The Ising Model

Spins on a lattice with pairwise couplings and local fields -- the Ising model is the canonical map for combinatorial optimization and quantum annealing.

Spins, couplings, and a field

The Ising model describes N binary spins s_i in {-1, +1} with energy H = - sum_{i

Optimization in disguise

Many combinatorial problems -- MaxCut, graph coloring, scheduling, portfolio selection -- map exactly onto finding the lowest-energy spin configuration of an Ising Hamiltonian (equivalently, a QUBO). Solving the optimization means finding the Ising ground state, which is NP-hard in general.

The quantum connection

Adding a transverse field turns it into the transverse-field Ising model, the driver Hamiltonian for quantum annealing and a standard benchmark for QAOA. Kronos's KDYN code simulates exactly this transverse-field Ising system as a stand-in for plasma-like dynamics, and reports the honest Trotter cost curve for doing so on a quantum computer.