A method for the numerical analysis of hybrid lumped-distributed superconducting quantum circuits

TL;DR

The paper presents a general, efficient framework for numerically analyzing superconducting quantum circuits that combine lumped elements and distributed CPW structures by modeling the linear network with admittance matrices and extracting quantum parameters via energy participation ratios. It introduces QuLTRA, a Python implementation that directly handles CPW lines and couplers, enabling fast calculation of mode frequencies, anharmonicities, cross-Kerr interactions, and Purcell decay without full electromagnetic simulations. Validations against HFSS/pyEPR and literature demonstrate excellent accuracy with orders-of-magnitude reductions in computation time, and the method supports advanced designs such as Purcell-protected readout, multimode ultra-strong coupling, and multiplexed qubit readout. The approach is generalizable to any linear circuit expressible as a network of multi-port components and offers significant value for rapid, reliable early-stage circuit optimization in circuit QED.

Abstract

We present a method for the numerical analysis of superconducting quantum circuits combining lumped elements, either linear or non-linear (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 ultra-strong coupling, and multiplexed qubit readout, illustrating how the method can support fast and reliable early-stage circuit design.

Figures (13)

• Figure 1: Cross-section of a multi-line CPW structure formed by two lines (L1 and L2) with (b) and without (a) a grounded central line.
• Figure 2: Test structures used for comparison between Ansys+pyEPR and QuLTRA. (a) Transmon qubit capacitively coupled to a $\lambda/2$ resonator taken from Qiskit Metal tutorials. (b) $\lambda/4$ resonator inductively coupled to a feedline terminated with 50-$\Omega$ loads at both ends. The red labels highlight the open/short terminations.
• Figure 3: Capacitance matrix (in fF) of the transmon qubit used in the first test example extracted by Ansys Q3D capacitance extractor. The pad labels correspond to the ones indicated in Fig. \ref{['qubit_lambda_a']}.
• Figure 4: Linewidth of the $\lambda/4$ resonator coupled to a feedline as a function of the length of the coupler obtained with QuLTRA (solid line, blue) and Ansys HFSS (dashed line, orange).
• Figure 5: Schematic layout of the notch Purcell filter for fast qubit readout proposed in Spring_2025, where a $\lambda/4$ readout resonator and a $\lambda/4$ filter resonator are coupled through a multi-line coupler. In the complete circuit port 1 and port 2 are capacitively connected to the qubit and to the readout transmission line, respectively, while the other two line ends are grounded.