Experimentally Verified, Fast Analytic, and Numerical Design of Superconducting Resonators in Flip-Chip Architectures

In superconducting quantum processors, the predictability of device parameters is of increasing importance as many laboratories scale up their systems to larger sizes in a 3-D-integrated architecture. In particular, the properties of superconducting resonators must be controlled well to ensure high-fidelity multiplexed readout of qubits. Here, we present a method, based on conformal mapping techniques, to predict a resonator's parameters directly from its 2-D cross-section, without computationally heavy and time-consuming 3-D simulation. We demonstrate the method's validity by comparing the calculated resonator frequency and coupling quality factor with those obtained through 3-D finite-element-method simulation and by measurement of 15 resonators in a flip-chip-integrated architecture. We achieve a discrepancy of less than 2% between designed and measured frequencies for 6-GHz resonators. We also propose a design method that reduces the sensitivity of the resonant frequency to variations in the interchip spacing.