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arXiv:2606.03515

A Voxel-Based Quantum Computing Method (VBQC) for Solid Mechanics Problem

Feng Wu, Yuxiang Yang, Li Zhu, Chen Li, Yansong Guo, Xu Guo

cs.CEmath.NAquant-phSubmitted Jun 2, 2026• Announced Jun 3, 2026• 0 citations
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Abstract

Quantum computing presents a promising method to overcome the efficiency and memory constraints in large-scale mechanical problems, with numerous successful applications demonstrated in fluid mechanics. However, solid mechanics problems usually require irregular grids for spatial discretization, due to the Lagrange formulations and complex boundaries, which makes the quantum simulation of the system matrix, e.g., the mass or stiffness matrix which is often referred to as the Hamiltonian in quantum computing, difficult to be effectively conducted. This study proposes a voxel-based quantum computing method (VBQC) for the quantum simulation of Hamiltonians in solid mechanics. VBQC applies voxel grids to discretize the spatial domain, thereby enabling the system matrix to exhibit the tridiagonal fractal property. Based on this property, the system matrix can be decomposed into three groups of fundamental matrices, $\mathbf{k}_{n}$, $\mathbf{c}_{n}$, and $\mathbf{q}_{n}$. This decomposition process is referred to as the KCQ decomposition. By integrating the KCQ decomposition with the quantum Fourier transform and the quantum multiplexer, VBQC enables efficient quantum simulation of Hamiltonians in solid mechanics. Three specific solid problems with different dimensions and numbers of variables are applied to preliminarily verify the correctness of the proposed VBQC for solid mechanics problems.

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Quantum computing presents a promising method to overcome the efficiency and memory constraints in large-scale mechanical problems, with numerous successful applications demonstrated in fluid mechanics. However, solid mechanics problems usually require irregular grids for spatial discretization, due to the Lagrange formulations and complex boundaries, which makes the quantum simulation of the system matrix, e.g., the mass or stiffness matrix which is often referred to as the Hamiltonian in quantum computing, difficult to be effectively conducted. This study proposes a voxel-based quantum computing method (VBQC) for the quantum simulation of Hamiltonians in solid mechanics. VBQC applies voxel grids to discretize the spatial domain, thereby enabling the system matrix to exhibit the tridiagonal fractal property. Based on this property, the system matrix can be decomposed into three groups of fundamental matrices, kn\mathbf{k}_{n}kn​, cn\mathbf{c}_{n}cn​, and qn\mathbf{q}_{n}qn​. This decomposition process is referred to as the KCQ decomposition. By integrating the KCQ decomposition with the quantum Fourier transform and the quantum multiplexer, VBQC enables efficient quantum simulation of Hamiltonians in solid mechanics. Three specific solid problems with different dimensions and numbers of variables are applied to preliminarily verify the correctness of the proposed VBQC for solid mechanics problems.
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.
This manuscript presents the Quantum Finite Element Method (Q-FEM) developed for use in noisy intermediate-scale quantum (NISQ) computers and employs the variational quantum linear solver (VQLS) algorithm. The proposed method leverages the classical FEM procedure to perform the unitary decomposition of the stiffness matrix and employs generator functions to design explicit quantum circuits corresponding to the unitaries. Q-FEM keeps the structure of the finite element discretization intact allowing for the use of variable element lengths and material coefficients in FEM discretization. The proposed method is tested on a steady-state heat equation discretized using linear and quadratic shape functions. Numerical verification studies are performed on the IBM QISKIT simulator and it is demonstrated that Q-FEM is effective in converging to the correct solution for a variety of problems and model discretizations, including with different element lengths, variable coefficients, and different boundary conditions. The formalism developed herein is general and can be extended to problems with higher dimensions. However, numerical examples also demonstrate that the number of parameters for the variational ansatz scale exponentially with the number of qubits, and increases the odds of convergence. Moreover, the deterioration of system conditioning with problem size results in barren plateaus and convergence difficulties.
Quantum computing offers the potential to solve computational mechanics problems with a more favourable complexity scaling than classically possible. To realise this potential, a quantum-compatible formulation of the finite element method is essential. We present such a formulation for Poisson problems with spatially varying coefficients on general domains discretised by an immersed uniform Cartesian grid. The resulting banded sparse system matrices are encoded as the sum of diagonal matrices with polynomial entries and their products with shift matrices. The sum of the matrices is formed using the linear combination of unitaries (LCU) technique. The resulting block-encoded system matrix can be implemented using a polylogarithmic number of elementary gates. For immersed boundaries, the block-encoded system matrix is further processed using a diagonal indicator matrix with zeros and ones. The indicator matrix is defined in terms of an integer-valued level-set function composed of simple geometric primitives via Boolean set operations. The required integer operations are computed on the fly using quantum arithmetic. The resulting block-encoded linear system of equations is solved using quantum singular value transformation (QSVT). The overall gate count, including the QSVT, is O(Ntot2/dpolylog⁡(Ntot))O(N_{\text{tot}}^{2/d} \operatorname{polylog}(N_{\text{tot}}))O(Ntot2/d​polylog(Ntot​)), where ddd is the problem dimension and NtotN_{\text{tot}}Ntot​ the number of grid points. In numerical examples that confirm the theoretical complexity estimates, the solutions converge approximately linearly for immersed problems and quadratically for simply connected domains.
The construction of quantum circuits to simulate Hamiltonian evolution is central to many quantum algorithms. State-of-the-art circuits are based on oracles whose implementation is often omitted, and the complexity of the algorithm is estimated by counting oracle queries. However, in practical applications, an oracle implementation contributes a large constant factor to the overall complexity of the algorithm. The key finding of this work is the efficient procedure for representation of a tridiagonal matrix in the Pauli basis, which allows one to construct a Hamiltonian evolution circuit without the use of oracles. The procedure represents a general tridiagonal matrix ${2}^{n}\ifmmode\times\else\texttimes\fi{}{2}^{n}$ by systematically determining all Pauli strings present in the decomposition, dividing them into commuting subsets. The efficiency is in the number of commuting subsets O(n)O(n)O(n). The method is demonstrated using the one-dimensional wave equation, verifying numerically that the gate complexity as a function of the number of qubits is lower than the oracle-based approach for n<15n<15n<15 and requires half the number of qubits. This method is applicable to other Hamiltonians based on the tridiagonal matrices.
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.
Quantum mechanics has introduced a new theoretical framework for the study of molecules, enabling the prediction of properties and dynamics through the solution of the Schrödinger equation applied to these systems. However, solving this equation is computationally expensive, which has led to the development of various mathematical frameworks and computational methods designed to balance the available resources with the desired level of accuracy. In particular, quantum computers have emerged as a promising technology with the potential to address these problems more efficiently in the coming decades, whether through reductions in memory, time, and energy consumption, i. e.\textit{i. e.}i. e., reductions in computational complexity or by enhancing precision. This research field is known as Quantum Simulation. Given the current technological limitations of quantum computers, Variational Quantum Algorithms (VQAs), especially the Variational Quantum Eigensolver (VQE), stand out as a feasible approach to demonstrating advantages over classical methods in the near term. This feasibility arises from their lower demand for quantum gates and the reduced depth of the circuits required for their implementation. In this work, we present the application of quantum mechanics to the study of molecules, provide an introduction to the fundamentals of quantum computing, and explore the integration of these fields by employing the VQE in molecular simulations. Finally, we discuss the spatial and temporal complexity associated with the algorithm, highlighting its implications and challenges.