SesQ: A Surface Electrostatic Simulator for Precise Energy Participation Ratio Simulation in Superconducting Qubits

Abstract

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.

Figures (14)

• Figure 1: Schematic of an $L$-layer structure defined in cylindrical coordinates $(\rho, z)$. The system consists of multiple regions with varying permittivities (from $\epsilon_1$ to $\epsilon_L$), bounded by a semi-infinite top layer and bottom layer. A superconductor surface is situated at the interface $z = 0$, where a source point charge $q(\rho^\prime,z^\prime)$ is placed. The electrostatic potential $\phi(\rho, z)$ can be calculated at any location in the system, depicted here as being inside the $l$-th layer.
• Figure 2: The cross-section of the dielectric loss interfaces. The structure consists of a substrate-metal interface (SM) sandwiched between a substrate and a superconducting metal conductor. The metal conductor is encapsulated by metal-air interfaces (MA, MA-I, and MA-II), while the surrounding regions are covered by substrate-air interfaces (SA). The coordinate system indicates the $x$, $y$, and $z$ axes, with the $y$-axis oriented into the plane.
• Figure 3: The simplified cross-section for the simulation. The thickness of the metal is approximated as zero. The dielectric constants of the SM and SA interfaces are assumed to be the same as the substrate, while the MA interface is assumed to be air.
• Figure 4: Homogeneous mesh refinement on a boundary triangle. Each triangle is refined into four congruent triangles, where the extra three vertices reside at the center of the edges. Multiple levels of refinement can be applied.
• Figure 5: Boundary layer refinement of a boundary triangle. The height of each refined triangle $t_i$ decreases exponentially to capture the singularity accurately.