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.
ABSTRACT We present a novel variational quantum framework for nonlinear partial differential equation (PDE) constrained optimization problems. The proposed work extends the recently introduced bi-level variational quantum PDE constrained optimization (BVQPCO) framework for linear PDE to a nonlinear setting by leveraging Carleman linearization (CL). CL framework allows one to transform a system of polynomial ordinary differential equations (ODE), i.e., ODE with polynomial vector field, into a system of infinite but linear ODE. For instance, such polynomial ODEs naturally arise when the PDE is semi-discretized in the spatial dimensions. By truncating the CL system to a finite order, one obtains a finite system of linear ODE to which the linear BVQPCO framework can be applied. In particular, the finite system of linear ODE is discretized in time and embedded as a system of linear equations. The variational quantum linear solver (VQLS) is used to solve the linear system for given optimization parameters and evaluate the design cost/objective function, and a classical black box optimizer is used to select the next set of parameter values based on this evaluated cost. We present detailed computational errors and complexity analysis and prove that under suitable assumptions, our proposed framework can provide potential advantages over classical techniques. We implement our framework using the PennyLane library and apply it to solve inverse Burgers’ problem. We also explore an alternative tensor product decomposition which exploits the sparsity/structure of linear system arising from PDE discretization to facilitate the computation of VQLS cost functions.
Efficiently solving nonlinear ordinary and partial differential equations using a quantum computer is a major challenge due its inherent linearity. To circumvent this challenge, the Carleman linearization method has been proposed to transform a nonlinear ordinary differential equation into a linear system of equations, the primary advantage being that existing quantum linear systems algorithms may then be applied to obtain a solution. However, this methodology also brings forth several major challenges that must be addressed to attain a quantum advantage. Herein, we address several of these challenges enabling us to solve the Carleman linearized one-dimensional Burgers' equation on real and simulated quantum hardware. All simulations were performed on BlueQubit's platform allowing for quantum circuits to be run on GPU or QPU's seamlessly. We first demonstrate that the Carleman linearized Burgers' equation can be efficiently loaded onto a quantum computer using the linear combination of non-unitaries method, an alternative to the linear combintaiton of unitaries approach. Once loaded, the linear system is then solved using the variational quantum linear solver. Since a naive implementation of this solver is hindered by the barren plateau phenomenon, we introduce a multigridding method to solve the problem in a series of stages with the solution of the previous stage acting as a warm start for the next stage. This approach is found to significantly improve the accuracy of the solution compared with a naive cold start. Finally, circuits with a combined number of spatial and temporal discretization points totaling up to
280≈1024 are transpiled onto real quantum hardware demonstrating that the proposed methodology could feasibly produce a quantum advantage on future hardware.
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.
The Variational Quantum Linear Solver (VQLS), a hybrid quantum-classical algorithm for solving linear systems, faces a practical scalability bottleneck: the Linear Combination of Unitaries (LCU) decomposition requires
O(L2) circuit evaluations per optimizer iteration, where
L can grow to
4n in the worst case for an
n-qubit system. We address this computational bottleneck through two complementary strategies. First, we present a distributed VQLS (D-VQLS) framework (https://code.ornl.gov/olcf-qcfd/DVQLS.git), built on NVIDIA CUDA-Q, that enables asynchronous, scalable distribution of the
O(L2) cost evaluations. Second, a fast Walsh--Hadamard transform (FWHT)-based Pauli decomposition with coefficient-amplitude pruning threshold
τ=0.01 curbs LCU growth for the structured Toeplitz family, reducing
L from
O(2n) to 64 for
n>6 and compressing the circuit complexity per optimizer iteration from
O(n4n) to
O(n). We derive the exact top-
L Frobenius error and connect it to worst-case solution error. For a 10-qubit tridiagonal Toeplitz system, the
L=64 pruning yields a
256× reduction---from 23 million to 90k circuits per optimizer iteration.
The D-VQLS framework is validated on the NERSC Perlmutter supercomputer using multi-node, multi-GPU ideal state-vector simulations, achieving over
99.99% fidelity against classical solutions on tridiagonal Toeplitz and Hele--Shaw flow benchmarks, with near-ideal strong scaling up to 24 GPUs and
95.3% weak scaling efficiency at 96 GPUs processing more than 360k circuits per optimizer iteration (from larger-
L pruning) for the 10-qubit system. Systematic profiling identifies the optimal resource allocation for distributed quantum circuit workloads, yielding a
2.52× speedup for the configurations studied.
From fluid flow and transport to collective dynamics, numerical simulation of nonlinear partial differential equations underpins modern scientific computing. Extending this capability to quantum computers remains a longstanding challenge because nonlinear and non-Hermitian evolution is fundamentally incompatible with conventional Hamiltonian-based quantum simulation. Here we experimentally realize the time evolution of nonlinear fluid dynamics on a quantum processor using a hybrid variational framework for the viscous and inviscid Burgers equations. Our approach directly encodes the nonlinear dynamics into a variational optimization procedure, avoiding the enlarged linear embeddings and truncation overhead associated with Carleman linearization-based quantum algorithms. We further demonstrate convection-dominated dynamics corresponding to Reynolds numbers of order
102. We encode the governing evolution into parametrized quantum circuits and iteratively reconstruct the time-dependent field through quantum-classical optimization. By introducing a zero-noise extrapolation method without additional circuit-folding overhead, we accurately execute deep error-circuits with entangling-gate counts beyond those typical of Hadamard test circuits. We accurately reconstruct the time evolution across multiple timesteps despite hardware noise and finite device coherence. Our results constitute, to our knowledge, the first experimental realization of nonlinear time propagation on a quantum processor, extending quantum simulation beyond predominantly linear settings and establishing a route toward quantum computation for nonlinear continuum dynamics.
Abstract \noindent This paper applies the Variational Quantum Linear Solver (VQLS) to the continuous-time Lyapunov equation
ATX+XA=−Q. Using the Kronecker product formulation, the matrix equation is transformed into the linear system
Hz=b, where
H=In⊗AT+AT⊗In and
z=vec(X). VQLS approximates the solution via parameterized quantum circuits optimized in a hybrid quantum-classical loop. Numerical experiments on four control system benchmarks with Kronecker dimensions up to
25×25 demonstrate residual norms as low as
10−9, with logarithmic qubit scaling (2-5 qubits) and shallow circuit depths (6-14 layers) suitable for near-term quantum devices. Compared to the HHL algorithm, VQLS offers reduced circuit complexity and qubit overhead, making it practical for noisy intermediate-scale quantum (NISQ) hardware. This paper demonstrates the potential of variational quantum algorithms for solving Lyapunov equations in control and dynamical systems applications.