An accurate and efficient numerical electromagnetic model for superconducting qubits is essential for characterizing and minimizing design-dependent dielectric losses. The energy participation ratio (EPR) is the commonly adopted metric used to evaluate these losses, but its calculation presents a severe multiscale computational challenge. Conventional finite element method (FEM) requires 3D volumetric meshing, leading to prohibitive computational costs and memory requirements when attempting to capture singular electric fields at nanometer-thin material interfaces. To address this bottleneck, we propose SesQ, a surface integral equation simulator tailored for the precise simulation of the EPR. By applying discretization on 2D surfaces, deriving a semi-analytical multilayer Green's function, and employing a dedicated non-conformal boundary mesh refinement scheme, SesQ accurately resolves singular edge fields without an explosive growth in the number of unknowns. Validations with analytically solvable models demonstrate that SesQ accelerates capacitance extraction by roughly two orders of magnitude compared to commercial FEM tools. While achieving comparable accuracy for capacitance extraction, SesQ delivers superior precision for EPR calculation. Simulations of practical transmon qubits further reveal that FEM approaches tend to significantly underestimate the EPR. Finally, the high efficiency of SesQ enables rapid iteration in the layout optimization, as demonstrated by minimizing the EPR of the qubit pattern, establishing the simulator as a powerful tool for the automated design of low-loss superconducting quantum circuits.
An accurate and efficient numerical electromagnetic model for superconducting qubits is essential for characterizing and minimizing design-dependent dielectric losses. The energy participation ratio (EPR) is the commonly adopted metric used to evaluate these losses, but its calculation presents a severe multiscale computational challenge. Conventional finite element method (FEM) requires 3D volumetric meshing, leading to prohibitive computational costs and memory requirements when attempting to capture singular electric fields at nanometer-thin material interfaces. To address this bottleneck, we propose SesQ, a surface integral equation simulator tailored for the precise simulation of the EPR. By applying discretization on 2D surfaces, deriving a semi-analytical multilayer Green's function, and employing a dedicated non-conformal boundary mesh refinement scheme, SesQ accurately resolves singular edge fields without an explosive growth in the number of unknowns. Validations with analytically solvable models demonstrate that SesQ accelerates capacitance extraction by roughly two orders of magnitude compared to commercial FEM tools. While achieving comparable accuracy for capacitance extraction, SesQ delivers superior precision for EPR calculation. Simulations of practical transmon qubits further reveal that FEM approaches tend to significantly underestimate the EPR. Finally, the high efficiency of SesQ enables rapid iteration in the layout optimization, as demonstrated by minimizing the EPR of the qubit pattern, establishing the simulator as a powerful tool for the automated design of low-loss superconducting quantum circuits.
Dielectric loss due to two-level systems is a limiting factor for superconducting qubit relaxation times. These losses arise mostly from nanometer-scale interfacial defect regions in superconducting devices with planar dimensions of microns to millimeters, thus making it resource intensive to accurately simulate the electric field density in these regions with traditional electromagnetic solvers. In this work, we demonstrate a fast boundary integral equation solver that allows precise simulation of electric field density in these thin regions, showing a speedup of around two orders of magnitude over traditional solvers, with relative errors around 10−7 for a ten-minute solution runtime. By computing participation ratios through Green's first identity without squaring the electric field, our approach is less susceptible to the field singularities near conductor corners. We apply this solver to a basic untrenched coplanar waveguide cross-section, showing that the common assumption of participation ratio linearity with dielectric constant holds well for some interfaces and not others; in particular, while the metal-air (MA) top and corner follow this linear relationship strongly, the MA sidewall does not. We then compare isotropic and anisotropic etching, showing that the MA sidewall and the metal-air-substrate triple junction are the most strongly affected. We are currently leveraging this solver to explore geometries that will uniquely isolate the participation ratios of the different dielectrics. Finally, we are working to combine this solver framework with a full 3D microwave solver to accurately calculate participation ratios for the thin dielectrics that are known sources of loss in superconducting qubits.
Superconducting circuits are being used for large-scale quantum devices, and a major challenge is to perform accurate numerical simulations of device parameters. One of the most advanced methods for analyzing superconducting circuit designs is the energy-participation-ratio (EPR) method, which constructs quantum Hamiltonians based on the energy distribution extracted from classical electromagnetic simulations. In the EPR approach, we extract linear terms from finite-element simulations and add nonlinear terms using the energy participation ratio extracted from the classical simulations. However, the EPR method relies on a low-order expansion of nonlinear terms, which is prohibitive for accurately describing highly anharmonic circuits. An example of such a circuit is the fluxonium qubit, which has recently attracted increasing attention due to its high lifetimes and low error rates. In this work, we extend the EPR approach to effectively address highly nonlinear superconducting circuits, and, as a proof of concept, we apply our approach to a fluxonium qubit. Specifically, we design, fabricate, and experimentally measure a fluxonium qubit coupled to a readout resonator. We compare the measured frequencies of both the qubit and the resonator to those extracted from the EPR analysis, and we find an excellent agreement. Furthermore, we compare the dispersive shift as a function of external flux obtained from experiments with our EPR analysis and a simpler lumped-element model. Our findings reveal that the EPR results closely align with the experimental data, providing more accurate estimations compared to the simplified lumped-element simulations.
Superconducting qubits are sensitive to a variety of loss mechanisms, which include dielectric loss from interfaces. The calculation of participation near the key interfaces of planar designs can be accomplished through an analytical description of the electric field density based on conformal mapping. In this way, a 2-D approximation to coplanar waveguide and capacitor designs produces values of the participation as a function of depth from the top metallization layer as well as the volume participation within a given thickness from this surface by reducing the problem to a surface integration over the region of interest. These quantities are compared to finite-element method numerical solutions, which validate the values at large distances from the coplanar metallization but diverge near the edges of the metallization features due to the singular nature of the electric fields. A simple approximation to the electric field energy at shallow depths (relative to the waveguide width) is also presented that closely replicates the numerical results based on conformal mapping and those reported in prior literature. These techniques are applied to the calculation of surface participation within a transmon qubit design, where the effects due to shunting capacitors can be easily integrated with those associated with metallization comprising the local environment of the qubit junction.
Electromagnetic simulations form an indispensable part of the design and optimization process for superconducting quantum devices. Although several commercial platforms exist, open-source alternatives optimized for high-performance computing remain limited. To address this gap, we introduce SQDMetal, a Python-based API that integrates Qiskit Metal (IBM), Gmsh, Palace (AWS), and Paraview (Kitware) into an open-source, highly parallel simulation workflow for superconducting quantum circuits. SQDMetal enables accurate, efficient, and scalable simulations while remaining community-driven and free from commercial constraints. In this work, we validate SQDMetal through mesh convergence studies which benchmark SQDMetal against COMSOL Multiphysics and Ansys, demonstrating excellent agreement for both eigenmode and electrostatic (capacitance) simulations. Furthermore, we simulate superconducting resonators and transmon qubits, showing reasonable agreement with experimental measurements. SQDMetal also supports advanced capabilities, including Hamiltonian extraction via the energy participation ratio (EPR) method, incorporation of kinetic inductance effects, and full 3D modelling of device geometry for improved predictive accuracy. By unifying open-source tools into a single framework, SQDMetal lowers the barriers to entry for community members seeking to access high-performance simulations to assist in the design and optimization of their devices.
Abstract Energy participation ratio (EPR) at the metal-substrate (MS) interface is a dominant decoherence source limiting the coherence time ( T 1 ) of planar superconducting transmon qubits. Traditional EPR simulations rely on idealized geometric models, ignoring process-induced morphological deviations that significantly impact actual device performance. To bridge this gap, we propose a unified data-driven framework. It automatically extracts geometric parameters from SEM cross-sections using computer vision (CV). These parameters are then imported into a calibrated 2D EPR simulation model. Sobol global sensitivity analysis is further applied to quantify the influences of key parameters. Parametric simulations reveal that the EPR scaling factor F ms generally decreases with increasing interface thickness, substrate over-etch depth, and trench angle, driven by mitigated edge electric field crowding. Sobol analysis further identifies substrate over-etch depth and trench angle as the dominant drivers of F ms , with a significant synergistic interaction between these two parameters. Evaluations across eight process-feasible geometric configurations indicate that thin interface layer combined with a deep substrate over-etch yields the largest predict reduction in interface EPR, corresponding to a projected relative enhancement in T 1 under the assumption of a constant interface loss tangent. We derive actionable process guidelines for planar Al-on-Si transmon qubits, prioritizing the co-tuning of trench parameters to balance MS-interface EPR reduction with manufacturing feasibility within the investigated parameter range. This framework provides a systematic pipeline for incorporating process-induced geometric deviations into EPR prediction, and offers a design-stage simulation tool applicable to planar superconducting structures with similar coplanar edge-dominated EPR characteristics.
We study the energy relaxation times (T1) of superconducting transmon qubits in 3D cavities as a function of dielectric participation ratios of material surfaces. This surface participation ratio, representing the fraction of electric field energy stored in a dissipative surface layer, is computed by a two-step finite-element simulation and experimentally varied by qubit geometry. With a clean electromagnetic environment and suppressed non-equilibrium quasiparticle density, we find an approximately proportional relation between the transmon relaxation rates and surface participation ratios. These results suggest dielectric dissipation arising from material interfaces is the major limiting factor for the T1 of transmons in 3D circuit quantum electrodynamics architecture. Our analysis also supports the notion of spatial discreteness of surface dielectric dissipation.