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arXiv:2607.00490 · IEEE Transactions on Microwave Theory and Techniques

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

Zhen Luo, Lea Richard, Ivan Tsitsilin, Christian M. F. Schneider, Marco Dietz, Stefan Filipp, Amelie Hagelauer

quant-phSubmitted Jul 1, 2026• Announced Jul 2, 2026• 1 citation
arXivHTMLPDF

Abstract

Superconducting quantum chips commonly utilize quarter-wavelength (λ/4) transmission line resonators as readout circuits. An analytical model for the accurate determination of resonance frequencies and coupling Q-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 λ/4 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 Q-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.

Journal reference
IEEE Transactions on Microwave Theory and Techniques, vol. 73, no. 10, pp. 8059-8070, Oct. 2025
DOI
10.1109/TMTT.2025.3578414
Comments
12 pages, 12 figures

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Superconducting quantum chips commonly utilize quarter-wavelength (λ/4) transmission line resonators as readout circuits. An analytical model for the accurate determination of resonance frequencies and coupling Q-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 λ/4 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 Q-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.
In superconducting quantum processors, the predictability of device parameters is of increasing importance as many labs scale up their systems to larger sizes in a 3D-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 2D cross-section, without computationally heavy and time-consuming 3D simulation. We demonstrate the method's validity by comparing the calculated resonator frequency and coupling quality factor with those obtained through 3D 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 inter-chip spacing.
In this paper, we present a novel design of a compact, double-fold half-wave superconducting coplanar waveguide resonator, specifically optimized for quantum readout. The geometry offers advantages in terms of scalability and allows for a more compact integration of qubits per unit area. The superconducting resonators are designed to operate within a fundamental frequency range of 1.9 GHz$-1.95GHz,utilizingbothgapandfingercouplingschemes.Thedevicesarefabricatedusingthinaluminumfilmsdepositedonhigh−resistivitysapphiresubstrates.Theproposedresonatorsexhibitqualityfactor(Q1.95 GHz, utilizing both gap and finger coupling schemes. The devices are fabricated using thin aluminum films deposited on high-resistivity sapphire substrates. The proposed resonators exhibit quality factor (Q1.95GHz,utilizingbothgapandfingercouplingschemes.Thedevicesarefabricatedusingthinaluminumfilmsdepositedonhigh−resistivitysapphiresubstrates.Theproposedresonatorsexhibitqualityfactor(Q_{L})of2.9×) of 2.9 ×)of2.9× 10^{5}$and 1.58 ×10310^{3}103for gap coupled and finger coupled, respectively, during experimental measurements. The simulated QL_{L}L​of single-folded meander line resonators are found to be 4.65 ×10510^{5}105and 258.42, for gap coupled and finger coupled, respectively, for comparison purpose at similar operating frequency. A comprehensive analysis of the microwave transmission spectra was conducted at cryogenic temperatures near 10 mK, across very high input drive powers ($-30dBmto30 dBm to30dBmto-10dBm).Experimentalresultsshowagoodagreementwithsimulationsandtheequivalentlumpedcircuitmodel.Notably,theproposeddesignreducesthephysicalfootprintoftheresonatordevicesupto28.210 dBm). Experimental results show a good agreement with simulations and the equivalent lumped circuit model. Notably, the proposed design reduces the physical footprint of the resonator devices up to 28.210dBm).Experimentalresultsshowagoodagreementwithsimulationsandtheequivalentlumpedcircuitmodel.Notably,theproposeddesignreducesthephysicalfootprintoftheresonatordevicesupto28.2\%comparedtoconventionalsingle−foldmeanderedstructures.TheresonatorsdemonstratestrongpotentialforintegratingqubitswithreadoutandcontrolbusesinquantuminformationprocessingwhichrequireaQcompared to conventional single-fold meandered structures. The resonators demonstrate strong potential for integrating qubits with readout and control buses in quantum information processing which require a Qcomparedtoconventionalsingle−foldmeanderedstructures.TheresonatorsdemonstratestrongpotentialforintegratingqubitswithreadoutandcontrolbusesinquantuminformationprocessingwhichrequireaQ_{L}$between10410^{4}104−-−10510^{5}105. They also being well-suited for cryogenic detector applications that utilize microwave kinetic inductance detector (MKID) arrays in frequency multiplexing for different coupling schemes.
We successfully designed and fabricated an absorption-type of superconducting coplanar waveguide (CPW) resonators. The resonators are made from a niobium film (about 160 nm thick) on a high-resistance Si substrate, and each resonator is fabricated as a meandered quarter-wavelength transmission line (one end is short to the ground and another end is capacitively coupled to a through feedline). With a vector network analyzer we measured the transmissions of the applied microwave through the resonators at ultra-low temperature. The obtained loaded quality factors are significantly high, i.e. up to ∼10 6 . When the temperature increases slowly from the base temperature (20 mK), the resonance frequencies of the resonators are blue shifted and the quality factors are lowered slightly. In principle, this type of device can integrate a series of CPW resonators with a common feedline, making it a promising candidate as the data bus for coupling distant solid-state qubits and the sensitive detector of single photons.
Superconducting resonators are powerful tools widely used in circuit Quantum Electrodynamics [1]. High-quality factors, tunability, and several implemented layouts make them ideal circuital elements for various applications from quantum metrology [2], to quantum computation [3] and quantum information [4]. Here, we report on the simulation and design of superconducting Coplanar Waveguide (CPW) resonators whose resonance frequencies are in the range of 4–8 GHz. Typical properties of superconducting resonators employed in standard transmon qubits can be achieved by designing a λ/4 resonator made by Niobium (Nb) and modeling the capacitive coupling to provide a high-quality factor useful for its future employments in more advanced quantum circuits [5]. Nb gives the advantage of better stability of the superconducting properties of the material against the application of a magnetic field, compared those achievable by Aluminum (Al), thus guaranteeing an additional degree of freedom in the implementation of complex quantum architectures [5]–[7].
We investigate analytically the coupling of a coplanar waveguide resonator to a coplanar waveguide feedline. Using a conformal mapping technique we obtain an expression for the characteristic mode impedances and coupling coefficients of an asymmetric multi-conductor transmission line. Leading order terms for the external quality factor and frequency shift are calculated. The obtained analytical results are relevant for designing circuit-QED quantum systems and frequency division multiplexing of superconducting bolometers, detectors and similar microwave-range multi-pixel devices.
The use of a transmission line resonator offers the possibility for different sensor readout methods, e.g. flux qubit and kinetic inductance detectors. Both cases require a high quality factor, Q L , of the cavity circuit to detect small deviations in the resonance frequency. Various superconducting niobium coplanar waveguide resonators of λ/4 and λ/2 wavelength for 3 and 6 GHz were developed. Different layouts for the coplanar resonators were designed, simulated and measured on common silicon and sapphire substrates at 4.2 K. We found a very good agreement between simulation and measurement results, with Q L between 10 4 and 10 5 at helium temperatures. The presented results show the dependences on the chosen cross section geometries, the coupling factor and the complex layout of the transmission line.