Extracting Electromagnetic Bare Mode Couplings in Large Superconducting Quantum Processors

AI summary

Overview

Research area: Superconducting quantum computing hardware, specifically full-wave electromagnetic modeling and characterization of qubit–package-mode couplings in large-scale processors.

Technical level: Advanced.

Scope: The paper presents, benchmarks, and cross-validates four numerical methods for extracting electromagnetic coupling strengths between transmon qubits and distributed package modes in a 10 × 10 qubit array.

What This Paper Is About

Large superconducting quantum processors are housed in metallic packages whose electromagnetic modes can couple to many qubits at once, causing decoherence and long-range crosstalk. Extracting these weak couplings accurately is a multi-scale modeling problem spanning nanometer-scale Josephson junctions to centimeter-scale packages. The paper compares four practical methods for extracting the “bare” coupling strengths using commercial 3D electromagnetic solvers and checks whether they agree.

Key Contributions

1. Presents four coupling-extraction methods—avoided level crossing, energy participation ratio (EPR), induced electromotive force (EMF), and impedance-matrix analysis—each using a different boundary-condition representation of the Josephson junction in a linear electromagnetic model.
2. Benchmarks all four methods on a realistic 10 × 10 array of planar transmons inside a metallic package, extracting couplings to the fundamental package mode.
3. Shows that the four methods agree closely: the maximum relative difference is 4.8% and the average relative difference is 2.9% across the array.
4. Explains the spatial pattern of qubit–package coupling physically, showing that it follows a derivative-of-the-mode-field profile rather than the electric-field profile of the unmodified TM₁₁₀ mode.

Main Findings

• Strong cross-method agreement: The four methods produce nearly equal coupling strengths, with a maximum relative difference of 4.8% and an average relative difference of 2.9%. The abstract describes this as a maximum relative difference of less than 5%.

• Small systematic differences: Avoided-crossing and induced-EMF methods tend to produce slightly larger couplings than the EPR and impedance-matrix methods.

• Computational efficiency varies: The induced-EMF method is the most convenient computationally because replacing junctions with open circuits pushes qubit modes to zero frequency, so only one full-wave eigenmode simulation is needed to extract couplings for all qubits. It does, however, require knowing the qubit capacitances.

• Accuracy trade-offs: EPR can be more accurate because its eigenmode simulation treats qubit current explicitly. Avoided-crossing and impedance-matrix methods can be more accurate in some situations because driven full-wave simulations are typically more accurate than eigenmode simulations.

• Representative avoided-crossing result: For the qubit at array coordinates (1,1), the minimum frequency separation gives 2|g|/2π = 12.6 MHz for the coupling to the fundamental package mode.

• Representative impedance-matrix result: For qubit (0,0), fitting the impedance matrix yields Cₙ = 173.8 fF, δL′ₙ = 0.32 nH, Cₘ = 132.0 fF, C_c = 100 aF, δL′ₘ = 4.57 nH, and M = −0.34 pH, giving g_C/2π = 2.07 MHz and g_L/2π = −0.25 MHz, hence g/2π = 2.42 MHz.

• Package-mode physics: The dielectric substrate converts the fundamental TM₁₁₀ mode into a longitudinal-section-magnetic mode, labeled LSM₁₁₀. This mode has a nonzero x-component of electric field at the dielectric surface, which interacts with the qubit’s x-directed electric dipole moment.

• Coupling spatial profile: The extracted coupling follows g = A cos(πx/Lₓ) sin(πy/L_y), fitted with a single parameter reported in the text as A/2π = 15.6 MHz (the figure caption states A = 15.6 MHz). This profile is proportional to ∂ₓE_z of the TM₁₁₀ mode, not to E_z itself.

• Symmetry: Couplings are antisymmetric about x = Lₓ/2 and symmetric about y = L_y/2. Minor deviations from exact symmetry, such as between qubits (0,0) and (9,0), are attributed to numerical inaccuracy.

• Validation geometry: The simulated package is 30 mm × 30 mm × 3 mm, partially filled with a 0.5 mm silicon substrate with εᵣ = 11.9. The transmon paddles are 0.5 mm × 1 mm, separated by 0.2 mm, with a 3 mm array period and 0.7 mm between paddle centers. The fundamental mode frequency is 7.07 GHz without dielectric, 6.49 GHz with dielectric, and 6.48 GHz after adding qubit paddles with junctions removed.

• Large couplings in the test structure: Couplings are on the order of 10 MHz because the simulated structure lacks the ground-plane metallization common in actual processors, so real devices are expected to have much smaller couplings.

• Capacitive and inductive parts cannot be separated: Although the EPR and impedance-matrix procedures appear to distinguish g_C and g_L, the paper shows this is not reliable. Changing the cavity port location changes the extracted g_C and g_L significantly while g remains the same, which is expected because the package mode is distributed.

• Sign information: EPR and induced-EMF methods can provide the sign of g, while the avoided-crossing method produces only |g|.

Methodology in Plain English

The researchers model the qubits as linear resonators by replacing each Josephson junction with a method-specific electromagnetic boundary condition. They use Ansys HFSS for full-wave eigenmode and driven simulations, and use circuit/network tools such as SPICE or scikit-rf for post-processing.

The four methods work as follows. In avoided crossing, the junction is replaced by a variable inductance and the qubit frequency is swept through the package-mode frequency; the minimum splitting between the two hybridized eigenfrequencies gives |g|. In EPR, the junction is a fixed inductance and the coupling is inferred from the fraction of the package mode’s inductive energy stored in the junction. In induced EMF, the junction is replaced by an open circuit, the package mode is excited, and the voltage induced across each qubit’s junction leads gives the coupling. In impedance-matrix analysis, the junction is replaced by a lumped port, the driven network response is simulated, and the impedance matrix is fitted to an equivalent two-resonator circuit model from which g is computed.

All methods are interpreted through a common two-mode model of two coupled LC resonators, with electric and magnetic coupling channels g_C and g_L. Under the rotating-wave approximation, the total coupling is g = g_C − g_L. The validation structure is a 10 × 10 transmon array in a metallic box, and couplings to the fundamental package mode are extracted and compared across the array, using symmetry to display only one quadrant containing 25 qubits.

Why This Matters

Accurate coupling extraction is essential for predicting and suppressing package-mode-induced decoherence and crosstalk in large superconducting processors. Crosstalk can effectively reduce surface-code distance and degrade quantum error correction, so hardware teams need reliable ways to model these weak, long-range interactions before building or calibrating a device. The paper’s cross-validation gives practical confidence that different commercial-solver workflows can produce consistent answers.

Real-world applications:

• Designing metallic packages and cavities for large superconducting quantum processors to control package-mode frequencies and spatial profiles.
• Predicting qubit–package crosstalk and choosing qubit placements near field nodes to reduce unwanted interactions.
• Extracting Hamiltonian parameters for control calibration and validation of processor electromagnetic models.
• Integrating full-wave electromagnetic simulation into industrial quantum-hardware design flows, including the use of commercial solvers and network-analysis tools.

Industry relevance: The work comes from Google Quantum AI and addresses a practical bottleneck for scaling superconducting processors to hundreds of qubits. It is directly relevant to quantum-hardware companies, electromagnetic simulation vendors, and teams building cryogenic packaging and control electronics for superconducting quantum computers.

Future Directions

• Experimental validation of the extracted couplings against measured device spectra and crosstalk in fabricated processors with realistic ground planes.
• Extending the comparison to qubit–qubit couplings, multiple package modes, and larger multi-port networks beyond the two-port examples discussed.
• Determining which method is most reliable in different regimes, especially when coupling efficiencies fall below 10⁻⁵ and when resonances are closely spaced.
• Developing ways to separate capacitive and inductive coupling contributions reliably, which the paper shows is not possible with the present distributed-mode methods, and incorporating nonlinear junction physics beyond the linearized treatment.

Target Audience

This paper is most useful to researchers and engineers working on superconducting quantum processor design, cryogenic packaging, electromagnetic simulation, and quantum-hardware calibration. It is also relevant to computational electromagnetics researchers interested in multi-scale modeling and to graduate students with a background in circuit QED or microwave engineering.

Authors’ abstract

High-fidelity control of superconducting quantum processors requires accurate characterization of electromagnetic coupling strengths among the device's constituent elements. Accurately extracting these couplings across large-scale architectures, presently featuring hundreds of qubits, poses a challenging multi-scale modeling problem. This requires resolving scales from the nanometer-scale geometry of Josephson junctions and their leads to the centimeter-scale size of the enclosing metallic packages. We present four numerical coupling extraction methods based on the avoided level crossing, the energy participation ratio, the induced electromotive force, and the impedance matrix. These methods are tailored to work with commercially available 3D electromagnetic solvers. We benchmark these techniques on a $10\times10$ array of transmon qubits, extracting their couplings to standing package modes. Our results show that these methods yield consistent coupling strengths with a maximum relative difference of less than 5%.