Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

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 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.

Automated summary

The study presents a quantum algorithm based on the homotopy perturbation method for solving n-dimensional nonlinear dissipative ordinary differential equations (ODEs), providing exponential improvement over the best classical algorithms or previous quantum algorithms.