Numerical solution of nonlinear Schrödinger equation by a hybrid pseudospectral-variational quantum algorithm

Area of Science:

• Quantum computing
• Computational physics
• Nonlinear dynamics

Background:

• The nonlinear Schrödinger equation (NLSE) models various wave phenomena.
• Accurate numerical solutions are crucial for understanding soliton propagation.
• Classical methods can face limitations in stability and efficiency.

Purpose of the Study:

• To develop and analyze a hybrid pseudospectral-variational quantum algorithm for the time-dependent 1D NLSE.
• To assess the algorithm's accuracy, stability, and efficiency compared to classical approaches.
• To investigate the impact of quantum circuit parameters on simulation outcomes.

Main Methods:

• A hybrid quantum algorithm combining pseudospectral and variational steps.
• Classical computation of Fourier transforms for the Hamiltonian term.
• First-order explicit time stepping for the nonlinear term within a variational block.
• Analysis of ansatz circuit expressibility and algorithm parameter influence.

Main Results:

• The quantum algorithm accurately reproduces analytical solutions for propagating solitons.
• A small root mean square error was achieved over extended time intervals.
• The method avoids numerical instabilities associated with higher-order integration schemes.
• Comparison with classical methods highlights the quantum approach's potential.

Conclusions:

• The hybrid pseudospectral-variational quantum algorithm offers a stable and accurate method for solving the NLSE.
• This approach demonstrates the potential of quantum computation for complex nonlinear physics problems.
• Further investigation into algorithm parameters can optimize performance for specific applications.