A Versatile Analytical Model for Fast and Accurate Determination of Feedline-Coupled Resonators for Superconducting Qubit Readout

Superconducting quantum chips commonly utilize quarter-wavelength (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda /4$</tex-math> </inline-formula>) transmission line resonators as readout circuits. An analytical model for the accurate determination of resonance frequencies and coupling <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i>-factors of feedline-coupled superconducting resonators is introduced. The model leverages four-port microwave network analysis, integrating boundary conditions and conformal mapping techniques to compute even- and odd-mode impedances in edge-coupled coplanar waveguide (CPW) structures. Its versatility allows application to both planar and 3-D heterogeneous architectures, making it a powerful tool for resonator design. To validate the model, a test chip with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda /4$</tex-math> </inline-formula> resonators of varying geometries is fabricated and measured in a cryogenic environment. Comparisons with finite element method (FEM) simulations and experimental measurements confirm the model’s accuracy, with resonance frequencies and coupling <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i>-factors aligning closely across configurations. This proposed model facilitates the design of superconducting resonators in readout circuits for more effective, scalable, and adaptable quantum computing architectures.