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IEEE Control Systems Letters

Exploring Chebyshev Spectral Method With Quantum Annealing via Carleman Linearization

Hieu Trung Nguyen, Anh Phuong Ngo

Published Jan 1, 2025• 0 citations
Publisher

Abstract

This paper explores the integration of Carleman linearization and quantum annealing to solve nonlinear ordinary differential equations (ODEs) under the Chebyshev spectral method. Using Carleman linearization, the computation of optimal Chebyshev coefficients is reformulated as a quadratic program and further transformed into a Quadratic Unconstrained Binary Optimization (QUBO) problem via binary expansion. This QUBO formulation is well-suited for quantum annealers, where solutions are obtained by minimizing the Hamiltonian of the corresponding Ising model. To address hardware limitations, we employ an iterative quantum annealing strategy with dynamically updated parameters. Case studies on the D-Wave platform demonstrate the potential of our approach to harness the strengths of Carleman linearization and Chebyshev approximation theory, offering continuous and differentiable solutions to nonlinear ODEs, especially promising as quantum resources become more economically viable.

DOI
10.1109/lcsys.2025.3638545

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This paper explores the integration of Carleman linearization and quantum annealing to solve nonlinear ordinary differential equations (ODEs) under the Chebyshev spectral method. Using Carleman linearization, the computation of optimal Chebyshev coefficients is reformulated as a quadratic program and further transformed into a Quadratic Unconstrained Binary Optimization (QUBO) problem via binary expansion. This QUBO formulation is well-suited for quantum annealers, where solutions are obtained by minimizing the Hamiltonian of the corresponding Ising model. To address hardware limitations, we employ an iterative quantum annealing strategy with dynamically updated parameters. Case studies on the D-Wave platform demonstrate the potential of our approach to harness the strengths of Carleman linearization and Chebyshev approximation theory, offering continuous and differentiable solutions to nonlinear ODEs, especially promising as quantum resources become more economically viable.
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Nonlinearity presents a significant challenge in developing quantum algorithms involving differential equations, prompting the exploration of various linearization techniques, including the well-known Carleman Linearization. Instead, this paper introduces the Koopman Spectral Linearization method tailored for nonlinear autonomous ordinary differential equations. This innovative linearization approach harnesses the interpolation methods and the Koopman Operator Theory to yield a lifted linear system. It promises to serve as an alternative approach that can be employed in scenarios where Carleman Linearization is traditionally applied. Numerical experiments demonstrate the effectiveness of this linearization approach for several commonly used nonlinear ordinary differential equations.
We investigate hybrid quantum–classical solvers for nonlinear boundary value problems using Chebyshev spectral collocation. Unlike prior methods such as H–DES, which repeatedly recompile circuits and encode the entire spectral basis on the quantum processor, our framework offloads only the residual minimisation to a quantum backend while retaining classical enforcement of boundary conditions. Two paradigms are considered: (i) gate-based residual minimisation on CUDA-Q using variational circuits to evaluate a Cubic Unconstrained Binary Optimisation (CUBO) cost, which naturally arises from the discretisation, and (ii) a Quadratic Unconstrained Binary Optimisation (QUBO) reformulation, which is required for execution on a quantum annealer, executed via a classical–quantum mapping. We further explore a CUBO extension on CUDA-Q and direct residual-to-energy mapping on annealers. Benchmarks confirm that the classical solver reproduces the analytic solution with spectral accuracy; among quantum-enhanced methods, the annealer-based QUBO yields the closest approximation. The gate-based CUBO solver improves upon a legacy variational baseline but exhibits a small interior bias due to limited circuit depth and precision. These findings underscore the complementary roles of annealers and gate-based devices in hybrid scientific computing and demonstrate a feasible workflow for the NISQ era rather than a speedup over classical methods. Recent progress in quantum algorithms for differential equations signals a rapidly maturing field with significant potential for practical quantum advantage.
Quantum algorithms offer an exponential advantage with respect to the number of dependent variables for solving certain nonlinear ordinary differential equations (ODEs). These algorithms typically begin by transforming the original nonlinear ODE into a higher-dimensional linear ODE using a linearization technique, most commonly Carleman linearization. Existing works restrict their analysis to ODEs where the nonlinearities are polynomial functions of the dependent variables, significantly limiting their applicability. In this work we construct an efficient quantum algorithm for solving ODEs with `Fourier' nonlinear terms expressible as du/dt=G0+G1eiud{\bf u}/dt = G_0 + G_1 e^{i{\bf u}}du/dt=G0​+G1​eiu, where u{\bf u}u denotes a vector of nnn complex variables evolving with ttt, G0G_0G0​ is an nnn-dimensional complex vector, G1G_1G1​ is an n×nn \times nn×n complex matrix and eiue^{i{\bf u}}eiu denotes the vector with entries {eiuj}\{e^{iu_j}\}{eiuj​}. To tackle the Fourier nonlinear term, which is not expressible as a finite sum of polynomials of u{\bf u}u, our algorithm employs a generalization of the Carleman linearization technique known as Koopman linearization. We also make other methodological advances towards relaxing the stringent dissipativity condition required for efficient solution extraction and towards integrated readout of classical quantities from the solution state. Our results open avenues to the development of efficient quantum algorithms for a significantly wider class of high-dimensional nonlinear ODEs, thereby broadening the scope of their applications.
The quantum annealing devices, which encode the solution to a computational problem in the ground state of a quantum Hamiltonian, are implemented in D-Wave systems with more than 2,000 qubits. However, quantum annealing can solve only a classical combinatorial optimization problem such as an Ising model, or equivalently, a quadratic unconstrained binary optimization (QUBO) problem. In this paper, we formulate the QUBO model to solve elliptic problems with Dirichlet and Neumann boundary conditions using the finite element method. In this formulation, we develop the objective function of quadratic binary variables represented by qubits and the system finds the binary string combination minimizing the objective function globally. Based on the QUBO formulation, we introduce an iterative algorithm to solve the elliptic problems. We discuss the validation of the modeling on the D-Wave quantum annealing system.