Fig. 1: A comparative illustration of GCIM and VQE on a two-electron four-spin-orbital system.

a A toy model consists of two-electron (one alpha electron and one beta electron) in four spin-orbitals (two alpha spin-orbitals and two beta spin-orbitals), where the spin-flip transition is assumed forbidden. b The projection of the exact wave function on each configuration (yellow shadow). A constrained optimization of the free parameters will put limits on the projection (dashed line). c Comparative demonstration between GCIM method and VQE. The GCIM method generates a set of non-orthogonal bases, called generating functions. Then, the GCIM explores the projection of the system on these generating functions and solves a corresponding generalized eigenvalue problem for the target state and its energy. The conventional VQE essentially explores the parameter subspace for a given wave function ansatz through numerical optimization that can be usually constrained by many factors ranging from ansatz inexactness to barren plateaus and others. For given Givens rotations that generated excited state configurations, the lowest eigenvalue obtained from the GCIM method guarantees a lower bound of the most optimal solution from the VQE. It is worth mentioning that: (i) for a standard (single-circuit) VQE, having one two-qubit Givens rotation acting on a standard quantum-chemistry reference (restricted Hartree-Fock) would not improve the energy estimate; (ii) “fermionic swap” gates⁷¹ would be required if the excitations included in the ansätze involve spin-orbitals that are not mapped onto adjacent qubits for a given quantum architecture.