Quantum Computing Approach to Atomic and Molecular Three-Body Systems

TL;DR

This work presents high-precision quantum simulations of non-Born–Oppenheimer three-body systems (H$_2^+$, HD$^+$, He, H$^-$) using NI-DUCC-VQE, a gradient-free, first-quantized VQE with a Lie-algebraic MCP of Pauli excitations. By combining a first-quantized Hamiltonian with a compact, layer-wise MCP ansatz, the method achieves energies with errors as low as $10^{-11}$ a.u. and near-unity state fidelities using only a few thousand function evaluations, outperforming gradient-based ADAPT-VQE in resource usage. The approach enables efficient qubit scaling ($n_e \cdot \log_2 N$) and avoids barren plateaus, making it a promising benchmark for NISQ devices and a platform to extend to larger basis sets and relativistic corrections. Future work includes extending to four-body systems and perturbative corrections beyond the nonrelativistic framework, leveraging the MCP-Lie algebra structure for scalable quantum simulations of electronic structure.

Abstract

We present high-precision quantum computing simulations of three-body atoms (He, H$^-$) and molecules (H$_2^+$, HD$^+$), the latter being studied beyond the Born-Oppenheimer approximation. The Non-Iterative Disentangled Unitary Coupled Cluster Variational Quantum Eigensolver (NI-DUCC-VQE) [M. Haidar et al., Quantum Sci. Technol. 10, 025031 (2025)] is used. By combining a first-quantized Hamiltonian with a Minimal Complete Pool (MCP) of Lie-algebraic excitations, we construct a compact ansatz with a gradient-independent construction, avoiding costly gradient evaluations and yielding efficient computational scaling with both basis size and electron number. It avoids barren plateaus and enables rapid convergence, achieving energy errors as low as 10$^{-11}$ a.u. with state fidelities only limited by arithmetic precision in only a few thousand function evaluations in all four systems. These results make three-body atoms and molecules excellent candidates for benchmarking and testing on current Noisy Intermediate-Scale Quantum (NISQ) devices. Further, our approach can be extended to more complex systems with larger basis sets, taking advantage of the efficient scaling of qubit requirements to study electronic correlations and non-adiabatic effects with high precision. We also demonstrate the applicability of NI-DUCC-VQE for simulating higher-order effects such as relativistic corrections and hyperfine interactions.

Figures (4)

• Figure 1: (a) Table comparing energies computed using classical computational methods with high-precision reference values. (b) Energy convergence plots for the ground states of H$_2^+$, HD$^+$, He, and H$^-$ obtained via the NI-DUCC-VQE algorithm, using $n=7$ qubits. The energy errors are computed as the difference between the NI-DUCC-VQE result and the reference energy given in (a). (c) Convergence of the fidelity, calculated as the overlap $\langle \Psi_j(\vec{\theta}^*) | \Psi_g \rangle$ between the NI-DUCC-VQE state at each optimization step $j$ and the exact ground-state eigenvector $|\Psi_g\rangle$ of the Hamiltonian $H$.
• Figure 2: Convergence plots for H$_2^+$ using NI-DUCC-VQE with 7 qubits ($N = 128$) and 8 qubits ($N = 256$) are shown in (a) and (c) for different numbers of layers $k$. The fidelity is displayed in (b) and (d).
• Figure 3: Convergence of Qubit-ADAPT-VQE for the H2+ molecule (7 qubits, $N=128$ basis functions), and resource comparison with NI-DUCC-VQE. Panel (a) shows the energy error (in a.u.) of Qubit-ADAPT-VQE relative to the reference value, together with the corresponding gradient norm $g$ (right axis) as the ansatz is iteratively expanded. In Qubit-ADAPT-VQE, one operator is added per iteration; the number of parameters, which is equal to the number of selected operators, thus corresponds to the number of iterations. In panel (b), the first six columns show the total number of required function evaluations in Qubit-ADAPT for the gradient norm $g$ to reach the indicated threshold values $\epsilon = 10^{-1}, 10^{-2}, \ldots$. The last column shows the number of function evaluations in NI-DUCC-VQE with $k=12$ layers. Function evaluations correspond to the optimization steps performed using the BFGS optimizer.
• Figure 4: Convergence of the delta-function expectation values $\langle \delta(\mathbf{r}_a) \rangle$ is shown for the ground states of H$_2^+$ and HD$^+$. The calculations are done with 7 qubits ($N=128$) and 12 NI-DUCC-VQE layers. The errors (vertical axis) are computed as the difference between the NI-DUCC-VQE result and the reference values given in Table 11 of Appendix \ref{['referencesclassical']}.