Energy participation ratio analysis for very anharmonic superconducting circuits

Abstract: Superconducting circuits are being employed for large-scale quantum devices, and a pertinent 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.

• The paper introduces an extended Energy Participation Ratio method that accurately models highly anharmonic circuits like the fluxonium qubit.
• It employs refined computational techniques to overcome limitations of low-order nonlinear expansions and improves dispersive shift predictions.
• Experimental validation using spectroscopy and dispersive readout demonstrates strong agreement with simulations, enhancing quantum circuit design.

Energy Participation Ratio Analysis for Very Anharmonic Superconducting Circuits

Overview

The paper "Energy participation ratio analysis for very anharmonic superconducting circuits" explores advanced computational methods for analyzing superconducting quantum circuits, with a focus on the fluxonium qubit. This work extends the traditional Energy Participation Ratio (EPR) method to accommodate circuits with significant nonlinearity, addressing the modeling challenges posed by highly anharmonic systems such as fluxoniums.

Energy Participation Ratio Method

The EPR method quantifies how much energy each element in a circuit contributes to the system's modes and forms the basis for constructing the quantum Hamiltonian from classical electromagnetic simulations. This paper refines the EPR method to handle the complexities of highly anharmonic circuits, mitigating the inaccuracies from low-order nonlinear expansions.

In classical EPR, the system's classical energy modes set the stage for the quantum Hamiltonian. This approach, however, faltered with highly nonlinear systems like the fluxonium qubit, due to its strong nonlinearity and complex inter-mode couplings, demanding more sophisticated treatment.

Figure 1: The fluxonium consists of a linear inductor with the inductive energy $E_L$, a Josephson junction with a Josephson energy $E_J$, and a capacitor with a charging energy $E_C$. The fluxonium is capacitively coupled to a readout resonator with angular frequency $\omega_r$.

Fluxonium Qubit Design

The fluxonium qubit architecture is designed to maximize coherence and fidelity, leveraging its large inductive component to mitigate charge noise. The qubit employs a Josephson junction for nonlinearity and a large inductor to stabilize qubit states. A significant innovation of this study lies in the accurate modeling of these components under the enhanced EPR framework.

Figure 2: (a) False-coloured optical image of a fluxonium device. (b) SEM image showing Al junctions. (c) SEM image focused on the central Josephson junction.

Experimental Validation

Experimentally, the fluxonium circuits were fabricated and characterized using a combination of spectroscopy and dispersive readout techniques. The resonator and qubit frequencies measured experimentally were in excellent agreement with the predictions of the extended EPR method.

Figure 3: (a) Resonator frequency vs. applied flux, comparing experimental (blue) and EPR simulations (red). (b) Qubit frequency vs. applied flux, showing experiment (blue) and simulation (red).

Dispersive Shift Analysis

A critical component of interaction in these quantum circuits is the dispersive shift between the qubit and the readout resonator. This study benchmarks the dispersive shift predictions from the extended EPR method against those from lumped element models and experimental data, demonstrating superior accuracy from the EPR method across varying flux biases.

Figure 4: Dispersive shift $\chi$ comparison between experimental, EPR simulations, and lumped element model results.

Conclusion

This work advances the modeling and simulation of complex superconducting circuits, enabling precise predictions of both linear and nonlinear circuit parameters. The extended EPR method effectively addresses the limitations of previous models, specifically for circuits exhibiting large anharmonicity. Future research directions may involve expanding this methodology to multi-qubit systems and investigating its applicability in large-scale quantum processors.

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The results presented here significantly contribute to the practical and theoretical understanding necessary for advancing superconducting quantum technologies, particularly those involving highly anharmonic qubit systems. This paper provides a comprehensive framework and validation for enhancing the fidelity and scalability of quantum circuits in cutting-edge quantum computing applications.