Exponential Fitting Techniques for the Numerical Simulation of Quantum Devices

ABSTRACT In recent years, the rapid advancement of quantum computing \cite{AA} has driven the need for numerical tools capable of accurately modeling the devices that enable its implementation. Among these, Josephson junctions play a crucial role in quantum computing, where they are used to realize qubits, and are also employed in practical applications such as parametric amplifiers, which are essential in quantum technologies for qubit control and readout \cite{BB, CC}. However, their dynamics are governed by nonlinear differential equations with highly oscillatory solutions, posing significant challenges for standard numerical methods. This research introduces a numerical method based on exponential fitting theory \cite{DD, EE} to address the challenges associated with the simulation of Josephson junctions. The developed method follows a predictor-corrector approach, incorporating both an explicit and an implicit formula, and leverages prior knowledge of key system parameters, such as oscillation frequency and damping coefficient, to construct a functional basis suitable for problem discretization. Tests conducted on various scenarios with strongly oscillatory solutions have demonstrated the effectiveness of this approach, achieving high accuracy in solving the differential equations governing Josephson junction dynamics. This study contributes to the development of specialized numerical tools for the modeling of superconducting devices and the improvement of simulation techniques in quantum computing.