

Min gap Δ_min

—

Outcome after sweep

not run yet

How it works

This is adiabatic quantum optimization — the mechanism behind quantum annealers (D-Wave) and the continuous-time limit of QAOA. A small MaxCut-style spin problem is encoded as an Ising-like Hamiltonian; the goal is to find the spin configuration that minimizes it, i.e. its ground state.

H(s) = (1-s) H_init + s H_problem,  s: 0 → 1
H_init     = -Σᵢ σˣᵢ            (trivial ground state: uniform superposition)
H_problem  = Σ⟨i,j⟩ Jᵢⱼ σᶻᵢ σᶻⱼ   (encodes the graph's cut-weight problem)

The system starts (s=0) in the easily-prepared ground state of H_init and s is swept to 1 over time T. The adiabatic theorem guarantees the system stays in the instantaneous ground state of H(s) — solving the optimization problem exactly — provided the sweep is slow compared to the inverse square of the smallest energy gap Δ_min between the ground and first excited state anywhere along the path:

T ≫ 1 / Δ_min²   (adiabatic condition)

This simulator diagonalizes the full 2ⁿ×2ⁿ Hamiltonian at each step (n≤4 spins) with a real Schrödinger-equation time integrator, so the gap curve and success probability you see are physically computed, not scripted:

Graph — picks the coupling pattern Jᵢⱼ; a frustrated triangle has a small minimum gap and is the hardest instance here.
Sweep rate — how fast s advances per second. Slow sweeps respect the adiabatic condition and land in the true ground state. Fast sweeps cause diabatic transitions at the minimum gap, leaving the system in an excited (wrong) state — exactly the failure mode real quantum annealers must avoid.
Gap plot — the instantaneous spectral gap Δ(s) as the Hamiltonian interpolates; the dip is where a real annealer is most vulnerable to noise and speed.
The 3D view shows each spin as an arrow on its own small Bloch-like sphere, rotating as the collective quantum state evolves under H(s).

s = 0.00 | P(ground) = 100%

Spin (qubit) arrows Antiferromagnetic coupling

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