This paper extends a recently proposed Variational Quantum Algorithm framework for nonlinear elasticity to a broader class of constitutive nonlinearities involving rational powers of the stretch. One-dimensional incompressible Ogden and Mooney-Rivlin models are employed as representative examples to demonstrate the proposed methodology. Nonlinear constitutive terms are transformed into forms compatible with the available quantum algorithmic primitives through the introduction of auxiliary variables and penalty constraints, yielding approximate solutions via a Variational Quantum Algorithm. An iterative correction strategy based on a sequence of Variational Quantum Algorithms is then introduced to improve solution accuracy. A Numerical example demonstrates the proposed approach.
This paper extends a recently proposed Variational Quantum Algorithm framework for nonlinear elasticity to a broader class of constitutive nonlinearities involving rational powers of the stretch. One-dimensional incompressible Ogden and Mooney-Rivlin models are employed as representative examples to demonstrate the proposed methodology. Nonlinear constitutive terms are transformed into forms compatible with the available quantum algorithmic primitives through the introduction of auxiliary variables and penalty constraints, yielding approximate solutions via a Variational Quantum Algorithm. An iterative correction strategy based on a sequence of Variational Quantum Algorithms is then introduced to improve solution accuracy. A Numerical example demonstrates the proposed approach.
This manuscript explores a variational quantum formulation for nonlinear elasticity problems arising from hyperelastic material models. The approach leverages the potential energy structure of hyperelasticity and employs a hybrid quantum classical framework in which the energy functional is evaluated using parameterized quantum circuits and optimized through classical routines. To enable a hybrid (classical quantum implementation), polynomial approximations of the nonlinear terms in strain energy density are introduced, yielding a representation compatible with variational quantum algorithms. The methodology is demonstrated on a special case of the NeoHookean material model in a one dimensional setting using finite element discretizations with first and second order shape functions and nonhomogeneous boundary conditions. Numerical experiments investigate the influence of the polynomial approximation order on the accuracy and efficiency of the proposed approach, illustrating its feasibility for near-term quantum devices.
A variational quantum algorithm for numerically solving partial differential equations (PDEs) on a quantum computer was proposed by Lubasch et al. [Phys. Rev. A 101, 010301(R) (2020)]. In this paper we generalize the method introduced by Lubasch et al. to cover a broader class of nonlinear PDEs as well as multidimensional PDEs and study the performance of the variational quantum algorithm on several example equations. Specifically, we show via numerical simulations that the algorithm can solve instances of the single-asset Black-Scholes equation with a nontrivial nonlinear volatility model, the double-asset Black-Scholes equation, the Buckmaster equation, and the deterministic Kardar-Parisi-Zhang equation. Our simulations use up to n=12 Ansatz qubits, computing PDE solutions with 2n grid points. We also perform proof-of-concept demonstrations with a trapped-ion quantum processor from IonQ [Nat. Commun. 10, 5464 (2019)], showing accurate computation of two representative expectation values needed for the calculation of a single time step of the nonlinear Black-Scholes equation. Through our classical simulations and demonstrations on quantum hardware, we identify and discuss several open challenges for using quantum variational methods to solve PDEs in a regime with a large number (much greater than 220) of grid points, but also a practical number of gates per circuit and circuit shots.
We present a variational quantum algorithm (VQA) to solve the nonlinear one-dimensional Bratu equation. By formulating the boundary value problem within a variational framework and encoding the solution in a parameterized quantum neural network (QNN), the problem reduces to an optimization task over quantum circuit parameters. The trial solution incorporates a predictor from the previous continuation step and boundary-enforcing terms, allowing the circuit to focus on minimizing the residual of the differential operator. Using a noiseless quantum simulator, we demonstrate that the method accurately captures both solution branches of the Bratu equation and shows excellent agreement with classical pseudo arc-length continuation results.
We develop a pivot-shifted Carleman linearization framework for quantum algorithms solving quadratic nonlinear ordinary differential equations. By shifting the dynamics by a pivot state prior to Carleman lifting, and combining this with a Lyapunov transform and rescaling, we enlarge the class of nonlinear systems that can be efficiently simulated on quantum computers. For systems that exhibit stability in the shifted coordinates, we establish long time convergence of the truncated Carleman embedding. We prove that the truncation order scales only logarithmically with the simulation time and target precision, and we derive end-to-end quantum query complexity bounds for preparing a state proportional to the final solution. By introducing a modified nonlinearity condition, this framework entirely removes the conventional lower bound requirement on the initial condition. For more general systems that remain unstable after shifting, we provide short time convergence guarantees that are similarly free from the initial condition constraints. Numerical experiments on the logistic and the Lotka-Volterra equations demonstrate that an appropriate pivot choice improves stability and accuracy, and yields exponential error decay with truncation order. These results show that pivot shifting provides a practical and theoretically justified route for extending Carleman-based quantum algorithms to a broader class of nonlinear dynamical 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 The classical distortion models similarly to temporal data analysis provide a way to predict the trends in the output of the algorithms and discard the anomalies in the output which may impact the accuracy of the method. In this paper, we introduce a quantum distortion modeling framework that enables variational quantum algorithms to operate effectively in the presence of errors without traditional error correction. Drawing inspirations from classical distortion-tolerant computing, we describe different distortions measures and formulation which can be used in variational quantum algorithms. In particular, we develop mathematical foundations for distortion metrics including energy progression, parameter stability, and state fidelity distortions, and demonstrate their applicability to variational quantum eigensolver, quantum approximate optimization algorithm, and quantum power method. We believe this paper will provide a milestone for distortion-aware quantum computing which can expand the practical applicability of the pre-fault-tolerant era devices to problems where approximate solutions provide sufficient value, representing a paradigm shift from exact error elimination to managed accuracy degradation.