The variational quantum eigensolver (VQE) approach is currently one of the most promising strategies for simulating chemical systems on quantum hardware. In this work, I will describe a new quantum algorithm and a new set of classical algorithms based on VQE. The quantum algorithm, ADAPT-VQE, shows promise in mitigating many of the known limitations of VQEs: Ansatz ambiguity, local minima, and barren plateaus are all addressed to varying degrees by ADAPT-VQE. The classical algorithm family, O2DX-UCCSD, draws inspiration from VQEs, but is classically solvable in polynomial time. This group of algorithms yields equations similar to those of the linearized coupled cluster theory (LCCSD) but is more systematically improvable and, for X = 3 or X = ∞, can break single bonds, which LCCSD cannot do. The overall aim of this work is to showcase the richness of the VQE algorithm and the breadth of its derivative applications.