Varia­tio­nal methods (e.g. VQE, VBE, VarQITE)

The Varia­tio­nal Quantum Eigen­sol­ver (VQE) is the founda­tio­nal NISQ algorithm for ground-state energy estima­tion. You prepare a parame­ter­i­zed quantum circuit, measure the energy, and hand it to a classi­cal optimi­zer to update the parame­ters itera­tively until conver­gence. It works on today’s hardware and uses shallow circuits, but the problems are real: barren plateaus make gradi­ents vanish in large systems, the optimi­zer frequently gets stuck in local minima, and thousands of circuit evalua­tions are needed per run with no guaran­tee of finding the global minimum.  Adaptive Deriva­tive-Assem­bled Pseudo-Trotter Varia­tio­nal Quantum Eigen­sol­ver (ADAPT-VQE) addres­ses the ansatz problem directly by building the circuit adaptively:  instead of fixing the circuit struc­ture upfront, it selects opera­tors one at a time from a prede­fi­ned pool, based on which gradi­ent is largest, growing only the gates that actually matter for the problem. This gives a much more compact and physi­cally motiva­ted ansatz compared to standard VQE, and tends to converge faster with fewer parame­ters. Sampled Quantum Diago­na­liza­tion (SQD) takes a diffe­rent philo­so­phy: By removing the optimiza­tion loop entirely, the quantum compu­ter is used purely as a sampler, genera­ting bitstrings that identify the most important basis confi­gu­ra­ti­ons, which are then used to build a small subspace in which the Hamil­to­nian is diago­na­li­zed classi­cally, avoiding barren plateaus and naturally yielding multi­ple eigenva­lues in a single run. Subspace-Krylov Quantum Diago­na­liza­tion (SKQD) extends this by augmen­ting the sampled confi­gu­ra­ti­ons with Krylov vectors genera­ted through real-time evolu­tion, which syste­ma­ti­cally explore the eigen­spec­trum with well-estab­lished conver­gence guaran­tees and better accuracy for systems where dynamic corre­la­tion plays an important role.