Variational quantum eigensolvers fail before optimization begins when strong correlation splits the molecular energy landscape into competing basins and the initial state selects a non-ground-state basin. We introduce a geometry-conditioned preconditioner
Peq:R↦θ0 constrained by the
SE(3) covariance of the molecular Hamiltonian, so that nuclear geometry is mapped directly into circuit parameters in the correlated ground-state basin. This basin localization changes the relevant gradient statistics from concentration controlled to curvature controlled. In statevector benchmarks on six stretched molecules,
Peq reduces Hartree--Fock initialization errors by factors of
38×--
6250×, reaches sub-mHa initialization in CO, LiH, and H
8, and places N
2, H
2O, and BeH
2 in the mHa-scale correlated basin. In disordered H
10 chains, equivariant basin targeting and stochastic escape reach unit success probability at fixed optimization budget. The procedure performs basin selection before the shot-limited quantum loop; the quantum circuit then refines correlation inside the selected basin.
Digital quantum simulation of strong correlation effects with iterative quantum phase estimation over the variational quantum eigensolver algorithm:
H4 on a circle as a case study
The iterative quantum phase estimation algorithm, applied to calculating the ground state energies of quantum chemical systems, is theoretically appealing in its wide scope of being able to handle both weakly and strongly correlated regimes. However, the goodness of the initial state that is sent as an input to the algorithm could strongly decide the quality of the results obtained. In this work, we generate the initial state by using the classical-quantum hybrid variational quantum eigensolver algorithm with unitary coupled cluster ansatz. We apply the procedure to obtain the ground state energies of the H4 molecule on a circle, as the system exhibits an interplay of dynamic as well as static correlation effects at different geometries. Furthermore, we argue on the importance of static correlation in construction of the reference determinant, and propose a minimally parametrized unitary coupled cluster ansatz, which drastically reduces number of variational parameters while incorporating the static correlation effects in the wavefunction. We demonstrate that a carefully and appropriately prepared initial state can greatly reduce the effects of noise due to sampling in the estimation of the desired eigenphase.
Simulating strongly correlated molecular systems on near-term quantum hardware remains challenging because current hardware offers limited quantum volume and moderate-fidelity qubits. One way to address this challenge is through bootstrap embedding (BE). Bootstrap embedding partitions molecules into smaller fragments that are then embedded in the ``bath'' of other fragments iteratively. Bootstrap embedding is appealing for quantum simulation because fragmentation reduces the qubit requirements of each fragment problem. In this work, we develop a quantum bootstrap embedding (QBE) workflow that uses variational quantum eigensolver (VQE) fragment solvers and examines the algorithmic choices that determine the success of the overall VQE-QBE procedure. To improve efficiency, we introduce FastAdaptVQE, a sparse-matrix-accelerated form of the adaptive variational quantum eigensolver (ADAPT-VQE) that replaces symbolic commutator evaluation with direct statevector calculations, and MatrixFreeAdaptVQE, a matrix-free extension that removes the sparse-matrix memory bottleneck that arises for larger fragments. We also explore modifying the ADAPT-VQE operator-selection step by replacing the purely greedy choice with a lookahead strategy. Benchmarks on H
4, F
2, and selected geometries of H
6 reach chemical accuracy, within 1 kcal/mol of BE results obtained with a full configuration interaction (FCI) solver. For
n-butane, density-matching residuals converged, but energy errors remained outside chemical accuracy. These results show that combining QBE with VQE can accurately compute molecular energies, while density convergence alone does not ensure energy accuracy. This work lays the foundation for extending these energy calculations to larger molecular systems and quantum materials on near-term quantum hardware.