In this chapter, we explore the energy-participation ratio (EPR) method, a pivotal technique for quantizing and analyzing superconducting circuits. The EPR method enables us to systematically quantify how energy is distributed across different components and vibrational modes in a physical system, such as superconducting qubits, resonators, or coupled systems. While the method has roots in the study of mechanical and vibrational systems, its application to electromagnetic structures—like resonators, cavities, and Josephson circuits—has been transformative. By identifying and quantifying the contribution of each mode to the total energy, the EPR method serves as a critical tool for understanding system dynamics, optimizing circuit designs, and enabling accurate quantization. This chapter demonstrates how the EPR method is applied to superconducting circuits, both conceptually and practically, with an emphasis on its implementation in Qiskit Metal, a comprehensive framework for modeling and simulating superconducting qubits.
In this chapter, we explore the energy-participation ratio (EPR) method, a pivotal technique for quantizing and analyzing superconducting circuits. The EPR method enables us to systematically quantify how energy is distributed across different components and vibrational modes in a physical system, such as superconducting qubits, resonators, or coupled systems. While the method has roots in the study of mechanical and vibrational systems, its application to electromagnetic structures—like resonators, cavities, and Josephson circuits—has been transformative. By identifying and quantifying the contribution of each mode to the total energy, the EPR method serves as a critical tool for understanding system dynamics, optimizing circuit designs, and enabling accurate quantization. This chapter demonstrates how the EPR method is applied to superconducting circuits, both conceptually and practically, with an emphasis on its implementation in Qiskit Metal, a comprehensive framework for modeling and simulating superconducting qubits.
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 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.
Abstract Energy participation ratio (EPR) at the metal-substrate (MS) interface is a dominant decoherence source limiting the coherence time ( T 1 ) of planar superconducting transmon qubits. Traditional EPR simulations rely on idealized geometric models, ignoring process-induced morphological deviations that significantly impact actual device performance. To bridge this gap, we propose a unified data-driven framework. It automatically extracts geometric parameters from SEM cross-sections using computer vision (CV). These parameters are then imported into a calibrated 2D EPR simulation model. Sobol global sensitivity analysis is further applied to quantify the influences of key parameters. Parametric simulations reveal that the EPR scaling factor F ms generally decreases with increasing interface thickness, substrate over-etch depth, and trench angle, driven by mitigated edge electric field crowding. Sobol analysis further identifies substrate over-etch depth and trench angle as the dominant drivers of F ms , with a significant synergistic interaction between these two parameters. Evaluations across eight process-feasible geometric configurations indicate that thin interface layer combined with a deep substrate over-etch yields the largest predict reduction in interface EPR, corresponding to a projected relative enhancement in T 1 under the assumption of a constant interface loss tangent. We derive actionable process guidelines for planar Al-on-Si transmon qubits, prioritizing the co-tuning of trench parameters to balance MS-interface EPR reduction with manufacturing feasibility within the investigated parameter range. This framework provides a systematic pipeline for incorporating process-induced geometric deviations into EPR prediction, and offers a design-stage simulation tool applicable to planar superconducting structures with similar coplanar edge-dominated EPR characteristics.
An accurate and efficient numerical electromagnetic model for superconducting qubits is essential for characterizing and minimizing design-dependent dielectric losses. The energy participation ratio (EPR) is the commonly adopted metric used to evaluate these losses, but its calculation presents a severe multiscale computational challenge. Conventional finite element method (FEM) requires 3D volumetric meshing, leading to prohibitive computational costs and memory requirements when attempting to capture singular electric fields at nanometer-thin material interfaces. To address this bottleneck, we propose SesQ, a surface integral equation simulator tailored for the precise simulation of the EPR. By applying discretization on 2D surfaces, deriving a semi-analytical multilayer Green's function, and employing a dedicated non-conformal boundary mesh refinement scheme, SesQ accurately resolves singular edge fields without an explosive growth in the number of unknowns. Validations with analytically solvable models demonstrate that SesQ accelerates capacitance extraction by roughly two orders of magnitude compared to commercial FEM tools. While achieving comparable accuracy for capacitance extraction, SesQ delivers superior precision for EPR calculation. Simulations of practical transmon qubits further reveal that FEM approaches tend to significantly underestimate the EPR. Finally, the high efficiency of SesQ enables rapid iteration in the layout optimization, as demonstrated by minimizing the EPR of the qubit pattern, establishing the simulator as a powerful tool for the automated design of low-loss superconducting quantum circuits.
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.
Over the past two decades, superconducting quantum circuits have become one of the essential platforms for realizing quantum computers. The Hamiltonian of a superconducting quantum circuit system is the key to describing the dynamic evolution of the system. For this reason, various methods for analyzing the Hamiltonian of a superconducting quantum circuit system have been proposed, among which the LOM (Lumped Oscillator Model) and the EPR (Energy Participation Ratio) methods are the most popular ones. To analyze and improve the design methods of superconducting quantum chips, this paper compares the similarities and differences of the LOM and the EPR quantification methods. We verify the applicability of these two theoretical approaches to the design of 2D transmon quantum chips. By comparing the theoretically simulated results and the experimentally measured data at extremely low temperature, the errors between the theoretical calculation and observed measurement values of the two methods were summarized. Results show that the LOM method has more parameter outputs in data diversity and the qubit frequency calculation in LOM is more accurate. The reason is that in LOM more coupling between different systems are taken into consideration. These analyses would have reference significance for the design of superconducting quantum chips.