◬ MIMZY · instrument № 15 · the analog–quantum suite · THE PATH
The Icosian Engine
In 1857 William Rowan Hamilton sold a wooden puzzle: visit all twenty corners of a dodecahedron exactly once. The same man gave physics the Hamiltonian — the energy operator Ĥ at the heart of quantum mechanics. This engine closes his loop: play his game, watch an exact solver walk the search tree, then anneal — the path found as the lowest-energy state of its own Hamiltonian, which is precisely how a quantum annealer attacks it.
⊙ EDUCATIONAL & SIMULATION — exact backtracking + simulated annealing of the genuine Ising objective (classically, and it says so)
graphseek
mode
The board
PLAY: click any corner to start. Visit every corner exactly once along the edges.
The ledger
vertices · edges
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visited
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solver nodes searched
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annealer step · temp
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energy E (broken links)
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the Hamiltonian being cooled:
Ĥ = A·Σᵥ(1−Σᵢ xv,i)² + A·Σᵢ(1−Σᵥ xv,i)²
+ B·Σ(u,v)∉E Σᵢ xu,ixv,i+1
ground state E = 0 ⟺ the path exists bits for the full QUBO here: — · this engine anneals on the permutation manifold (same objective, feasible slice)
How real is this — and is the pairing really chef's kiss? Yes, with receipts. Same Hamilton: the Ĥ of quantum mechanics and the Hamiltonian path are both his. The encoding is genuine: the energy function shown is the standard Ising/QUBO formulation (Lucas 2014) used on real quantum annealers — the path literally becomes the ground state of a Hamiltonian, and cooling finds it. The honest limits: Hamiltonian path is NP-complete; gate-model quantum computers offer only Grover's √ speedup for it (no known exponential advantage), and this page's annealer is a classical simulation of the annealing idea, not quantum hardware. The Petersen graph is the classic counterexample — the exact solver proves by exhaustion (a few hundred nodes) that no Hamiltonian cycle exists, while a path does: choose it and watch both truths demonstrated. Verified before shipping: dodecahedron cycle found in 66 search nodes; Petersen cycle refuted in 274; the annealer reaches E = 0 on the dodecahedron path.