⚛️ quantum physics

Extracting Electromagnetic Bare Mode Couplings in Large Superconducting Quantum Processors

This paper presents and benchmarks four numerical methods for accurately extracting electromagnetic bare mode couplings in large-scale superconducting quantum processors, demonstrating their consistency and effectiveness on a 10x10 transmon qubit array with a maximum relative difference of less than 5%.

Original authors: Reza Molavi, Ebrahim Forati, Yaxing Zhang, Andrey R. Klots, Juan Atalaya, Brandon W. Langley, Dogan A. Timucin, Moein Nazari, Ghazi Khan, Zlatko K. Minev, Alexander N. Korotkov, Michel H. Devoret

Published 2026-09-22

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Reza Molavi, Ebrahim Forati, Yaxing Zhang, Andrey R. Klots, Juan Atalaya, Brandon W. Langley, Dogan A. Timucin, Moein Nazari, Ghazi Khan, Zlatko K. Minev, Alexander N. Korotkov, Michel H. Devoret

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

To build a computer that can solve problems beyond the reach of any machine today, scientists are turning to the strange rules of quantum mechanics. These machines, known as quantum processors, rely on tiny circuits that can exist in multiple states at once, but they are incredibly fragile. To keep them working, engineers must isolate them from the noisy world outside, often placing them inside large, sealed metal boxes. However, these boxes are not just empty containers; they are resonant chambers that can trap electromagnetic waves, much like a guitar body traps sound. When the frequencies of the computer's internal components accidentally match the frequencies of these trapped waves, the two begin to interact. This interaction can cause errors, scrambling the delicate information the computer is trying to hold. As these processors grow larger, containing hundreds of these components, predicting and controlling these unwanted interactions becomes a massive challenge, requiring a way to measure invisible forces across scales that range from the width of a single atom to the size of a shoebox.

In a recent study, researchers at Google Quantum AI and their collaborators tackled this problem by developing and testing four different ways to measure how strongly the computer's components talk to the metal box surrounding them. The team focused on a specific type of quantum bit, or qubit, arranged in a grid of one hundred by one hundred, housed inside a metallic package. Their goal was to map out exactly how much each qubit couples to the standing waves inside the box, a process that had previously been difficult to do with high precision across such a large array. To solve this, they created a digital twin of the processor and ran four distinct types of computer simulations, each using a different mathematical approach to extract the coupling strength.

The first method involved tuning the qubits until they matched the frequency of the box's waves, watching how their energy levels split apart when they got close. The second method measured how much of the box's energy was stored inside the tiny junction of each qubit. The third approach looked at the voltage generated across the qubit when the box's waves were excited, treating the qubit as an open circuit. The fourth method analyzed how the system responded to electrical signals sent into it, building a model of the circuit from the resulting data. By running these simulations on a commercial electromagnetic solver, the team was able to calculate the coupling for every single qubit in the grid.

The results were strikingly consistent. Despite the different mathematical paths taken, all four methods produced nearly identical maps of the interactions. The strongest disagreement between any of the methods was less than five percent, a level of agreement that gives engineers high confidence in their models. The simulations revealed that the strength of the connection between a qubit and the box depends heavily on where the qubit is located. The coupling was not uniform; instead, it followed a specific pattern that changed based on the qubit's position relative to the center of the box. The researchers found that this pattern was driven by the electric field of the box's waves interacting with the qubit's own electric dipole, a tiny separation of positive and negative charge within the circuit.

One of the most useful findings was that the researchers could predict the coupling strength for the entire grid by understanding the shape of the electric field inside the box. They discovered that the interaction was strongest where the electric field changed most rapidly across the surface of the chip. This insight allows designers to predict where errors might occur simply by looking at the geometry of the box and the placement of the components. Furthermore, the study showed that while the box's waves have a complex three-dimensional shape, the interaction with the flat, planar qubits is dominated by the electric field running parallel to the surface of the chip, rather than the field pointing up and down.

This work provides a reliable toolkit for engineers building the next generation of quantum computers. Before this study, accurately modeling these interactions in a large system was a computational bottleneck, often requiring simplifications that could miss critical details. Now, with four validated methods that agree with each other, designers can choose the approach that best fits their specific needs, whether they prioritize speed or the ability to model complex current flows. The study confirms that even in a system as complex as a hundred-by-hundred grid of qubits, the rules governing their interaction with the environment are consistent and predictable. By mapping these invisible connections, the researchers have provided a clear path toward designing processors that can suppress errors and operate with the high fidelity required for practical quantum computing.

Technical Summary: Extracting Electromagnetic Bare Mode Couplings in Large Superconducting Quantum Processors

Problem Statement
The realization of fault-tolerant superconducting quantum computers requires high-fidelity control of coupled quantum systems. A critical challenge in scaling these systems to hundreds of qubits is the accurate characterization of electromagnetic coupling strengths between localized qubit modes and distributed electromagnetic "package modes" (cavity modes) within the metallic enclosure. These package modes can induce qubit decoherence and long-range crosstalk, potentially degrading quantum error correction (QEC) performance.

Extracting these minute couplings (often with efficiencies below $10^{-5}$ ) presents a formidable multi-scale modeling problem. Computational models must simultaneously resolve the nanometer-scale geometry of Josephson junctions (JJs) and their leads while capturing the centimeter-scale full-wave dynamics of the metallic package. Traditional approaches relying on explicit overlap integrals of "bare" mode fields are computationally expensive and highly sensitive to numerical errors in complex, multi-scale 3D domains.

Methodology
The authors present and benchmark four distinct numerical methods for extracting the total coupling strength ( $g$ ) between qubits and package modes. These methods utilize commercially available 3D electromagnetic solvers (specifically Ansys HFSS) and replace the nonlinear Josephson junctions with method-specific linear boundary conditions, modeling qubits as linear resonators. The four methods are:

Avoided-Crossing Method: The JJ is replaced by a variable inductance. The qubit frequency is swept through the package mode resonance, and the coupling $g$ is extracted from the minimum frequency splitting ( $2|g|$ ) of the resulting hybridized eigenmodes.
Energy Participation Ratio (EPR) Method: The JJ is modeled as a fixed inductance. The coupling is derived from the fraction of the package eigenmode's inductive energy stored within the qubit's JJ. This method relies on the relationship between the participation ratio and the coupling efficiency.
Induced Electromotive Force (EMF) Method: The JJs are replaced by open circuits. The package mode is excited, and the coupling is calculated from the voltage induced across the open qubit ports. This approach effectively pushes qubit modes to zero frequency, allowing the solver to target only the relevant package modes.
Impedance-Matrix Method: The JJs are replaced by lumped ports. The system's scattering parameters (S-parameters) or impedance matrix are simulated. A lumped circuit model (incorporating geometric capacitances, inductances, and mutual inductance) is fitted to the simulated impedance data to extract circuit parameters and subsequently the coupling $g$ .

Key Contributions and Results
The authors validated these four methods on a realistic simulation of a $10 \times 10$ array of planar transmon qubits housed in a $30 \text{ mm} \times 30 \text{ mm} \times 3 \text{ mm}$ metallic cavity with a silicon substrate. The fundamental package mode was identified as a modified Transverse Magnetic (TM) mode, specifically an LSM (Longitudinal-Section-Magnetic) mode due to the dielectric substrate.

Consistency: All four methods yielded consistent coupling strengths. The maximum relative difference between the methods across the entire qubit array was less than 5% (specifically 4.8%), with an average relative difference of 2.9%.
Spatial Profile: The extracted coupling strengths exhibited a specific spatial dependence: $g \propto \cos(\pi x/L_x) \sin(\pi y/L_y)$ . This profile was found to be proportional to the $x$ -derivative of the package mode's electric field ( $\partial_x E_z$ ), rather than the field amplitude itself.
Physical Mechanism: The paper demonstrates that this coupling profile arises from the interaction between the transmon's electric dipole moment (oriented along the $x$ -axis due to the paddle geometry) and the in-plane ( $x$ -component) electric field of the package mode at the dielectric surface. The authors validated this via an electric dipole interaction model, estimating the coupling coefficient $A$ to be 14.2 MHz, which closely matches the numerically extracted value of 15.6 MHz.
Efficiency: The Induced-EMF method was identified as the most computationally efficient, requiring only a single full-wave eigenmode simulation to extract couplings for all qubits simultaneously. However, the EPR method was noted for potentially higher accuracy in specific scenarios due to its explicit treatment of qubit currents.

Significance
The paper establishes a robust, flexible computational framework for characterizing qubit-package interactions in large-scale superconducting processors. By demonstrating that four distinct numerical approaches converge to the same physical result with high precision, the work provides confidence in the accuracy of these extraction techniques.

The primary significance lies in the ability to systematically map the spatial dependence of these couplings. Understanding that the coupling is driven by the in-plane electric field component interacting with the qubit's dipole moment allows for the engineering of geometric coupling nulls. This insight is crucial for designing dense superconducting quantum processors where minimizing crosstalk and decoherence from package modes is essential for maintaining the performance of surface code error correction. The methods described are directly applicable to current hardware architectures featuring hundreds of qubits.

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