A robust quantum nonlinear solver based on the asymptotic numerical method

TL;DR

The paper tackles efficient quantum solution of nonlinear problems by marrying the asymptotic numerical method (ANM) with quantum linear solvers. By linearizing R(u,λ)=0 along a solution path using Taylor expansions, qANM reduces each nonlinear step to solving a sequence of linear systems K u_p = F with K = D_u R(u_0,λ_0); the authors implement two quantum solvers, VQLS and the quantum Jacobi method (q-Jacobi), and analyze their complexities. They validate the approach on Qiskit and demonstrate robustness on an Euler–Bernoulli beam and a buckling scenario, including a hardware demonstration on Quafu achieving ~98% path accuracy, indicating practical feasibility on NISQ devices. The work also introduces a speculative quantum matrix-inversion pathway (VQMI) to further reduce linear solves per step and discusses error-mitigation needs, highlighting qANM’s potential to advance quantum-enabled scientific computing in nonlinear mechanics and beyond.

Abstract

Quantum computing offers a promising new avenue for advancing computational methods in science and engineering. In this work, we introduce the quantum asymptotic numerical method, a novel quantum nonlinear solver that combines Taylor series expansions with quantum linear solvers to efficiently address nonlinear problems. By linearizing nonlinear problems using the Taylor series, the method transforms them into sequences of linear equations solvable by quantum algorithms, thus extending the convergence region for solutions and simultaneously leveraging quantum computational advantages. Numerical tests on the quantum simulator Qiskit confirm the convergence and accuracy of the method in solving nonlinear problems. Additionally, we apply the proposed method to a beam buckling problem, demonstrating its robustness in handling strongly nonlinear problems and its potential advantages in quantum resource requirements. Furthermore, we perform experiments on a superconducting quantum processor from Quafu, successfully achieving up to 98% accuracy in the obtained nonlinear solution path. We believe this work contributes to the utility of quantum computing in scientific computing applications.

Figures (19)

• Figure 1: (a) Illustration of the hardware-efficient ansatz kandala2017hardware. The ansatz begins with single-qubit $\bm{R}_y$ rotations, followed by layers of entangling gates and parameterized rotations. Each $\bm{R}_y$ gate is associated with a parameter in $\bm{\theta}$, which is optimized during the algorithm. (b) Quantum circuit for computing the cost function $C(\bm{\theta})$ in VQLS.
• Figure 2: Quantum circuit for computing the inner product between a row of $\bm{M}$ and the vector $\bm{u}^{(k)}$ using the modified Hadamard test. The circuit involves state preparation of $\ket{\bm{\Psi}}$, application of the Hadamard gate $\bm{H}$, and measurement of the ancilla qubit.
• Figure 3: Performance comparison of VQLS and q-Jacobi in solving $\bm{K} \bm{u} = \bm{F}_0$. (a) Cost function $C(\bm{\theta})$ versus iterations in VQLS. (b) Tolerance $Tol$ versus iterations in q-Jacobi. (c) Solutions $\bm{u}$ obtained by VQLS and q-Jacobi compared to the reference solution.
• Figure 4: Accuracy of solutions obtained by VQLS and q-Jacobi for different right-hand side vectors $\bm{F}_j$.
• Figure 5: Impact of the number of shots $n_s$ on the performance of q-Jacobi for solving $\bm{K} \bm{u} = \bm{F}_0$. (a) Accuracy of the solution versus $n_s$, with the classical Jacobi method accuracy as a reference (blue dashed line). (b) Number of iterations required for convergence versus $n_s$, compared to the classical Jacobi method (blue dashed line).