Circle fit optimization for resonator quality factor measurements: point redistribution for maximal accuracy

The control of material loss mechanisms is playing an increasingly important role for improving coherence times of superconducting quantum devices. Such material losses can be characterized through the measurement of planar superconducting resonators, which reflect losses through the resonance's quality factor $Q_l$. The resonance quality factor consists of both internal (material) losses as well as coupling losses when resonance photons escape back into the measurement circuit. The combined losses are then described as $Q_l^{-1} = \mathrm{Re}\{Q_c^{-1}\} + Q_i^{-1}$, where $Q_c$ and $Q_i$ reflect the coupling and internal quality factors of the resonator, respectively. To separate the relative contributions of $Q_i$ and $Q_c$ to $Q_l$, diameter-correcting circle fits use algebraic or geometric means to fit the resonance signal on the complex plane. However, such circle fits can produce varied results, so to address this issue, we use a combination of simulation and experiment to determine the reliability of a fitting algorithm across a wide range of quality factor values from $Q_i\ll Q_c$ to $Q_c\ll Q_i$. In addition, we develop a novel measurement protocol that can not only reduce fitting errors by factors $\gtrsim 2$ but also mitigates the influence of the measurement background on the fit results. This technique can be generalized for other resonance systems beyond superconducting resonators.