VQE
A hybrid algorithm for finding a molecule's lowest energy state — the quantum part computes, the classical part tunes the parameters.
The Variational Quantum Eigensolver rests on the variational principle: whatever guess you make yields an energy that never falls below the true one. You prepare a state with tunable parameters, measure the energy, and let a classical optimiser propose better parameters. You iterate until the value stops dropping.
The quantum circuits are kept short, which is the whole point on noisy hardware. Chemistry is the obvious application, since molecular electronic structure is exactly the kind of quantum problem that scales badly classically. The method has been demonstrated on small molecules, but the optimisation has proved hard to scale — the landscape is full of flat regions where the optimiser loses direction.