A Surface Integral Equation Method for the Superconducting Quantum Device

Computing the Hamiltonian of a superconducting quantum device from its layout is a necessary step for making use of the device as a quantum processor. A conventional approach is to use the finite element method (FEM) to extract the linear capacitance matrix from the electrostatic simulation for the circuit quantization. However, this method is computationally intensive for large-scale systems. In particular, the calculations of the surface losses, quantified by the participation ratios, have been computationally challenging for arbitrary designs due to the difficulties to capture the singular electrical field in such a multi-scale problem. Here, we apply a much more accurate and efficient method, the surface integral equation (SIE) method. It only requires the discretization of the conductive surfaces, as opposed to the discretization of the truncated volume in the FEM. Consequently, the number of unknowns is greatly reduced, resulting in a significant saving in the simulation time. We further introduce a non-conformal mesh scheme to reinforce a large mesh density near the superconductor boundaries without affecting the interior mesh density. As a result, the electric field singularity is much better captured with only slight increases in the computational time and the memory usages. We benchmark our method with an analytically solvable coplanar capacitor. To achieve the same accuracy in the computation of the capacitance matrix and the participation ratios, our method is accelerated by tens or hundreds of times respectively, compared to the FEM simulations. This method paves the way for the accurate and efficient design optimization of the superconducting quantum devices.