The Variational Quantum Eigensolver (VQE)

The Variational Quantum Eigensolver (VQE) is the flagship example. A parameterized quantum circuit, called an ansatz, with tunable angles θ, prepares a candidate state |ψ(θ)⟩ on the quantum computer.

The quantum computer then measures the energy expectation value ⟨ψ(θ)|H|ψ(θ)⟩ for the Hamiltonian H of interest — most often the electronic Hamiltonian of a molecule, making VQE one of the leading near-term approaches to quantum chemistry simulation.

A classical optimizer running on an ordinary computer looks at that estimated energy, adjusts θ in an attempt to lower it further, and the loop repeats: quantum hardware handles the expensive part, evaluating the energy on a genuinely quantum state, while classical hardware handles the comparatively cheap part, deciding how to adjust the parameters.

The objective is sound precisely because of the variational principle: for any candidate state |ψ(θ)⟩ whatsoever, ⟨ψ(θ)|H|ψ(θ)⟩ is guaranteed to be greater than or equal to the true ground-state energy. So driving the measured energy down, however the optimizer chooses to do it, can never overshoot past the real answer — the classical loop is always searching in the right direction.

This hybrid division of labor is exactly why VQE tolerates substantially more noise than Shor's or Grover's algorithms: a somewhat noisy energy estimate still nudges the classical optimizer in roughly the right direction, whereas an algorithm like Shor's needs its entire long circuit to be essentially error-free from start to finish.

The ansatz's design is itself an active research question: a chemically-motivated ansatz (built from the molecule's known orbital structure) tends to need fewer parameters but more circuit depth, while a hardware-efficient ansatz (built from whatever gates a given chip runs natively and cheaply) needs less depth but often more parameters to reach the same accuracy — the right tradeoff depends on both the target molecule and the specific hardware available.