⚛️ quantum physics

Molecular Dynamics with Nuclear Effects on Quantum Computers

This paper introduces and validates a novel hybrid quantum-classical algorithm that incorporates nuclear quantum effects into ab-initio molecular dynamics simulations using a variational quantum eigensolver, demonstrating improved accuracy in vibrational spectra and barrierless proton shuttling for systems like the Zundel ion with hardware-efficient circuits suitable for near-term quantum devices.

Original authors: Lukas Haßfurth, Juliane Heitkämper, Elias Walter, Birger Horstmann

Published 2026-10-02

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CC BY 4.0

Original authors: Lukas Haßfurth, Juliane Heitkämper, Elias Walter, Birger Horstmann

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Atoms are the building blocks of matter, but they are not the solid, static spheres often depicted in school diagrams. At the heart of every atom lies a nucleus, and surrounding it are clouds of electrons. In most chemical calculations, scientists treat the heavy nuclei as fixed points while the lighter electrons zip around them, a simplification that works well for many substances. However, when it comes to hydrogen, the lightest element, this simplification breaks down. Because hydrogen nuclei are so light, they do not behave like tiny, stationary marbles; instead, they jitter and tunnel through energy barriers in a way that only quantum physics can describe. This behavior, known as nuclear quantum effects, is critical for understanding how protons move in biological processes like photosynthesis and how water molecules stick together. Yet, simulating this jittery, quantum nature of atomic nuclei on traditional supercomputers is often impossible, as the mathematical complexity grows too fast to handle.

A team of researchers at the German Aerospace Center and the Helmholtz Institute Ulm has taken a significant step toward solving this problem by moving the simulation onto a quantum computer. They developed a new hybrid method that combines the power of a quantum processor with the reliability of a classical computer. In their approach, the quantum computer acts as a highly specialized calculator that determines the energy and the forces acting on the atoms at any given moment, while the classical computer handles the movement of the atoms themselves. Crucially, this method treats certain atomic nuclei not as fixed points, but as quantum objects that can exist in multiple places at once, allowing for a more realistic picture of how molecules vibrate and react.

To test their algorithm, the researchers simulated three specific systems: a simple hydrogen molecule, a water molecule, and the Zundel ion, which is a cluster of two water molecules sharing a single proton. The Zundel ion is particularly important because it serves as a minimal model for proton transfer, a fundamental process in chemistry and biology. The team found that when they ignored the quantum nature of the nuclei, their simulations produced results that were often inaccurate. However, once they included these nuclear quantum effects, the behavior of the molecules changed dramatically. In the case of the Zundel ion, the inclusion of these effects removed the energy barrier that usually prevents the proton from moving freely between the two water molecules. Instead of being trapped in a valley, the proton's movement became effectively barrierless, allowing it to shuttle back and forth with ease. This shift aligned much more closely with experimental data, significantly reducing errors in the predicted vibrational frequencies of the molecules.

The researchers also explored how to make these simulations practical for the quantum computers available today, which are still in their early, noisy stages. They tested different ways of encoding the problem, comparing a highly accurate but complex method against simpler, more compact approaches designed for current hardware. They discovered that while the complex method provided the most precise results, the simpler, compact methods were surprisingly effective. For the water molecule and the hydrogen molecule, these streamlined approaches produced results that were nearly as good as the complex ones, requiring far fewer computational resources. This suggests that the algorithm could be run on near-term quantum devices, even those with limited power.

However, the path to perfect accuracy was not without its challenges. The team found that the success of the simulation depended heavily on how they chose to represent the quantum states of the nuclei. For the water molecule, a simple representation was sufficient, but for the more complex Zundel ion, they needed to include more detailed descriptions of the nuclear orbitals to capture the correct interactions. When they used a more detailed description, the simulation successfully reproduced a specific double-peak pattern in the ion's vibrational spectrum that matches what is seen in real-world experiments. Without this extra detail, the simulation missed these crucial features. The study concludes that while current quantum computers are not yet powerful enough to simulate every detail of large chemical systems, this hybrid approach offers a viable path forward. By carefully selecting which parts of the system to simulate on the quantum computer and which to handle classically, scientists can begin to model the quantum behavior of atomic nuclei with a level of accuracy that was previously out of reach.

Technical Summary: Molecular Dynamics with Nuclear Effects on Quantum Computers

Problem Statement
Nuclear quantum effects (NQEs), such as proton tunneling and zero-point energy, are critical for accurately describing processes like proton transfer and hydrogen bonding. However, incorporating these effects into quantum chemistry calculations on classical hardware is often computationally prohibitive, as it can cause the full-configuration interaction (FCI) space to grow exponentially. While quantum computers offer a potential solution due to their linear scaling in qubit requirements relative to system size, demonstrating a practical quantum advantage for dynamic simulations involving NQEs remains a challenge. Existing algorithms often lack the inclusion of nuclear quantum effects, the simulation of dynamics, or rely on pre-determined trajectories.

Methodology
The authors propose a novel hybrid quantum-classical algorithm for ab-initio molecular dynamics (AIMD) that integrates NQEs via the Nuclear-Electronic Orbital (NEO) framework.

NEO Framework: Selected light nuclei (specifically protons in this work) are treated quantum-mechanically alongside electrons using nuclear orbitals analogous to electronic atomic orbitals. This avoids the standard Born-Oppenheimer separation for these specific nuclei.
Hybrid Dynamics Algorithm: The algorithm follows a second-quantization approach based on classical Born-Oppenheimer dynamics.
- Quantum Component: A Variational Quantum Eigensolver (VQE) is employed on the quantum computer to determine the instantaneous ground state energy ( $E_0$ ) and the forces acting on the nuclei. The forces are calculated using the Hellmann-Feynman theorem by evaluating the expectation value of the force operator $\nabla_I H(R)$ on the ground state $|\psi_0(R)\rangle$ .
- Classical Component: Molecular geometries are updated classically using the velocity Verlet algorithm based on the forces computed by the quantum processor.
Active Space Techniques: To address the limited qubit count of Noisy Intermediate-Scale Quantum (NISQ) devices, the authors implement a Hartree-Fock embedding. An active space is selected for the quantum computer while inactive orbitals are treated at the Hartree-Fock level. The selection criteria include single-orbital entropies, variability along the reaction path, and symmetry preservation.
Ansatz Selection: The study utilizes both chemically motivated Unitary Coupled Cluster (UCCSD/NEOUCCSD) ansatzes and hardware-efficient ansatzes (Real Amplitudes and Tiled Unitary Product Ansatz - tUPS) to assess feasibility on near-term devices.
Basis Sets: The simulations employ the 6-31G electronic basis set and specific nuclear basis sets derived from DZSPDN (DZSNB for s-type only; DZSPNB for s- and p-type).

Key Contributions

Algorithm Implementation: The work presents the first implementation of a hybrid quantum-classical AIMD algorithm that explicitly incorporates nuclear quantum effects using the NEO method on a quantum computer.
Active Space Strategy: The authors develop a specific protocol for selecting active spaces in NEO dynamics that accounts for orbital variability and symmetry, ensuring physical consistency during dynamic processes.
Hardware Efficiency: The study demonstrates that compact, hardware-efficient ansatzes can achieve results comparable to more accurate UCCSD ansatzes, suggesting feasibility for execution on near-term quantum hardware.
Basis Set Sensitivity: The paper highlights the critical dependence of simulation accuracy on the choice of nuclear basis sets (specifically the necessity of p-type orbitals for complex mode coupling) and active spaces.

Results
The algorithm was validated through simulations of three systems: the hydrogen molecule ( $H_2$ ), water ( $H_2O$ ), and the Zundel ion ( $H_5O_2^+$ ).

Hydrogen ( $H_2$ ): The inclusion of NQEs via NEO corrected the overestimation of vibrational frequencies seen in purely electronic methods, though it led to a slight underestimation. Hardware-efficient ansatzes (tUPS and Real Amplitudes) achieved errors below 1% in the NEO case, attributed to a fortuitous cancellation of errors between the ansatz overestimation and NEO underestimation.
Water ( $H_2O$ ): NEO simulations yielded lower relative errors than purely electronic methods for all normal modes (bending, symmetric stretch, antisymmetric stretch). The bending mode, involving close proton-proton interaction, remained challenging but showed improvement with NEO.
Zundel Ion ( $H_5O_2^+$ ):
- Potential Energy Surface: In the purely electronic case, the proton transfer potential exhibits a double-well structure with a barrier. Upon inclusion of NQEs, the proton shuttling movement becomes effectively barrierless, transforming the potential into a single-well. This indicates that the zero-point motion of the shared proton exerts a notable influence, placing the vibrational ground state above the purely electronic barrier.
- Spectra: Purely electronic simulations failed to capture the characteristic spectral doublet around 1000 cm $^{-1}$ and the coupling between proton transfer and water wagging modes.
- NEO Success: NEO simulations with p-type nuclear orbitals successfully reproduced the spectral doublet and the coupling between the proton transfer mode and water wagging/stretching modes. The reduced NEOUCCSD ansatz (selecting 17 excitations out of 63) was necessary to solve the optimization problem for the larger active space.
- Accuracy: The NEO results showed a maximum error of less than 40 cm $^{-1}$ per peak compared to the exact NEOCASCI solution within the active space, with a qualitative match to experimental predissociation spectra.

Significance and Claims
The paper claims that their hybrid algorithm successfully demonstrates the feasibility of simulating molecular dynamics with nuclear quantum effects on quantum computers. The authors emphasize that:

NQEs are essential for correctly modeling proton transfer in the Zundel ion, specifically in removing the artificial energy barrier and enabling correct mode coupling.
While UCCSD provides the highest accuracy, hardware-efficient ansatzes offer a viable path for near-term devices, provided the active space and nuclear basis sets are carefully chosen.
The choice of active space and nuclear basis set (specifically the inclusion of p-orbitals for the Zundel ion) is more critical to accuracy than the specific ansatz used.
The work establishes a foundation for future studies on larger systems, noting that further improvements in resource usage or accuracy could be achieved by transitioning from canonical to natural orbital bases.

The authors remain modest regarding the current state of the technology, acknowledging that the results rely on noiseless statevector simulators and that significant quantum advantage is yet to be demonstrated on physical hardware. The work serves as a proof-of-concept for extending quantum chemistry methods to include nuclear degrees of freedom dynamically.

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