The advent of fully variational quantum eigensolvers using a hybrid multiresolution approach

TL;DR

This paper introduces a fully variational quantum chemistry framework that jointly optimizes a basis-set-free multiresolution (multiwavelet) orbital representation with a variational quantum eigensolver for the many-body state. By alternating orbital refinement (via surrogate potentials and natural orbital formation) and VQE-based energy optimization, the method delivers continuous, high-precision energy profiles while potentially reducing the active-space size and qubit counts. Numerical demonstrations on cyclohexane, H2, LiH, BeH2, and H4 show favorable energy accuracy and convergence properties, while also exposing limitations of certain ansätze (e.g., SPA) for complex bonding. The approach provides a scalable pathway to near-basis-set-limit quantum chemistry on hardware, compatible with classical solvers and adaptable to larger systems through MRA-PNOs and orbital refinement.

Abstract

In electronic structure theory, variational methods offer a valuable paradigm for approximating electronic ground states. However, for historical reasons, this principle is mostly restricted to model chemistries in pre-defined fixed basis sets. Especially in quantum computation, these model chemistries are far from an accurate description of the initial electronic Hamiltonian. This work demonstrates a \textit{fully} variational approach to the electronic structure problem by optimizing the orbitals that represent the second-quantized Hamiltonian, alongside a quantum circuit that generates the many-electron wavefunction. To this end, the orbitals are represented within an adaptive multi-wavelet format, guaranteeing numerical precision. We then present explicit numerical protocols and highlight the quantum circuit's role in determining the optimal orbital basis.

Figures (8)

• Figure 1: Flowchart of the proposed multiresolution quantum framework. Based on initial HF orbitals and pair natural orbitals as well as their integrals with the molecular Hamiltonian (pink), energy and RDMs are determined using a VQE ansatz (orange) alternating with an orbital refinement step (green).
• Figure 2: Applicability: Full calculation of the Cyclohexane molecule in an active space of 36 electrons and 36 spatial (72 spin) orbitals with different orbitals and many-body methods. This serves as a stand-in of the system sizes still treatable with the method. We show the energy difference (atomic units) with respect to DMRG[BD=1000]/MRA(36,72). In this example, MRA-PNOs are computationally cheapest while still comparable in numerical precision.
• Figure 3: Illustration on the H$_2$ molecule: Comparison of the H$_2$ dissociation calculated with SPA+GSD(2,4) and adaptive MRA orbitals after several optimization steps starting from pair natural orbitals as an initial guess. Top row: total energies. Bottom row: occupation numbers.
• Figure 4: Effect of selective refinement: Orbitals of an initial orbital set (in the form of PNOs) are gradually refined while the others remain in their initial form (up to orthogonalization effects). Left: Absolute errors with respect to FCI/aug-cc-pV5Z for H$_2$, Center: Absolute errors with respect to FCI/aug-cc-pVQZ for LiH, Right: Absolute errors of LiH (same reference values) but here the number of fully-refined MRA orbitals is gradually increased. The red line, SPA+GS/MRA(2,18), is identical to the plot in the center. For H$_2$ (left) the difference of the reference values to the next basis set (aug-cc-pV5Z) is consistently below the millihartree threshold. For LiH aug-cc-pVTZ was chosen as a reference as it contains a similar amount of orbitals on Li as aug-cc-pVQZ on H.
• Figure 5: BeH$_2$ in MRA orbitals determined by VQEs (SPA+GS). Left: Absolute energies. Center: Energy errors relative to FCI/cc-pVDZ(4,46). Right: CASSCF/aug-cc-pVTZ with different sizes labeled as ($N_e$, $2N_o$) using the number of spin-orbitals $2N_o$ for consistency in notation.