We present a method for the numerical analysis of superconducting quantum circuits combining lumped elements, either linear or nonlinear (i.e., Josephson junctions), and distributed coplanar waveguide (CPW) structures. CPW transmission lines and multiline couplers are directly modeled without discretizing them into lumped-element equivalents, and the circuit Hamiltonian parameters are extracted by using the energy-participation ratio (EPR) method. This approach enables fast and accurate extraction of mode frequencies, anharmonicities, cross-Kerr interactions, and Purcell decay rates without relying on full electromagnetic simulations, while naturally accounting for higher-order modes of distributed components. We have implemented the proposed method in a Python framework, QuLTRA (Quantum hybrid Lumped and TRansmission lines circuits Analyzer), which we have used to validate the approach against full electromagnetic simulations (Ansys HFSS, pyEPR), existing circuit-analysis tools (QuCAT), and designs reported in the literature. The comparisons show excellent agreement with orders-of-magnitude reductions in computational time relative to full-wave solvers. We demonstrate applications including Purcell-protected readout, multimode ultrastrong coupling, and multiplexed qubit readout, illustrating how the method can support fast and reliable early-stage circuit design.
We present a method for the numerical analysis of superconducting quantum circuits combining lumped elements, either linear or nonlinear (i.e., Josephson junctions), and distributed coplanar waveguide (CPW) structures. CPW transmission lines and multiline couplers are directly modeled without discretizing them into lumped-element equivalents, and the circuit Hamiltonian parameters are extracted by using the energy-participation ratio (EPR) method. This approach enables fast and accurate extraction of mode frequencies, anharmonicities, cross-Kerr interactions, and Purcell decay rates without relying on full electromagnetic simulations, while naturally accounting for higher-order modes of distributed components. We have implemented the proposed method in a Python framework, QuLTRA (Quantum hybrid Lumped and TRansmission lines circuits Analyzer), which we have used to validate the approach against full electromagnetic simulations (Ansys HFSS, pyEPR), existing circuit-analysis tools (QuCAT), and designs reported in the literature. The comparisons show excellent agreement with orders-of-magnitude reductions in computational time relative to full-wave solvers. We demonstrate applications including Purcell-protected readout, multimode ultrastrong coupling, and multiplexed qubit readout, illustrating how the method can support fast and reliable early-stage circuit design.
Superconducting circuits are being used for large-scale quantum devices, and a major challenge is to perform accurate numerical simulations of device parameters. One of the most advanced methods for analyzing superconducting circuit designs is the energy-participation-ratio (EPR) method, which constructs quantum Hamiltonians based on the energy distribution extracted from classical electromagnetic simulations. In the EPR approach, we extract linear terms from finite-element simulations and add nonlinear terms using the energy participation ratio extracted from the classical simulations. However, the EPR method relies on a low-order expansion of nonlinear terms, which is prohibitive for accurately describing highly anharmonic circuits. An example of such a circuit is the fluxonium qubit, which has recently attracted increasing attention due to its high lifetimes and low error rates. In this work, we extend the EPR approach to effectively address highly nonlinear superconducting circuits, and, as a proof of concept, we apply our approach to a fluxonium qubit. Specifically, we design, fabricate, and experimentally measure a fluxonium qubit coupled to a readout resonator. We compare the measured frequencies of both the qubit and the resonator to those extracted from the EPR analysis, and we find an excellent agreement. Furthermore, we compare the dispersive shift as a function of external flux obtained from experiments with our EPR analysis and a simpler lumped-element model. Our findings reveal that the EPR results closely align with the experimental data, providing more accurate estimations compared to the simplified lumped-element simulations.
Superconducting quantum circuit design is reliant on accurately mapping design parameters to a quantum Hamiltonian. Designers typically rely on computationally intensive finite-element electromagnetic simulations. However, effective circuit-level models can in principle capture much of the relevant physics for these systems, reducing the need for finite-element simulations. A barrier to implementing these simpler models has been the prevalence of distributed elements such as coplanar waveguides resonators in device designs. Techniques for modeling these distributed elements have been developed, but may be difficult to scale, reduce intuition, and give inaccurate results under strong coupling. In this work we describe a simple effective circuit-level model that faithfully reproduces the scattering parameters, and subsequently can predict certain Hamiltonian parameters, for a coupled, distributed element within a broader two-port network even up to strong coupling. Our approach does not explicitly require an electromagnetic simulation. Our model, along with other common lumped models used in black box quantization, is publicly available to researchers via an open-source code package, simpleLOMs.
Abstract Superconducting microwave circuits incorporating nonlinear devices, such as Josephson junctions, are a leading platform for emerging quantum technologies. Increasing circuit complexity further requires efficient methods for the calculation and optimization of the spectrum, nonlinear interactions, and dissipation in multi-mode distributed quantum circuits. Here we present a method based on the energy-participation ratio (EPR) of a dissipative or nonlinear element in an electromagnetic mode. The EPR, a number between zero and one, quantifies how much of the mode energy is stored in each element. The EPRs obey universal constraints and are calculated from one electromagnetic-eigenmode simulation. They lead directly to the system quantum Hamiltonian and dissipative parameters. The method provides an intuitive and simple-to-use tool to quantize multi-junction circuits. We experimentally tested this method on a variety of Josephson circuits and demonstrated agreement within several percents for nonlinear couplings and modal Hamiltonian parameters, spanning five orders of magnitude in energy, across a dozen samples.
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.
Time-dependent electromagnetic drives are fundamental for controlling complex quantum systems, including superconducting Josephson circuits. In these devices, accurate time-dependent Hamiltonian models are imperative for predicting their dynamics and designing high-fidelity quantum operations. Existing numerical methods, such as black-box quantization (BBQ) and energy-participation ratio (EPR), excel at modeling the static Hamiltonians of Josephson circuits. However, these techniques do not fully capture the behavior of driven circuits stimulated by external microwave drives, nor do they include a generalized approach to account for the inevitable noise and dissipation that enter through microwave ports. Here, we introduce numerical techniques that leverage classical microwave simulations, efficiently executable in finite-element solvers, to obtain the time-dependent Hamiltonian of microwave-driven superconducting circuits with arbitrary geometries under charge, flux, or mixed electromagnetic modulation. Importantly, our techniques do not rely on a lumped-element description of the superconducting circuit, in contrast to previous approaches to tackling this problem. We demonstrate the versatility of our approach by characterizing the driven properties of realistic circuit devices in complex electromagnetic environments, including coherent dynamics due to charge and flux modulation, as well as drive-induced relaxation and dephasing. Our techniques offer a powerful toolbox for optimizing circuit designs and advancing practical applications in superconducting quantum computing.
In the design of superconducting quantum chips, the control and readout of the quantum bit (qubit) are realized through interactions with other components in the circuit, such as drive lines, flux lines and readout resonator. Depending on the design, these interactions employ either an inductive or capacitive coupling scheme. Note that every interaction governs critical performance metrics for the chip, including gate fidelity, readout speed, and coherence times, a more effective and precise approach to extract the strength of the coupling is necessary. For most qubit designs, the structure of the qubit typically remains in the electrically small regime. Therefore, simulation software based on static fields and the Method of Moments (MoM), such as Q3D, is still applicable in most instances. However, as the number of qubits on a chip increases, Q3D faces challenges in effectively addressing the complex coupling and crosstalk issues present on the chip. In this work, we extend the lumped-element-based qubit model by using microwave network parameters and thus enable the simulation of the qubit for the Finite Element Method (FEM)-based 3D electromagnetic simulation software, such as HFSS. This approach is suitable for the determination of the capacitive coupling strength between the qubit and the drive line, as well as readout circuits. Furthermore, to address the inductive coupling between the qubit and the flux line, we propose a method in this work that directly computes the mutual inductance, which is implemented by determining the magnetic flux through the DC SQUID area, generated by the current port excitation in HFSS. The proposed methodologies not only effectively extracts the key qubit design parameters, but also offer broader applicability for analysing complex crosstalk in multi-qubit chip designs.