The variational quantum nonlinear solver (VQNLS), a quantum-enhanced computational method for nonlinear dynamical systems, converts solution tasks into global spatiotemporal optimization via parameterized ansatz circuits, thereby eliminating the need for iterative quantum state tomography. Herein, we present an extended implementation scheme of the VQNLS, which is compatible with most numerical discretization schemes and boundary conditions. The method is validated using a nonlinear ordinary differential equation (ODE) and the Burgers' equation using quantum virtual machines and superconducting quantum computers. Experimental results show that (1) the VQNLS can effectively solve nonlinear equations. On an ideal simulator, the ODE's solution achieves a relative error of only 0.17% when the loss function converges to 1×10−6. For the Burgers' equation, VQNLS captures key solution features with an average relative error below 5% on quantum virtual machines, while the average relative error increases to 7.91% on superconducting quantum computers. (2) VQNLS is highly sensitive to the loss function. (3) An appropriate additional loss function mitigates this issue by improving training efficiency and suppressing errors. At a fixed loss threshold of 1×10−3, it reduces training steps from 100 to 56 and cuts the maximum relative error from 15.3% to 4.1%. (4) Furthermore, under the same loss threshold, VQNLS's accuracy improves with the computational scale, with the average relative error decreasing monotonically in a power-law manner. Despite inherent limitations, including high error sensitivity and the challenge of selecting high-performance ansatz circuits, we consider that VQNLS provides a promising framework for engineering-oriented approximate simulations of nonlinear systems.
The variational quantum nonlinear solver (VQNLS), a quantum-enhanced computational method for nonlinear dynamical systems, converts solution tasks into global spatiotemporal optimization via parameterized ansatz circuits, thereby eliminating the need for iterative quantum state tomography. Herein, we present an extended implementation scheme of the VQNLS, which is compatible with most numerical discretization schemes and boundary conditions. The method is validated using a nonlinear ordinary differential equation (ODE) and the Burgers' equation using quantum virtual machines and superconducting quantum computers. Experimental results show that (1) the VQNLS can effectively solve nonlinear equations. On an ideal simulator, the ODE's solution achieves a relative error of only 0.17% when the loss function converges to 1×10−6. For the Burgers' equation, VQNLS captures key solution features with an average relative error below 5% on quantum virtual machines, while the average relative error increases to 7.91% on superconducting quantum computers. (2) VQNLS is highly sensitive to the loss function. (3) An appropriate additional loss function mitigates this issue by improving training efficiency and suppressing errors. At a fixed loss threshold of 1×10−3, it reduces training steps from 100 to 56 and cuts the maximum relative error from 15.3% to 4.1%. (4) Furthermore, under the same loss threshold, VQNLS's accuracy improves with the computational scale, with the average relative error decreasing monotonically in a power-law manner. Despite inherent limitations, including high error sensitivity and the challenge of selecting high-performance ansatz circuits, we consider that VQNLS provides a promising framework for engineering-oriented approximate simulations of nonlinear systems.
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.
The solution for non-linear, complex partial differential Equations (PDEs) is achieved through numerical approximations, which yield a linear system of equations. This approach is prevalent in Computational Fluid Dynamics (CFD), but it restricts the mesh size since the solution of the linear system becomes computationally intractable when the mesh resolution increases. The reliance on the ability of High-Performance Computers (HPC) to scale up and meet these requirements is myopic; such very high-fidelity simulations require a paradigm shift in computing. This paper presents an exploration of quantum methodologies aimed at achieving high accuracy in solving such a large system of equations. Leveraging recent works in Quantum Linear Solver Algorithms (QLSA) and variational algorithms suitable for Quantum Simulation in HPC, we aspire to push the boundaries of CFD-relevant problems that can be solved on hybrid quantum-classical framework. To this end, we consider the 2D, transient, incompressible, viscous, non-linear coupled Burgers equation as a test problem and investigate the accuracy of our approach by comparing results with a classical linear system of equation solvers, such as the Generalized Minimal RESidual method (GMRES). Through rigorous testing, our findings demonstrate that our quantum methods yield results comparable in accuracy to traditional approaches. Additionally, we demonstrate the accuracy, scalability, and consistency of our quantum method. Lastly, we present an insightful estimation of the resources our quantum algorithm needs to solve systems with nearly 2 billion mesh points.
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.
As quantum computing technology continues to develop, quantum algorithms have the potential to outperform classical algorithms in future for certain computational tasks. One of these potential applications of quantum computing is numerically solving differential equations and integral equations. In this thesis the variational quantum linear solver algorithm (VQLS) proposed by Bravo-Prieto et al. is adapted to solve the Laplace and Poisson equations, ordinary differential equation initial value problems (IVPs) with both constant and function coefficients, and a Fredholm integral equation of the second type. As this thesis demonstrates, variational quantum algorithms for solving differential equations and integral equations such as VQLS are highly promising for use on Noisy Intermediate-Scale Quantum (NISQ) devices. The quantum circuit components of the VQLS algorithm were simulated using IBM Quantum's statevector simulator with the optimized solution circuits implemented on the IBM Quantum Guadalupe device (in most cases). A discussion of errors and convergence is also included. The application of VQLS to these differential equations and integral equations is novel and, to my knowledge, work using VQLS to solve the Poisson equation has only recently been published once by authors Liu et al. which was conducted concurrently and independently of this research.
Solving nonlinear partial differential equations is important in various scientific and engineering applications. Recently, quantum computing was introduced as an alternative computational paradigm for solving nonlinear partial differential equations. In this paper, a new method called the variational quantum homotopy perturbation method is proposed to improve the scalability of solving nonlinear partial differential equations through two aspects. First, the dimension of the Hilbert space remains the same after the nonlinear partial differential equation is linearized through the homotopy perturbation. Second, the solutions are obtained with a variational quantum simulation framework, where the number of qubits is decreased with functional encoding and the depth of parametrized circuits is reduced. The additional contribution of this paper is the introduction of new criteria for selecting the homotopy series truncation order and circuit depth for cost-effective variational quantum homotopy perturbation method. The proposed approach is demonstrated with several examples, including the vorticity transport equation and the reduced magnetohydrodynamics equations.
This paper proposes a variational quantum algorithm to address the large-scale and low computational efficiency issues of high-dimensional partial differential equations (PDEs) in climate science. In this method, parameterized quantum circuits encode the discrete solutions of PDEs into quantum states. Then, through repeated quantum measurements, we can obtain the expectation values of the operators. By iteratively adjusting the circuit parameters, a hybrid quantum-classical optimization loop can reduce the error in the solution. The Schrödinger equation and the Poisson equation are benchmark partial differential equations for numerical experiments, and they are four-dimensional and five-dimensional, respectively. According to the above results, the convergence speed is relatively fast; after 100 iterations, the median relative errors of all test cases are less than 1.6% and 3.1%, respectively, with the final target loss ranging between 0.014 and 0.045. The above comparison indicates that in high-dimensional and ill-posed problems, quantum solvers can achieve the same or better accuracy and robustness compared to traditional methods. Resource analysis indicates that quantum algorithms use less memory and fewer sub-exponential growth gates and runtime in larger systems. According to the above findings, variational quantum solvers can be used for complex scientific computing problems. In addition, they hope that future climate models will be more accurate and efficient.