Measurement-Efficient Variational Quantum Linear Solver for Carleman-Linearized Nonlinear Dynamics

TL;DR

This paper introduces a hybrid quantum-classical approach using Carleman linearization and VQLS to efficiently solve nonlinear dynamics like the Duffing equation on quantum hardware.

Contribution

It demonstrates the effectiveness of combining Carleman linearization with VQLS across multiple platforms for solving nonlinear differential equations.

Findings

Carleman linearization accurately approximates the Duffing equation.

VQLS achieves near-unity fidelity in hardware benchmarks.

Topology-agnostic ansatz and optimized Hermitianization improve quantum state recovery.

Abstract

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.