Varia­tio­nal optimiza­tion (e.g. QAOA, VQE)

A varia­tio­nal quantum algorithm (e.g., the Varia­tio­nal Quantum Eigen­sol­ver, VQE, or the Quantum Appro­xi­mate Optimiza­tion Algorithm, QAOA) uses a parame­ter­i­zed quantum circuit to generate a quantum state. The goal is to minimize a functional—typically the expec­ta­tion value of an observable—of the genera­ted quantum state. To optimize the parame­ters, a classi­cal optimi­zer is employed that searches for parame­ter values leading to lower objec­tive function values and, ideally, conver­gence to the global minimum. QAOA follows a very speci­fic circuit construc­tion inspi­red by quantum anneal­ing. VQE, in contrast, offers signi­fi­cantly more freedom in the choice of ansatz. In both cases, one can choose between shallow or deeper ansatz circuits with diffe­rent numbers of parame­ters, each with corre­spon­ding effects on expres­si­vity, traina­bi­lity, and robust­ness to noise.