Opportunities and Challenges of Computational Electromagnetics Methods for Superconducting Circuit Quantum Device Modeling: A Practical Review

TL;DR

This review addresses the challenge of modeling superconducting circuit quantum devices with computational electromagnetics (CEM) by surveying the major numerical methods (FEM, MoM, FDM, and hybrids) and their suitability for multiscale, unconventional-material structures typical of cQED. It explains how matrix solvers, conditioning, and eigenvalue extraction influence the reliability and efficiency of simulations, and it highlights the predominance of FEM in current practice while recognizing the potential of MoM with fast algorithms and PEEC in large-scale or specialized scenarios. The paper also discusses practical considerations such as higher-order basis functions, adaptive mesh refinement, and the multiscale difficulties posed by thin superconducting layers and nanoscale features, offering concrete guidance for improving simulations and reducing gaps between design and measurement. Finally, it outlines future directions including material benchmarking, advanced preconditioning, domain-decomposition strategies, and design-for-simulability approaches to enable scalable, accurate modeling of increasingly complex cQED devices with CEM tools.

Abstract

High-fidelity numerical methods that model the physical layout of a device are essential for the design of many technologies. For methods that characterize electromagnetic effects, these numerical methods are referred to as computational electromagnetics (CEM) methods. Although the CEM research field is mature, emerging applications can still stress the capabilities of the techniques in use today. The design of superconducting circuit quantum devices falls in this category due to the unconventional material properties and important features of the devices covering nanometer to centimeter scales. Such multiscale devices can stress the fundamental properties of CEM tools which can lead to an increase in simulation times, a loss in accuracy, or even cause no solution to be reliably found. While these challenges are being investigated by CEM researchers, knowledge about them is limited in the broader community of users of these CEM tools. This review is meant to serve as a practical introduction to the fundamental aspects of the major CEM techniques that a researcher may need to choose between to model a device, as well as provide insight into what steps they may take to alleviate some of their challenges. Our focus is on highlighting the main concepts without rigorously deriving all the details, which can be found in many textbooks and articles. After covering the fundamentals, we discuss more advanced topics related to the challenges of modeling multiscale devices with specific examples from superconducting circuit quantum devices. We conclude with a discussion on future research directions that will be valuable for improving the ability to successfully design increasingly more sophisticated superconducting circuit quantum devices. Although our focus and examples are taken from this area, researchers from other fields will still benefit from the details discussed here.

Figures (13)

• Figure 1: Notional computation times for different algorithms with varying computational complexities using a single-core processor (adapted from jin2011theory). While parallelization can make solving problems with billions of unknowns possible, this only occurs for algorithms with suitable computational complexity, which is usually $\sim \! O(N\log N)$ for practical algorithms.
• Figure 2: Example of an FDM grid using non-uniform spatial step sizes. A discretized cylinder is shown in solid black, where the staircasing approximation is apparent. The requirement for rectangular cells also forces the mesh to change along only one coordinate axis at a time, causing the "bands" of highly refined cells even far outside of the object of interest.
• Figure 3: Example of a vector basis function, often called a vector edge element, for a 2D triangular mesh (left) and in a single tetrahedron (right). The edge the basis function is associated with is shown as a dashed red line. A critical property of this basis function is it has a constant tangential value along the edge it is associated with and is purely normal to all other edges of the mesh element. This allows the tangential continuity of fields to be exactly captured. Further, the normal components across a boundary can be discontinuous, which also preserves the correct EM boundary conditions.
• Figure 4: Examples of how specialized Green's functions can reduce the number of unknowns for different problems. (Top) A cross-sectional view of a multi-layer circuit where a Green's function for layered media can be used so that only the circuit traces require basis functions. (Bottom) A top-view of a coplanar waveguide structure where a different specialized Green's function can be used so that only the gaps between conductors require basis functions.
• Figure 5: Example of a "partitioned mesh" of a coplanar waveguide resonator with airbridges distributed along the resonator. For visualization, only the mesh on the surface of the device is shown. The mesh is partitioned into eight sub-domains (denoted by different colors) with approximately the same number of tetrahedrons in each using METIS karypis1998fast. Although the size of this mesh does not require a domain decomposition method to be utilized, such mesh partitioning is a critical step in such methods, as well as for other uses such as parallelizing preconditioners.