Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

Abstract

While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. In our work, based on the homotopy perturbation method, we propose a quantum algorithm for solving $n$-dimensional nonlinear dissipative ordinary differential equations (ODEs). Our algorithm first converts the original nonlinear ODEs into the other nonlinear ODEs which can be embedded into finite-dimensional linear ODEs. Then we solve the embedded linear ODEs with quantum linear ODEs algorithm and obtain a state $\epsilon$-close to the normalized exact solution of the original nonlinear ODEs with success probability $\Omega (1)$. The complexity of our algorithm is $O(g\eta T\mathrm{poly}(\log (nT/\epsilon )))$, where $\eta$, $g$ measure the decay of the solution. Our algorithm provides exponential improvement over the best classical algorithms or previous quantum algorithms in $n$ or $\epsilon$.

arXiv comment26 pages, published in New Journal of Physics

References in corpus (21)

• Quantum algorithm for solving linear systems of equations
• Quantum Amplitude Amplification and Estimation
• Quantum algorithm for systems of linear equations with exponentially improved dependence on precision
• Variational quantum algorithms for nonlinear problems
• Preconditioned quantum linear system algorithm
• High-order quantum algorithm for solving linear differential equations
• Efficient quantum algorithm for dissipative nonlinear differential equations
• Quantum algorithm for linear differential equations with exponentially improved dependence on precision
• Solving nonlinear differential equations with differentiable quantum circuits
• Quantum algorithms and the finite element method
• Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing
• Quantum algorithm and circuit design solving the Poisson equation
• Quantum Algorithm for Simulating the Wave Equation
• Quantum spectral methods for differential equations
• Koopman-von Neumann Approach to Quantum Simulation of Nonlinear Classical Dynamics
• Quantum algorithm for non-homogeneous linear partial differential equations
• A Quantum Algorithm for Solving Linear Differential Equations: Theory and Experiment
• Quantum Algorithm for the Vlasov Equation
• Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms
• Explicit Error Bounds for Carleman Linearization
• Efficient state initialization by a quantum spectral filtering algorithm

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