Variational quantum algorithm

A variational quantum algorithm prepares a state from an ansatz, estimates a cost on a quantum device, and updates the parameters with a classical optimizer. McClean, Romero, Babbush, and Aspuru-Guzik set out the pattern for Hamiltonians in 2016, after the original VQE experiment. QAOA has the same outer loop for a combinatorial cost.

The shared shape is not a shared theorem. VQE has a variational upper bound on an energy and a theory of how many shots the estimate takes. It does not have a proved convergence time. QAOA has an approximation ratio for one case of MaxCut at depth 1. It does not estimate molecular energies. Barren plateaus, also analyzed by McClean and coauthors, are a warning about gradients of random parameterized circuits, not a reason to treat every ansatz as doomed or every ansatz as safe. The NISQ label describes the hardware era these algorithms were written for. It is not evidence that the loop scales.