Introduction: Quantum computing has the potential to transform scientific computing by providing efficient algorithms for solving large-scale numerical problems that arise in engineering and materials science. However, integrating quantum algorithms into multiscale mechanics remains largely unexplored due to the complexity of coupling atomistic and continuum models and the nonlinear nature of molecular dynamics. In this work, we develop a unified hybrid classical–quantum framework for molecular dynamics and multiscale materials modeling based on the Variational Quantum Linear Solver (VQLS). Materials and methods: The proposed framework reformulates three representative mechanics problems as quantum-compatible linear systems: static atomistic–continuum coupling through reduced stiffness equations, dynamic atomistic–continuum coupling using a boundary-reduced Berry–Childs–Ostrander–Wang formulation, and nonlinear molecular dynamics through Carleman linearization. These systems are solved using shot-free quantum-circuit simulations within a common VQLS framework. Results: For the static problem with 1024 free displacement degrees of freedom, the VQLS solution achieves R 2 = 0.99784 and a relative L 2 error of 2.70 % . For the dynamic problem, the five-qubit restarted systems produce maximum final-time displacement and velocity differences of 1.39 × 10 − 4 nm and 6.74 × 10 − 5 nm/ps, respectively. For the nonlinear problem, comparison of the six-qubit VQLS trajectory with a high-accuracy integration of the full Lennard–Jones equations over 150 time steps gives maximum displacement and velocity differences of 4.09 × 10 − 4 nm and 2.87 × 10 − 3 nm/ps, respectively. The reported velocities are reconstructed from the displacement histories using a common backward-difference rule with the prescribed zero initial velocity. The numerical comparisons quantify agreement with classical reference trajectories and distinguish selected modeling, temporal-discretization, and variational-solution errors. Conclusions: By providing a unified quantum linear-system formulation for representative multiscale mechanics problems, this deterministic, shot-free, simulator-based study establishes a methodological proof of concept for integrating VQLS formulations with computational materials models, rather than demonstrating near-term hardware performance or scalable quantum computational advantage.
Introduction: Quantum computing has the potential to transform scientific computing by providing efficient algorithms for solving large-scale numerical problems that arise in engineering and materials science. However, integrating quantum algorithms into multiscale mechanics remains largely unexplored due to the complexity of coupling atomistic and continuum models and the nonlinear nature of molecular dynamics. In this work, we develop a unified hybrid classical–quantum framework for molecular dynamics and multiscale materials modeling based on the Variational Quantum Linear Solver (VQLS). Materials and methods: The proposed framework reformulates three representative mechanics problems as quantum-compatible linear systems: static atomistic–continuum coupling through reduced stiffness equations, dynamic atomistic–continuum coupling using a boundary-reduced Berry–Childs–Ostrander–Wang formulation, and nonlinear molecular dynamics through Carleman linearization. These systems are solved using shot-free quantum-circuit simulations within a common VQLS framework. Results: For the static problem with 1024 free displacement degrees of freedom, the VQLS solution achieves R 2 = 0.99784 and a relative L 2 error of 2.70 % . For the dynamic problem, the five-qubit restarted systems produce maximum final-time displacement and velocity differences of 1.39 × 10 − 4 nm and 6.74 × 10 − 5 nm/ps, respectively. For the nonlinear problem, comparison of the six-qubit VQLS trajectory with a high-accuracy integration of the full Lennard–Jones equations over 150 time steps gives maximum displacement and velocity differences of 4.09 × 10 − 4 nm and 2.87 × 10 − 3 nm/ps, respectively. The reported velocities are reconstructed from the displacement histories using a common backward-difference rule with the prescribed zero initial velocity. The numerical comparisons quantify agreement with classical reference trajectories and distinguish selected modeling, temporal-discretization, and variational-solution errors. Conclusions: By providing a unified quantum linear-system formulation for representative multiscale mechanics problems, this deterministic, shot-free, simulator-based study establishes a methodological proof of concept for integrating VQLS formulations with computational materials models, rather than demonstrating near-term hardware performance or scalable quantum computational advantage.
We present hybrid quantum-classical pipelines for solving the Duffing equation that leverage Carleman linearization and the Variational Quantum Linear Solver (VQLS). First, we demonstrate that Carleman linearization accurately approximates the weakly nonlinear Duffing equation, with errors diminishing as the truncation order increases. Next, across IBM and Xanadu platforms, we deploy VQLS with symmetry-grouped Hadamard Test evaluations under both global and local cost formulations, compare distinct Hermitianization within a common cost framework, and benchmark hardware-efficient ansatz architectures under a fixed Hermitianization. Across block-banded test cases, each method achieves near-unity fidelity and vanishing relative residuals. These results show that topology-agnostic ansatz, optimized Hermitianization, and efficient cost formulation enable VQLS to recover quantum states proportional to classical solutions for Carleman-structured systems, providing a portable recipe for quantum-in-the-loop simulation of nonlinear dynamics.
Recent developments in quantum computing have triggered a paradigm shift extending from algorithmic theory to applied engineering fields, particularly in the modeling and optimization of complex energy systems. This study presents a hybrid quantum-classical computational framework designed for the multiscale and multiphysics modeling of nanotechnology-driven energy systems. The illustrative case involves a proton exchange membrane fuel cell (PEMFC) utilizing nanostructured materials, governed by electrochemical reactions, transport phenomena, degradation mechanisms, and adaptive control dynamics. The proposed framework integrates classical numerical solvers with quantum algorithms such as the Variational Quantum Eigensolver (VQE), Quantum Approximate Optimization Algorithm (QAOA), and Harrow-Hassidim-Lloyd (HHL) solver. A modular orchestration layer ensures data interoperability and coherence between quantum and classical components. This integration improves computational efficiency in high-dimensional, nonlinear problems and opens a path toward domain-specific quantum advantage in energy systems engineering. The results demonstrate the feasibility of incorporating quantum computing into classical modeling pipelines and suggest promising directions for future research in adaptive, scalable, and high-fidelity simulation environments.
Exploring the potential applications of quantum computers in material design and drug discovery is attracting more and more attention after quantum advantage has been demonstrated using Gaussian boson sampling. However, quantum resource requirements in material and (bio)molecular simulations are far beyond the capacity of near-term quantum devices. In this work, multiscale quantum computing is proposed for quantum simulations of complex systems by integrating multiple computational methods at different scales of resolution. In this framework, most computational methods can be implemented in an efficient way on classical computers, leaving the critical portion of the computation to quantum computers. The simulation scale of quantum computing strongly depends on available quantum resources. As a near-term scheme, we integrate adaptive variational quantum eigensolver algorithms, second-order Møller-Plesset perturbation theory and Hartree-Fock theory within the framework of the many-body expansion fragmentation approach. This new algorithm is applied to model systems consisting of hundreds of orbitals with decent accuracy on the classical simulator. This work should encourage further studies on quantum computing for solving practical material and biochemistry problems.
ConspectusSimulating molecular dynamics (MD) within a comprehensive quantum framework has been a long-standing challenge in computational chemistry. An exponential scaling of computational cost renders solving the time dependent Schrödinger equation (TDSE) of a molecular Hamiltonian, including both electronic and nuclear degrees of freedom (DOFs), as well as their couplings, infeasible for more than a few DOFs. In the Born-Oppenheimer (BO), or adiabatic, picture, electronic and nuclear parts of the wave function are decoupled and treated separately. Within this framework, the nuclear wave function evolves along potential energy surfaces (PESs) computed as solutions to the electronic Schrödinger equation parametrized in the nuclear DOFs. This approximation, together with increasingly elaborate numerical approaches to solve the nuclear time dependent Schrödinger equation (TDSE), enabled the treatment of up to a few dozens of degrees of freedom (DOFs). However, for particular applications, such as photochemistry, the BO approximation breaks down. In this regime of non-adiabatic dynamics, solving the full molecular problem including electron-nuclear couplings becomes essential, further increasing the complexity of the numerical solution. Although valuable methods such as multiconfigurational time-dependent Hartree (MCTDH) have been proposed for the solution of the coupled electron-nuclear dynamics, they remain hampered by an exponential scaling in the number of nuclear DOFs and by the difficulty of finding universal variational forms.In this Account, we present a perspective on novel quantum computational algorithms, aiming to alleviate the exponential scaling inherent to the simulation of many-body quantum dynamics. In particular, we focus on the derivation and application of quantum algorithms for adiabatic and non-adiabatic quantum dynamics, which include efficient approaches for the calculation of the BO potential energy surfaces (PESs). Thereafter, we study the time-evolution of a model system consisting of two coupled PESs in first and second quantization. In a first application, we discuss a recently introduced quantum algorithm for the evolution of a wavepacket in first quantization and exploit the potential quantum advantage of mapping its spatial grid representation to logarithmically many qubits. For the second demonstration, we move to the second quantization framework and review the scaling properties of two alternative time-evolution algorithms, namely, a variational quantum algorithm (VQA) (based on the McLachlan variational principle) and conventional Trotter-type evolution (based on a Lie-Trotter-Suzuki formula). Both methods clearly demonstrate the potential of quantum algorithms and their favorable scaling compared to the available classical approaches. However, a clear demonstration of quantum advantage in the context of molecular quantum dynamics may require the implementation of these algorithms in fault-tolerant quantum computers, while their application in near-term, noisy quantum devices is still unclear and deserves further investigation.
In solid mechanics, finite element discretization of large-scale static problems produces large sparse linear systems whose solution requires substantial computation time and memory. The variational quantum linear solver (VQLS) offers a hybrid quantum-classical route, but its use in quantum finite element analysis is limited by the decomposition of nonunitary matrices, barren plateaus, and the measurement cost of expectation values. We propose a voxel-based block variational quantum linear solver (Voxel-BVQLS) that combines structured matrix decomposition, the principle of minimum potential energy, and batched quantum tests. First, we construct an LCU decomposition of the stiffness matrix from the recursive block-banded structure of voxel-grid finite element matrices, with the number of unitary terms bounded independently of the problem size. Second, we optimize the ansatz parameters using a minimum-potential-energy objective in place of a conventional VQLS loss function, thereby mitigating barren plateaus in the studied problems. Third, we introduce a block-Hadamard test whose circuit directly estimates weighted sums of multiple inner products, reducing the number of circuit configurations required per iteration. We assessed the proposed method in noiseless classical simulations using three examples. These examples show that the method reduces both the number of unitary terms in the LCU decomposition and the number of iterations required to converge, while still yielding solutions of finite accuracy. Voxel-BVQLS thus provides a structured hybrid quantum-classical framework for quantum finite element analysis on regular grids.
Accurate modelling of fluid flow in porous media is crucial for applications including hydrology, geological CO₂ storage, hydrogen storage, geothermal energy, and hydrocarbon recovery. The inherent heterogeneity of subsurface formations, characterized by spatial permeability contrasts, fractures, and vugs, leads to large and often ill-conditioned linear systems when discretized at high resolution. Although upscaling techniques reduce computational cost, they may obscure fine-scale features that significantly influence flow behavior, particularly in the presence of small but highly conductive fractures. This study investigates the Variational Quantum Linear Solver (VQLS) as an alternative framework for solving discretized Darcy flow equations in one-, two-, and three-dimensional (1D, 2D, and 3D) homogeneous and heterogeneous systems. VQLS approximates the solution to linear systems of the form Ax = b by preparing a quantum state proportional to the classical solution vector. The framework incorporates a hybrid quantum-classical cost formulation, which strengthens optimization in ill-conditioned systems, and a curvature-aware Adaptive Dynamic Ansatz, which enables principled circuit depth selection without manual tuning. Five experimental modes are evaluated, including fixed-depth training, hybrid cost formulation, adaptive ansatz growth, depth sweep analysis, and hybrid weight sweep. All experiments are conducted using noiseless statevector simulation. The results examine convergence behavior, solution accuracy, and algorithmic limitations in heterogeneous media, providing insight into the practical potential and current constraints of variational quantum solvers for porous media flow modelling.