XY Model — quantum simulation of magnetic materials

Unlike optimization benchmarks (VRP, SMPP) where classical brute-force solves the toy instance in milliseconds, the XY model exhibits the exponential classical scaling that motivates quantum computing in the first place: storing the full wavefunction of an N-site chain requires 2ᴺ complex amplitudes. At N=30 that's ~10⁹ amplitudes (~10 GB); at N=50 it exceeds all storage humanity has produced. Quantum computers store the same state in N qubits.

For this particular Hamiltonian we are lucky: the Jordan-Wigner map converts the XY model to a system of free fermions whose spectrum has a closed analytical form. That gives us the ideal benchmark — an exact classical reference at any N — while the underlying physics is the same class of Hamiltonian that governs strongly-correlated magnetic systems where classical simulation genuinely fails.

Defense relevance spans three AFRL / NASA physics regimes: (1) radar-absorbing magnetic materials for low-observable platforms, (2) quantum spin liquids underlying next-generation quantum sensors and magnetometers, and (3) the baseline Hamiltonian class from which richer defense-relevant models (2D Heisenberg, Fermi-Hubbard, Kitaev chains) are built.

Hamiltonian: H = -J·Σᵢ[(1+γ)/2·XᵢXᵢ₊₁ + (1-γ)/2·YᵢYᵢ₊₁] - h·Σᵢ Zᵢ with periodic boundary conditions, γ=0.5 anisotropy, N ∈ {4, 6, 8}. Constructed as a Qiskit SparsePauliOp under standard little-endian ordering.

Exact reference: Jordan-Wigner + Bogoliubov gives single-particle energies εₖ = √((h - J·cos k)² + (Jγ·sin k)²), with the ground state energy the minimum over both Neveu-Schwarz (antiperiodic-fermion) and Ramond (periodic-fermion) sectors. VQE: hardware-efficient RealAmplitudes ansatz depth-3, COBYLA optimizer (maxiter=300), 10 random restarts. Statevector simulation, no shot noise.