Nonadiabatic molecular dynamics on quantum computers: challenges and opportunities

Eduarda Sangiogo-Gil; Leticia González

doi:10.1515/pac-2025-0599

Article Open Access

Nonadiabatic molecular dynamics on quantum computers: challenges and opportunities

Eduarda Sangiogo-Gil and Leticia González

Published/Copyright: November 6, 2025

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From the journal Pure and Applied Chemistry Volume 97 Issue 11

Abstract

In this Perspective, we discuss how quantum computers may advance the simulation of nonadiabatic molecular dynamics, a framework central to describing excited-state processes in photochemistry, biology, and materials science. Classical approaches span from exponentially scaling full quantum dynamics to more approximate mixed quantum–classical techniques such as surface hopping and Ehrenfest dynamics. Hybrid quantum–classical algorithms – particularly those based on the variational quantum eigensolver – offer a transformative alternative by providing access to the key electronic properties needed to drive nonadiabatic molecular dynamics simulations, including energies, gradients, and nonadiabatic couplings. We examine recent proof-of-principle quantum simulations of reduced model systems which, despite being restricted to small molecules and limited active spaces due to constraints of qubit number and device noise, already showcase the potential of quantum devices to capture phenomena such as conical intersections and ultrafast relaxation. Although practical applications are not yet feasible in the present noisy intermediate-scale quantum era, these efforts underline the conceptual and methodological advances of quantum algorithms paving the way for large-scale quantum simulations of nonadiabatic processes. Framed within the 2025 International Year of Quantum Science and Technology, such progress exemplifies how quantum computing may open new horizons for chemistry and beyond.

Keywords: Noadiabatic molecular dynamics; quantum chemistry; quantum computing; quantum mechanics; quantum science and technology

Introduction

The year 2025 has been designated the International Year of Quantum Science and Technology, marking a century of transformative developments in quantum theory and its profound impact on our understanding of nature and on modern technology. ¹ This milestone is not only a celebration of past achievements, but also a call to explore the frontiers that quantum ideas are now opening. Among these frontiers, understanding and accurately simulating the excited state dynamics of molecules – central to photochemistry, materials science, and biology – remains a formidable challenge. ² Quantum computers offer a fundamentally new way to tackle this problem, promising to capture electronic and nuclear motion with a fidelity inaccessible to classical devices. In this context, leveraging quantum algorithms to simulate light-driven molecular processes exemplifies the transformative potential of quantum technologies. At the core of these challenges is the coupling between electronic and nuclear motion – nonadiabatic molecular dynamics (NAMD) – which govern key processes across chemistry, materials, and biological systems. ³ ^, ⁴ ^, ⁵ ^, ⁶ Yet simulating such dynamical processes is notoriously difficult, as they involve transitions between multiple electronic states, typically mediated by conical intersections^[1] or avoided crossings,^[2] where the Born–Oppenheimer approximation breaks down and the separation between electronic and nuclear motion becomes invalid. ⁷ These transitions play a critical role in phenomena such as nonradiative relaxation, ⁸ ^, ⁹ ^, ¹⁰ photoisomerization, ¹¹ ^, ¹² ^, ¹³ ^, ¹⁴ ^, ¹⁵ ultrafast proton transfer, ¹⁶ ^, ¹⁷ ^, ¹⁸ or photodissociation. ¹⁹ ^, ²⁰ ^, ²¹

To capture these complex phenomena, a wide array of NAMD methods have been developed for classical computers. ² ^, ²² These span a broad spectrum: at one end lie highly accurate full quantum dynamics approaches, such as real-space grid-based solvers for the time-dependent Schrödinger equation, including discrete variable representation and sparse-grid techniques, ²³ ^, ²⁴ as well as the multiconfigurational time-dependent Hartree family of methods. ²⁵ ^, ²⁶ At the other end are more approximate yet computationally efficient mixed quantum–classical (MQC) approaches, such as trajectory surface hopping (TSH) ²⁷ ^, ²⁸ ^, ²⁹ and Ehrenfest dynamics, ³⁰ ^, ³¹ in which nuclear motion is described by classical trajectories and electronic evolution is treated quantum mechanically.

While full quantum approaches offer high accuracy through explicit wave function propagation, their computational cost scales exponentially with system size, rendering them infeasible for large, realistic molecular systems. To remain tractable, these methods often require severe simplifications, such as reduced dimensionality or assumptions of separability in the potential energy landscape. In contrast, MQC methods are more scalable but introduce fundamental limitations. These include zero-point energy leakage, ³² ^, ³³ ^, ³⁴ artificial coherence, ³⁵ and reliance on ad hoc corrections – such as decoherence corrections ¹⁴ ^, ³⁶ ^, ³⁷ or velocity adjustments ³⁸ ^, ³⁹ ^, ⁴⁰ – to compensate for their classical treatment of nuclei. This trade-off between accuracy and efficiency highlights the inherent limitations of classical computing in simulating nonadiabatic dynamics, and underscores the need for alternative computational paradigms.

The advent of quantum computing opens new perspectives. ⁴¹ ^, ⁴² ^, ⁴³ ^, ⁴⁴ By harnessing the exponentially large Hilbert space of quantum bits (qubits), quantum devices can naturally represent and manipulate the quantum states of both electrons and nuclei, transcending the classical limitations. In this spirit, a growing suite of quantum algorithms has emerged for NAMD, ⁴⁵ ^, ⁴⁶ ^, ⁴⁷ ^, ⁴⁸ ^, ⁴⁹ ^, ⁵⁰ ranging from fully quantum simulations to hybrid approaches. ⁵¹ ^, ⁵² ^, ⁵³ ^, ⁵⁴ By hybrid approaches, it is meant that quantum or hybrid quantum–classical routines are employed to compute key quantities – such as electronic energies, energy gradients, and nonadiabatic couplings – which are then fed into classical trajectory-based dynamics. These hybrid strategies offer a viable near-term path for leveraging today’s noisy intermediate-scale quantum (NISQ) hardware for realistic physical simulations. ⁵⁵ ^, ⁵⁶

In this Perspective, we discuss the potential of quantum algorithms to transform the simulation of NAMD. While proof-of-principle demonstrations already exist, practical implementations that outperform simulations in classical devices remain a future goal. We focus on both full quantum and hybrid quantum–classical methodologies. By framing these approaches within the broader vision of the International Year of Quantum Science, we aim to highlight not only the opportunities that quantum technologies offer for addressing chemically relevant NAMD problems but also the limitations and open questions that must be resolved to unlock their full potential. As quantum science enters its second century, the application of quantum computing to NAMD represents a promising direction, whose practical realization will still require sustained advances in algorithms, hardware, and theory.

In the following, we explore how hybrid quantum–classical algorithms, with particular emphasis on the Variational Quantum Eigensolver (VQE) and its extensions, can be employed to compute the static molecular properties required for nonadiabatic molecular dynamics, namely excited-state energies, nuclear gradients, and nonadiabatic coupling vectors. We review recent proof-of-concept demonstrations, discuss methodological strategies for integrating quantum routines into MQC approaches for dynamics, and analyze their potential and limitations in the near- and long-term. By doing so, we aim to provide a comprehensive perspective on the current state of quantum algorithms for excited-state dynamics and to outline pathways toward their application in realistic chemical systems.

Mixed quantum–classical dynamics

The time evolution of a nonrelativistic molecular system is governed by the time-dependent Schrödinger equation: i ℏ ∂ ∂ t Ψ ( r , R , t ) = H ˆ mol ( r , R ) Ψ ( r , R , t ) , where Ψ(r, R, t) is the molecular wave function, depending on electronic coordinates r, nuclear coordinates R, and time t. H ˆ mol ( r , R ) is the molecular Hamiltonian which comprises nuclear and electronic kinetic energies, together with nuclear–nuclear, electron–electron, and nuclear–electron Coulomb interactions. Nowadays, solving this equation exactly using classical hardware is very expensive due to the nonlocal nature of quantum mechanics and the exponential scaling of computational cost with system size.

A common approach for studying realistic chemical systems, even in complex environments, is the MQC approximation, where nuclei are treated classically and electrons quantum mechanically. This approximation is often justified by the large mass difference between nuclei and electrons, which leads to smaller nuclear quantum level spacings and makes quantum effects such as tunneling or interference negligible in many cases. MQC dynamics is particularly suitable for simulating nonadiabatic processes, where the energy gaps between electronic states are on the order of nuclear kinetic energies, allowing efficient transitions between states. ⁵⁷ ^, ⁵⁸

Perhaps the most traditional and widely used examples of such methods are Ehrenfest-like ³⁰ dynamics and TSH-like ²⁸ approaches, both of which couple classical nuclear motion with the quantum propagation of the electronic wave function. In both approaches, the electronic wave function is expressed as a linear combination of adiabatic electronic states:

(1) Φ ( t ) = ∑ m a m ( t ) φ m ( r ; R )

where φ _m(r; R) are eigenfunctions of the electronic Hamiltonian H ˆ e l ( r , R ) at the instantaneous nuclear configuration R ≡ R(t). Substituting this expansion into the time-dependent Schrödinger equation for the electrons leads to the following equations of motion for the time-dependent coefficients a _m(t):

(2) d a m ( t ) d t = − i ℏ E m ( R ) a m ( t ) − ∑ ν ∑ l , l ≠ m d m l ν ( R ) ⋅ v ν a l ( t ) ,

where E _m(R) is the adiabatic energy of state m, v _ν denotes the nuclear velocities of a given nucleus ν, and d m l ν ( R ) the first-order nonadiabatic coupling vector (NACV).

The forces driving nuclear motion differ in the two approaches. In TSH, the nuclei evolve on a single adiabatic potential energy surface corresponding to the active state (i.e., the electronic state on which the system is propagating at a given time), such that the force on nucleus ν is given by:

(3) F ν = − ∇ R ν E m

where E _m is the energy of the active electronic state m. In TSH, transitions between electronic states are enabled through stochastic hops from one potential energy surface to another, with the hopping probabilities governed by the nonadiabatic couplings.

In contrast, in Ehrenfest dynamics the nuclei experience a mean-field force, which is a weighted average over all electronic states:

(4) F ν = − ∑ m | a m ( t ) | 2 ∇ R ν E m − ∑ m , l a l * ( t ) a m ( t ) ( E m − E l ) d m l ν ,

where the first term corresponds to the population-weighted average of the adiabatic gradients, and the second term arises from nonadiabatic couplings between states.

The nuclei in these dynamics are propagated according to Newton’s equations of motion, with the forces evaluated from eqs. 3 or 4, depending on the chosen method. In practice, the nuclear trajectories are integrated using symplectic algorithms – most commonly Verlet-type schemes – which provide good energy conservation and long-term stability.

In MQC methods, the main computational bottleneck in terms of computational cost lies in evaluating the electronic properties required at each time step, such as adiabatic energies, energy gradients, and NACVs. An important aspect is that these quantities can be computed on-the-fly: for each new nuclear geometry generated along the trajectory, an electronic structure calculation is performed to obtain the energies, gradients, and couplings. The resulting information is then used both to propagate the electronic coefficients and to evaluate the forces acting on the nuclei. Quantum algorithms offer an appealing alternative for computing these quantities. In particular, hybrid quantum–classical algorithms (where part of the computation is performed on quantum hardware and part on classical hardware), such as VQE-based approaches, represent a promising strategy (as discussed in the next section). Even on near-term devices, i.e., NISQ processors that lack full error correction but are already available with tens to a few hundred qubits, these methods can already provide ground- and excited-state energies, energy gradients, and NACVs, making them attractive tools for nonadiabatic dynamics simulations.

In addition to those quantities, quantum algorithms can provide “CI-type” wave functions, which can be used to compute the wave function overlaps between consecutive time steps – required, for example, in the local diabatization scheme to propagate the electronic coefficients in TSH and in other NAMD approaches. ⁵⁹ ^, ⁶⁰ The computation of the wavefunction overlap between two consecutive time steps can be easily obtained because the variationally optimized quantum states can be naturally expressed in a configuration–interaction (CI)-like form. Such explicit access to wave functions constitutes a distinct advantage over machine learning techniques, where wave functions are not available and the propagation of electronic coefficients may therefore be less reliable.

Several proof-of-concept demonstrations highlight this potential. Gandon and co-workers employed the quantum subspace expansion and quantum equation-of-motion algorithms within a TSH framework to simulate the nonadiabatic dynamics of the H + H₂ collision system (three nuclei, three electrons). ⁴⁹ In a related effort, we recently proposed a hybrid quantum–classical framework in which the required electronic quantities are computed on quantum devices using VQE and variational quantum deflation (VQD), ⁴⁷ and then interfaced with the SHARC ⁶¹ ^, ⁶² molecular dynamics package via the TEQUILA ⁶³ quantum computing framework. Applications to the cis–trans photoisomerization of methanimine and the excited-state relaxation of ethylene demonstrated qualitatively accurate results in agreement with both experiment and high-level classical simulations. ⁴⁷ Further contributions include the work of Hirai, who employed hardware–efficient ansätze to simulate excited-state dynamics in hydrogen and methanimine. ⁴⁶ In addition, Bultrini and Vendrell applied a time-dependent variational quantum approach combined with classical Ehrenfest dynamics to the Shin–Metiu model. ⁵⁰

Together, these studies demonstrate that MQC methods combined with variational quantum algorithms are progressing from minimal model systems toward more chemically relevant, albeit still small, molecular systems, underscoring the potential of quantum computing for NAMD simulations.

In the following section, we discuss how the “static quantities” required for propagating MQC dynamics, i.e. the energies, energy gradients and NACVs, can be obtained using quantum algorithms, with a particular focus on VQE-based approaches.

Quantum computation of static molecular quantum properties

One of the most anticipated applications of quantum computing is electronic structure theory. This potential arises from the intrinsic ability of quantum computers to represent and manipulate the quantum states of many-electron systems with exponential efficiency compared to classical devices. Among the most prominent approaches, the already mentioned VQE and quantum phase estimation (QPE) ⁶⁴ ^, ⁶⁵ ^, ⁶⁶ have emerged as central algorithms for determining molecular ground and excited-state energies. VQE is a hybrid quantum–classical algorithm and it is particularly suitable for near-term NISQ devices, as it employs the variational principle to optimize a parameterized quantum circuit. QPE, in contrast, requires deeper circuits and fault-tolerant hardware,^[3] but in principle can provide energy eigenvalues to arbitrary precision through phase encoding. These algorithms offer promising routes for tackling strongly correlated systems that remain challenging for classical methods.

However, significant obstacles persist. Present-day quantum hardware is limited by the number of available qubit, short coherence times, and noisy gate operations. Moreover, hybrid algorithms like VQE can suffer from barren optimization landscapes, sensitivity to ansätz choice, and the overhead introduced by fermion-to-qubit mappings. Despite these hurdles, rapid progress in quantum algorithm design, quantum hardware, and error mitigation techniques continues to move the field forward, steadily improving the feasibility of quantum simulations for chemically relevant systems. ⁶⁷ ^, ⁶⁸ Leveraging these developments to compute the electronic energies, energy gradients and NACVs needed for NAMD offers a promising pathway to simulate complex molecular dynamics beyond classical limits. Although the VQE algorithm was developed more recently as a practical alternative to fully quantum algorithms such as QPE, it is conceptually easier to understand, requires shallower quantum circuits, and is directly applicable on current NISQ hardware. Moreover, VQE can be used to obtain high-quality trial wave functions that may serve as input states for more advanced algorithms like QPE.

In this section, we delve deeper into VQE, outlining its theoretical foundations, practical considerations, and the respective advantages and limitations. We show in the following how the static properties needed for propagating MQC dynamics can be obtained within this algorithm.

The Variational Quantum Eigensolver (VQE)

The VQE is a hybrid quantum–classical algorithm designed to estimate the eigenvalues and corresponding eigenstates of a given Hamiltonian. ⁶⁹ ^, ⁷⁰ While originally developed to compute ground state energies, VQE has since been extended to calculate excited states using various strategies, making it increasingly relevant for simulating the electronic structure quantities that underlie NAMD simulations.

VQE is based on the variational principle, which provides an upper bound to the ground state energy of a Hamiltonian. ⁷¹ Specifically, for a Hamiltonian H ˆ and a parameterized, normalized trial wave function |Ψ(θ)⟩, the ground state energy E _gs satisfies:

(5) E g s ≤ 〈 Ψ ( θ ) | H ˆ | Ψ ( θ ) 〉

where the trial wave function Ψ(θ) is assumed to be normalized, and θ is the variational parameter. Therefore, the objective of the VQE is to find the set of θ that minimizes the expectation value of H ˆ . The trial wave function is constructed using a quantum circuit known as the ansätz. This involves applying a series of quantum gates (unitary operations) to a reference state Ψ₀. Thus, the trial wave function Ψ(θ) can be represented as U(θ)|Ψ₀⟩, where U(θ) denotes the ansätz.

The VQE algorithm has shown to be an interesting approach for solving quantum chemistry problems. In this context, the molecular Hamiltonian in second quantization can be written as:

(6) H ˆ ( R ) = ∑ r s h r s ( R ) a ˆ r † a ˆ s + 1 2 ∑ pqrs g pqrs ( R ) a ˆ p † a ˆ q † a ˆ r a ˆ s + E N N ( R )

where h _rs and g _pqrs denote the one- and two-electron integrals, respectively, and are commonly obtained from a Hartree–Fock (HF) calculation, which is usually performed on a classical device. The collective vector of nuclear coordinates R = ( R ₁, R ₂, …, R _N) of N nuclei in R ^3N simply parametrizes the electronic Hamiltonian. The position vector of a single nucleus ν ∈ {1, …, N} is denoted by R _ν = (R _ν,x, R _ν,y, R _ν,z). The operators a ˆ r † and a ˆ r represent the fermionic creation and annihilation operators for electrons in the HF spin-orbitals. The indices r, s, t, u are used to label general (occupied or virtual) molecular orbitals. The term E _NN( R ) describes the nuclear repulsion energy.

Observables suitable for direct measurements on a quantum device are tensor products of spin operators, commonly referred to as Pauli operators. However, electronic structure problems in quantum chemistry are naturally described in terms of fermionic operators, which obey anti-commutation relations and do not directly correspond to qubit operations. Therefore, in order to simulate a fermionic system such as a molecule on a quantum computer, the fermionic Hamiltonian must first be mapped to a qubit Hamiltonian. This process is known as Hamiltonian mapping, and several encoding strategies have been developed for this purpose.

One of the most straightforward and traditional mappings is based on the Jordan–Wigner transformation, proposed in 1928, ⁷² which maps fermionic creation and annihilation operators onto strings of Pauli operators (note that expectation values of Pauli operators are easily implementable in quantum algorithms) by introducing sequences of Z operators to enforce the fermionic anti-commutation relations.

While the Jordan–Wigner mapping is conceptually simple and particularly efficient for one-dimensional systems, it often produces long Pauli strings in larger or more complex systems. ⁷³ ^, ⁷⁴ ^, ⁷⁵ This leads to non-local qubit interactions, which in turn increase circuit depth and make simulations more susceptible to noise on near-term quantum hardware. To address these limitations, several alternative mappings have been introduced. The Bravyi–Kitaev transformation provides a more balanced treatment of operator locality and fermionic parity, typically reducing the average length of Pauli strings and enabling shallower circuits for many molecular Hamiltonians. ⁷⁶ ^, ⁷⁷ The parity mapping, on the other hand, encodes occupation parity directly into qubit states, often yielding shorter Pauli strings than Jordan–Wigner and offering particular advantages in problems where particle number conservation can be exploited. Further optimizations can be achieved through qubit tapering techniques, which reduce the number of qubits required by leveraging symmetries in the Hamiltonian. ⁷⁸ ^, ⁷⁹ For example, Z 2 symmetries associated with particle number or spin conservation can be used to systematically remove qubits from the representation, thereby lowering resource requirements without loss of accuracy.

After applying one of these transformations, the molecular Hamiltonian H ˆ can be written as a linear combination of tensor products of Pauli matrices, also known as Pauli strings:

(7) H ˆ = ∑ α P w α P ˆ α , w α ∈ R .

Here, each P ˆ α is a tensor product of Pauli operators acting on different qubits (e.g., X ₁ Z ₂ Y ₃ I ₄…), and w _α are real coefficients determined by the specific molecular system and basis set used. This form is suitable for variational quantum algorithms like VQE, as each Pauli string can be measured individually on a quantum device, and the full Hamiltonian expectation value is reconstructed as a weighted sum of these individual measurements.

Therefore, the expectation value of the molecular Hamiltonian with the VQE can be calculated with:

(8) E g s ≤ min θ ∑ α P w α 〈 Ψ 0 | U † ( θ ) P ˆ α U ( θ ) | Ψ 0 〉

where the hybrid nature of the VQE algorithm becomes clearly apparent: each term E P α = 〈 Ψ 0 | U † ( θ ) P ˆ α U ( θ ) | Ψ 0 〉 corresponds to the expectation value of a Pauli string P ˆ α and can be computed on a quantum device, while the summation and minimization E g s = min θ ∑ α w α E P α are computed on a classical computer. The reference wave function |Ψ₀⟩ is often the HF wave function, while the parameters θ are optimized classically. A schematic representation of the VQE algorithm is shown in Fig. 1.

Fig. 1:

Schematic representation of the Variational Quantum Eigensolver (VQE) algorithm. The molecular Hamiltonian is first computed classically (e.g., using a quantum chemistry package) and mapped onto qubits. The quantum processing unit (QPU) prepares a parameterized trial wave function (ansätz) and measures the expectation values of the Hamiltonian terms. These measurement results are sent to the classical central processing unit (CPU), which evaluates the total energy and updates the parameters θ. The process is iterated between the QPU and CPU until convergence of the energy and the optimal parameters is achieved.

In practice, a variety of classical optimization algorithms are employed to minimize the VQE cost function. Common gradient-free optimizers include COBYLA (Constrained Optimization BY Linear Approximation), ⁸⁰ Nelder–Mead, ⁸¹ and Powell, ⁸² which are widely used due to their robustness to noise and the fact that they do not require gradient evaluations. It is important to emphasize that here the term gradient refers to the derivative of the energy E _gs with respect to the variational parameters θ, and not to the energy gradients with respect to the nuclear coordinates needed for the computation of the nuclear forces in MQC. In contrast, gradient-based methods such as BFGS (Broyden–Fletcher–Goldfarb–Shanno), ⁸³ ^, ⁸⁴ ^, ⁸⁵ ^, ⁸⁶ Adam ⁸⁷ and NatGrad ⁸⁸ are typically employed in noiseless simulations or in situations where reliable parameter gradients can be computed. Recently, several benchmark studies have compared the performance of these optimizers in the context of VQE; for a more detailed discussion, see Refs. [89], [90], [91.

The quality of the energy obtained from eq. 8 also depends critically on the choice of the ansätz. In recent years, several ansätze have been proposed within the framework of VQE for quantum chemistry applications. Among the most widely used are coupled cluster (CC)-based ansätze, particularly the unitary coupled-cluster (UCC) approach and its various extensions, ⁹² ^, ⁹³ ^, ⁹⁴ ^, ⁹⁵ as well as the hardware–efficient ansätze. ⁹⁶ These ansätze can be broadly categorized into two classes: chemistry-inspired and hardware–efficient.

Chemistry-inspired ansätze, such as the unitary CC family, originate from established quantum chemistry methods and incorporate physically motivated electron correlation directly into the wave function. These approaches are valued for their expressive power and systematic improvability but typically produce deep circuits, which are difficult to execute on near-term quantum hardware due to noise and limited coherence times. In contrast, hardware-efficient ansätze are tailored to the native gate set and connectivity of quantum devices, often consisting of repeated layers of parameterized single-qubit rotations combined with entangling operations. Such circuits are relatively shallow and more practical to implement on noisy devices, yet they are more susceptible to challenges like barren plateaus and generally lack the chemical interpretability and accuracy of chemistry-inspired ansätze. ⁹⁷ ^, ⁹⁸

The number of qubits required by an ansatz is generally proportional to the number of spin-orbitals included in the calculation. For instance, the hydrogen molecule (H₂) in the STO-3G basis set has 4 spin-orbitals, requiring 4 qubits to represent the system on a quantum computer. However, as molecular size increases, even when using minimal basis sets, the number of spin-orbitals grows rapidly, making it impractical to include the full orbital space in a quantum simulation. For example, seemingly small molecule like ethanol (C₂H₆O) requires 21 spatial orbitals in a STO-3G basis set. When accounting for spin, this becomes 42 spin-orbitals, demanding 42 qubits just for its basic representation. Using a larger basis set like cc-pVDZ for the same molecule requires more than hundreds of spin-orbitals. Current quantum hardware typically provides devices with hundreds to a few thousand physical qubits – still far from sufficient to simulate medium- or large-sized chemical systems without employing orbital reduction techniques or problem-specific encodings. The situation is even more demanding when considering logical qubits. Because of the overhead of quantum error correction, each logical qubit may require on the order of 10² − 10³ physical qubits, depending on the hardware error rates and the chosen error-correcting code. ⁹⁹ ^, ¹⁰⁰ Thus, while present devices are suitable for proof-of-concept demonstrations on small molecules, chemically relevant applications will likely only become feasible on future fault-tolerant architectures with hundreds to thousands of logical qubits.

In practical VQE implementations, the measurement overhead is often dominated by the large number of Pauli terms, especially when computing derivatives for gradients and nonadiabatic couplings. The empirical scaling of shot budgets has therefore become a key focus for optimizing measurement efficiency in variational quantum algorithms. In particular, the number of shots grows linearly with the number of Hamiltonian terms, leading to significant resource requirements. To mitigate this, several strategies have been developed. Term grouping exploits commuting sets of Pauli terms to measure multiple operators simultaneously, reducing the total number of measurements. ¹⁰¹ Classical shadowing uses randomized measurements to efficiently reconstruct expectation values of many observables from a limited number of shots. ¹⁰² Low-rank factorization approximates measurement operators with low-rank representations, further decreasing the complexity of the measurement process. ¹⁰³

Simulating quantum algorithms on classical computers is also extremely difficult. Even the fastest supercomputers can only handle about 50–60 qubits, since the memory and computational requirements grow exponentially with each additional qubit. As mentioned, the number of qubits required to represent a quantum system is directly tied to the number of spin-orbitals, or qubit states, that must be included. Since each qubit can exist in two states (0 or 1), the size of the quantum state space grows exponentially with the number of qubits. Specifically, a system of n qubits is described by a vector in a Hilbert space of dimension 2ⁿ. As a result, the memory needed to store such a state also scales exponentially. For example, storing the state of a 50–qubit quantum system requires 2⁵⁰ complex numbers, which amounts to 16 petabytes of memory. This is calculated by recognizing that each complex number requires 16 bytes (for two 8-byte double–precision numbers), leading to a total of 2⁵⁰ × 16 = 2⁵⁴ bytes. Since one petabyte is defined as 1 PB = 10¹⁵ bytes, the total memory required is 2 54 1 0 15 ≈ 16 PB . This exponential growth makes classical simulations of large quantum systems extremely demanding, and often impractical, due to memory limitations.

To mitigate this problem, a common approach is to select a subset of spin-orbitals, known as the active spin-orbitals, for inclusion in the ansätz, hereby reducing the size of the Hilbert space explored by the calculation. ¹⁰⁴ The selection of the active spin-orbitals is a critical step, as it defines the portion of the electronic Hilbert space that the quantum algorithm will explore and, thus determines the degree of electronic correlation captured by the ansätz. The active spin-orbitals typically consist of molecular orbitals – often those corresponding to the highest occupied molecular orbital and the lowest unoccupied molecular orbital, as well as those nearby in energy – that are normally the most relevant to the chemical process or property of interest. The remaining core and virtual orbitals, which have minimal impact on the chemical properties being studied, are typically frozen or excluded from the calculation to reduce computational cost. The choice of active spin-orbitals is important because if it is too small, essential electronic correlations may be neglected, leading to inaccurate results. In the context of NAMD, a careful selection of the active space is particularly important, since it directly affects the accuracy of the computed excited states, energy gaps, and nonadiabatic couplings, which in turn influence the fidelity of the simulated molecular dynamics. Therefore, find a balance between computational efficiency and accuracy is of primary importance, particularly for large molecular systems.

In summary, both quantum and classical simulators face several limitations. While quantum devices are still in the early stages of development, capable of handling a few dozen to a few hundred qubits, classical simulators are also constraint to the exponential scaling of memory and processing demands. For larger systems, only approximate methods or truncated active spin-orbitals are feasible, as full-scale simulations of quantum systems with many qubits remain impractical.

Excited state energies

The standard VQE algorithm, widely applied to molecular systems, is primarily designed to compute the ground state of the electronic wave function by minimizing the average energy of a parameterized quantum state. Therefore, despite its success and versatility in ground state applications, the algorithm in its unmodified form does not inherently lend itself to excited-state calculations; nevertheless, as already discussed in Section “Mixed quantum–classical dynamics”, the propagation of MQC dynamics necessarily depends on the availability of excited state energies, making the development or incorporation of suitable extensions to VQE indispensable for such simulations.

Several extensions and strategies have been proposed, inspired by classical quantum chemistry techniques, to adapt VQE for excited state estimation on near-term quantum devices. These include symmetry adapted approaches, which target excited states by restricting the ansätz to specific symmetry sectors; penalty-based methods, ¹⁰⁵ ^, ¹⁰⁶ ^, ¹⁰⁷ ^, ¹⁰⁸ such as VQD and orthogonally constrained VQE (OC-VQE), ⁹⁴ which modify the cost function to enforce orthogonality or quantum number constraints. For example, the VQD method, instead of minimizing the energy alone, it minimizes a modified cost function designed to enforce orthogonality to previously computed lower-energy states. For example, an excited state energy can be obtained via:

(9) E 1 ( ϕ ) = min ϕ 〈 Ψ ( ϕ ) | H ˆ ( R ) | Ψ ( ϕ ) 〉 + λ | Ψ ( ϕ ) | Ψ ( θ ) | 2

where Ψ(θ) is the set of angles which were optimized for the previous state, for example the ground state wavefunction obtained with VQE, and λ is a hyperparameter that penalizes overlap with the ground state. Following, ¹⁰⁹ the overlap term can be expressed as another expectation value:

(10) | Ψ ( ϕ ) | Ψ ( θ ) | 2 = 〈 Φ ( ϕ ) | P 0 | Φ ( ϕ ) 〉

with |Φ(ϕ)⟩ = U(ϕ)|Ψ(ϕ)⟩ and P ₀ = |0⟩⟨0| representing the all-qubit zero state. The parameter λ should be chosen large enough to enforce orthogonality, ideally exceeding the energy gap between the ground and excited states, yet not so large as to favor higher unwanted excitations. ¹¹⁰ A common approach is to adopt a value of λ equal in magnitude to the ground state energy obtained from VQE, which is sufficient for bound excited states. ¹⁰⁹ Importantly, the ansätz used here conserves both electron number and spin, avoiding spin contamination and obviating additional penalty terms. Despite its conceptual simplicity, VQD faces practical challenges due to its inherently sequential nature: each excited state depends on the accurate deflation of all lower-energy states. This sequential requirement increases computational overhead and can complicate optimization, potentially causing convergence to suboptimal states or the omission of certain excitations, particularly in complex systems or under noisy conditions.

Beyond penalty-based approaches, other strategies employ subspace expansion methods, like quantum subspace expansion and quantum equation-of-motion, ⁴⁸ which construct low-energy excited states via linear combinations of perturbations on the ground state wave function. Additional approaches such as Witnessing Eigenstates (WAVES) further broaden the toolkit. ¹¹¹ While these methods show significant promise, excited state algorithms for VQE are still an active area of research, and ongoing work is focused on improving their accuracy, scalability, and compatibility with NISQ devices.

Nuclear gradients and nonadiabatic coupling vectors

The forces acting on the nuclei in MQC dynamics (see eqs. 3 and 4) require the derivatives of the electronic energy with respect to the nuclear coordinates. For a given nucleus ν, the gradient of the ground-state energy is defined as:

This expression contains three contributions: the first term is the Hellmann-Feynman force, which depends only on the derivative of the Hamiltonian; the second and third terms are the so-called Pulay forces, which arise from the dependence of the wave function on the nuclear coordinates.

In many practical implementations, only the Hellmann-Feynman term is retained, ⁴⁵ ^, ⁴⁶ ^, ⁴⁷ ^, ⁴⁹ ^, ¹¹² leading to the approximation:

(12) ∇ R ν E g s ≈ 〈 Ψ ( θ ) | ∇ R ν H ˆ ( R ) | Ψ ( θ ) 〉 .

This simplifies the computation considerably, since it requires only the evaluation of Hamiltonian derivatives, which can be obtained from a HF-like calculation on a classical computer, followed by mapping into the quantum framework. In this way, the forces are directly given by the expectation value of the Hamiltonian gradient.

The drawback of neglecting Pulay forces is a loss of accuracy: these terms are known to correct errors caused by basis set incompleteness. ¹¹³ In the limit of a complete basis set, Pulay forces vanish. However, when finite basis sets are employed, their contribution can be significant. The calculation of Pulay forces is generally more involved, as it depends explicitly on the choice of the ansätz (U(θ)) used to represent the quantum states, and requires careful treatment of wave function derivatives. Some approaches exist to compute them for specific ansätze, but this remains a more complex task. An alternative way to avoid Pulay forces altogether is to adopt a perturbation-dependent basis set, which ensures that only the Hellmann–Feynman term contributes to the force expression. ¹¹⁴ However, while a perturbation-dependent basis can formally eliminate Pulay terms, practical constraints in VQE implementations – such as ansätz design, tapering, and qubit mappings – make it difficult to fully resolve Pulay effects on NISQ devices.

In a similar spirit, the NACV can also be approximated in terms of Hamiltonian derivatives: ¹¹⁵

(13) d m l ν ≈ Ψ ( θ m ) ∇ R ν H ˆ ( R ) Ψ ( θ l ) E ( θ l ) − E ( θ m )

where Ψ(θ _m) and Ψ(θ _l) are the variationally optimized wave functions (for instance via VQE or VQD) for electronic states m and l, with energies E(θ _m) and E(θ _l), respectively. Once the wave functions and their corresponding gradients are available, evaluating NACVs becomes a straightforward extension, with a computational cost similar to that of energy gradient calculations (eq. 12).

Conical intersections with VQE

Conical intersections are regions where two or more potential energy surfaces become degenerate, forming a multidimensional topological feature that arises only in systems with more than more than two degrees of freedom. They exhibit a characteristic double-cone topology around the point of degeneracy, which leads to geometric (Berry) phase^[4] effects and allows for ultrafast transitions between states. Because they act as funnels for radiationless relaxation, conical intersections play a central role in nonadiabatic dynamics and govern the behavior of many photochemical and photobiological processes. However, their strong multireference character^[5] makes them inaccessible to single-reference electronic-structure methods, requiring multiconfigurational approaches such as CASSCF (Complete Active Space Self-Consistent Field) to describe the correct topology and coupling. Quantum algorithms, especially state-averaged and multistate VQE-based methods, provide a promising framework to capture these features by directly encoding correlated wave functions. Even when a CC ansätz is used within VQE, the variational optimization can yield seemingly reasonable results – particularly in small active spaces – because the ansätz can partially mimic multiconfigurational character and avoids some of the instabilities of classical CC methods. ⁴⁷ However, further studies are needed to determine whether the exact topological structure of the intersection can be captured correctly.

In this respect, recent studies suggest that future quantum hardware could in principle access the complex topologies of conical intersections in molecular systems, as recently illustrated in works exploring Berry-phase detection with VQAs, variance-based contracted quantum eigensolvers, and VQE-based strategies for small molecules like methanimine. For example, Koridon et al. demonstrated a hybrid variational algorithm for detecting conical intersections based on Berry-phase signatures (see Fig. 2). ¹¹⁶ They applied their method to methanimine using a unitary coupled-cluster doubles (UCCD) ansätz containing only the active-space double excitation operator in a minimal STO-3G basis with a CAS(2,2) active space (2 electrons in 2 orbitals, mapped to 4 qubits). By propagating this ansätz along a closed nuclear coordinate loop and updating the parameters with Newton–Raphson steps, they identified conical intersections through the appearance of a quantized Berry phase of π, and when the loop did not enclose a conical intersection, the phase vanished (see Fig. 2, panel B).

Fig. 2:

Adapted figure from Ref. 116] with permission from Quantum – the open journal for quantum science. Panel A: Structure of the methanimine molecule, with the bending angle and dihedral coordinate highlighted as the internal coordinates used to probe the conical intersection. Panel B: Energy gap between the S ₀ and S ₁ states (i.e., E ₁ and E ₂) as a function of the dihedral and bending coordinates of methanimine. The conical intersection region and the three closed loops used to test the algorithm are indicated. The algorithm yields a Berry phase of π for loops enclosing the conical intersection, as expected.

Other studies, that does not compute the Berry phase but instead detect conical intersections from the near-degeneracy of adiabatic energies, have characterized conical intersections in larger systems. ¹¹⁷ ^, ¹¹⁸ For instance, Wang et al. investigated cytosine by combining state-averaged CAS(4,3) active spaces with VQD using a two-local ansätz and the variance-based contracted quantum eigensolver, identifying the ππ*/S ₀ intersection when the ground and excited states converged to degenerate energies within 0.0005 Hartree. ¹¹⁸

These results illustrate that even with a compact ansätz and small active space, VQE-based methods can capture the topological signature of a conical intersection, providing a robust proof of principle for quantum algorithms for the correct description of conical intersections. We emphasize that accurately representing multiple coupled states and their interactions near conical intersections is essential for reliable NAMD simulations.

Example of MQC dynamics using VQE

A representative proof-of-concept of MQC dynamics using VQE was recently reported by us for methanimine (see Fig. 3A), one of the smallest imine systems displaying ultrafast photoisomerization and internal conversion pathways. ⁴⁷ In this study, the required electronic-structure quantities – including excited-state energies, energy gradients, and electronic wave function overlaps between consecutive time steps – were obtained using VQE-based algorithms. In particular, ground state properties were computed with VQE, while excited-state properties were obtained via VQD. All calculations were performed with the TEQUILA library, ⁶³ which was interfaced with the surface hopping dynamics framework implemented in SHARC. ⁶¹ ^, ⁶²

Fig. 3:

(Panel A) Methanimine molecule which carbon (C) atoms in gray, nitrogen (N) atom in blue and hydrogen (H) atoms in light gray. (Panel B) Potential energy surfaces calculated with VQE/VQD for a typical reactive cis–trans trajectory, illustrating (a) the initial planar configuration, (b) the hopping structure with the N–H bond vector twisted out of the plane, and (c) the photoisomerized structure. (Panel C) Time-resolved adiabatic state populations. (Panel D) HNC angles and HNCH dihedrals at the S ₀/S ₁ hopping geometries.

The simulations demonstrated that even with reduced active spaces and a UCC-type ansatz, VQE was capable of reproducing key qualitative features of the dynamics. Upon excitation to the nπ* state, the molecule relaxes by twisting the N–H bond vector out of the molecular plane (see Fig. 2, bending angle α), reaching the conical intersection region between S ₁ and S ₀. This is illustrated in Fig. 3B, which shows the potential energy surfaces calculated with VQE/VQD for a representative cis–trans trajectory of methanimine. In particular, structure (b) corresponds to the hopping geometry, where the N–H bond vector is twisted out of the plane, as expected for the conical intersection. The obtained hopping geometries (which in principle are very close to the conical intersection geometries) are in very good agreement with previous theoretical predictions, yielding average HNC angles of ∼ 108 ° and HNCH dihedrals of ∼ 100 ° , consistent with earlier multireference studies. ¹¹⁹ ^, ¹²⁰ This agreement is further illustrated in Fig. 3D, which shows the distribution of HNC angles and HNCH dihedrals at the S ₀/S ₁ hopping geometries. Most hops occur when both internal coordinates lie in the range of 80–120°, confirming that our approach accurately reproduces the conical intersection geometries and remains stable in regions where the ground and the first excited state are very close in energy.

The timescales are likewise consistent with previous results obtained in purely classical simulations. As shown in Fig. 3C, the S ₁ population exhibits an initial plateau of ∼30 fs before undergoing exponential decay, with an extracted lifetime of ∼167 fs. These values are comparable to those obtained from TSH simulations based on semiempirical energy surfaces, ¹²¹ highlighting the reliability of VQE/VQD-based TSH.

Concluding outlook

Recent quantum-computing algorithms for NAMD exploit the large Hilbert space of quantum hardware to represent both electronic and nuclear degrees of freedom within a unified framework. In such approaches, the electronic states are typically encoded in qubits (or qudits, i.e., d-dimensional quantum systems), while nuclear motion is mapped onto bosonic modes or discretized bases, and the coupled electron–nuclear dynamics are propagated through sequences of quantum operations such as Suzuki–Trotter decompositions or variational time-evolution schemes. ⁴⁸ ^, ⁵¹ ^, ¹²² ^, ¹²³ Analog implementations have already demonstrated the feasibility of simulating vibronic-coupling models on current devices, while digital algorithms – requiring fault-tolerant quantum computers – could in principle simulate the full time-dependent Schrödinger equation with controllable accuracy. ⁵² ^, ⁵³ ^, ⁵⁴ In practice, however, these simulations demand significantly deeper circuits than ground- or excited-state calculations based on VQE, limiting their use to highly reduced models with only a few degrees of freedom. A comprehensive perspective on full quantum dynamics simulations for NAMD, including comparisons of first- vs. second-quantization encodings and Trotter-based vs. variational propagation schemes, is provided by Ollitrault, Miessen, and Tavernelli. ¹²⁴

NAMD represents one of the most compelling frontiers for quantum computing in chemistry. These processes inherently couple electronic and nuclear motion across exponentially scaling Hilbert spaces, which classical algorithms can only approximate. Quantum devices, at least in principle, provide a natural framework for representing such states and could enable a more balanced description of coupled electron–nuclear dynamics. Recent advances – from MQC simulations interfaced with VQE-based algorithms to fully quantum simulations of reduced vibronic models – illustrate steady progress toward this goal and highlight the potential of quantum hardware to capture phenomena such as ultrafast relaxation through conical intersections.

Despite this progress, formidable challenges remain. Scaling quantum algorithms to chemically realistic systems requires dramatic increases in both qubit count and quality. Larger active spin-orbital spaces, essential for strongly correlated excited states, already demand tens to hundreds of logical qubits; including nuclear motion pushes this requirement even further. Accurate time propagation also requires deep quantum circuits, which exacerbate hardware noise and decoherence. As a result, current demonstrations remain restricted to small molecules, truncated active spaces, or simplified Hamiltonians. While invaluable as proofs of concept, they do not yet rival the robustness and efficiency of established NAMD simulations on classical devices.

In the near term, the most promising opportunities lie in hybrid quantum–classical strategies. Here, quantum routines are used to compute key electronic properties – excited-state energies, gradients, and nonadiabatic couplings – which are then integrated into classical trajectory-based propagation. This approach capitalizes on the strengths of quantum algorithms while avoiding the need for fully quantum nuclear propagation, thereby offering a pragmatic route for NAMD simulations on NISQ devices. Encouraging demonstrations, such as trajectory surface hopping with quantum-evaluated electronic inputs, show the potential of this direction. Nevertheless, executing even hybrid dynamics directly on hardware remains difficult: noise-induced errors compromise energy conservation, often leading to unphysical trajectories. This underscores the urgent need for error-mitigation strategies and resource-efficient ansätze.

As mentioned, one of the main bottlenecks in MQC dynamics lies in the computation of electronic properties, such as energies, gradients, nonadiabatic coupling vectors, and overlap matrices between electronic states at different time steps. These quantities are naturally well-suited for quantum hardware, given its intrinsic ability to handle wavefunction-based information. Alternatively, recent years have seen many efforts to accelerate MQC simulations using machine learning techniques. ¹²⁵ ^, ¹²⁶ ^, ¹²⁷ ^, ¹²⁸ These approaches show promise even if they often require extensive training. In particular, accurately learning nonadiabatic coupling vectors remains extremely challenging. ¹²⁹ ^, ¹³⁰ Furthermore, in most machine learning frameworks the electronic wavefunction is not explicitly represented, which can lead to unreliable propagation of the electronic state, especially near nontrivial crossings. This issue is further compounded by the reliance on approximate methods for computing nonadiabatic couplings, which have been shown to lose accuracy when more than two states are involved. ¹³¹ Time will say, whether quantum algorithms may, in the future, offer a compelling and more reliable pathway to overcome these challenges.

Looking further ahead, the advent of fault-tolerant quantum computers promises a transformative leap. With thousands of logical qubits and robust error correction, it may become possible to perform rigorous full-quantum simulations of coupled electron–nuclear dynamics in chemically realistic systems. Such simulations would not only achieve unprecedented accuracy in describing excited-state processes but could also reveal subtle quantum nuclear effects beyond the reach of current classical methods.

Beyond the technical hurdles, conceptual challenges must also be addressed. Benchmarking quantum simulations against high-level classical methods and experimental observables will be essential to ensure reliability. In this regard, small molecules and model Hamiltonians will continue to serve as critical testing grounds for assessing accuracy, stability, and scalability.

Finally, the broader implications should not be overlooked. Nonadiabatic processes are central across diverse fields from chemistry and biology to materials science. If quantum computing can eventually provide qualitatively new insights into these processes – for example, by simulating chromophore aggregates, photoenzymes, or quantum coherence in biological light harvesting – it would have far-reaching impact across chemistry, biology, and materials science. Yet it is important to remain realistic: fully time-dependent quantum simulations of large chromophores or functional materials remain, for now, a distant goal rather than an imminent possibility.

In conclusion, quantum algorithms for NAMD should currently be regarded primarily as exploratory frameworks: valuable testbeds for algorithmic innovation, platforms for benchmarking hybrid strategies, and conceptual bridges between quantum computing and theoretical chemistry. As hardware improves and algorithms become more resource-efficient, these efforts will help chart the path toward practical, chemically realistic simulations.

Corresponding authors: Eduarda Sangiogo-Gil, Institute of Theoretical Chemistry, University of Vienna, Währinger Str. 17, A-1090 Vienna, Austria, e-mail: eduarda.sangiogo.gil@univie.ac.at; and Leticia González, Institute of Theoretical Chemistry, University of Vienna, Währinger Str. 17, A-1090 Vienna, Austria; and Research Platform on Accelerating Photoreaction Discovery (ViRAPID), University of Vienna, 1090 Vienna, Austria, e-mail: leticia.gonzalez@univie.ac.at

Article note: A collection of invited papers to celebrate the UN’s proclamation of 2025 as the International Year of Quantum Science and Technology.

Acknowledgments

University of Vienna for continuous support.

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: ESG: Data curation, Writing – original draft, LG: Writing – review & editing.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: The author states no conflict of interest.
Research funding: None.
Data availability: Does not apply.

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Received: 2025-08-28

Accepted: 2025-10-03

Published Online: 2025-11-06

Published in Print: 2025-11-25

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Keywords for this article

Noadiabatic molecular dynamics; quantum chemistry; quantum computing; quantum mechanics; quantum science and technology

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