Neural-Network Inverse Design of SRF Cavities and Transmons for Bosonic Quantum Computation

Technical Summary: Neural-Network Inverse Design of SRF Cavities and Transmons for Bosonic Quantum Computation

Problem Statement
The design of superconducting radio-frequency (SRF) cavities coupled to transmon qubits for bosonic quantum computation involves a complex inverse problem: recovering device geometries that produce specific electromagnetic (EM) and coupling targets. This mapping is generally one-to-many, meaning multiple geometries can yield similar performance, or conversely, finding the specific geometry for a target response is non-trivial. As design parameter spaces scale, the cost of conventional iterative simulation (e.g., using COMSOL or HFSS) becomes prohibitive. Specifically, the coupling strength between a qubit and a cavity depends sensitively on both the transmon's geometry and its position within the cavity's EM field. Traditional optimization methods, such as adjoint-based shape determination or evolutionary searches, often tune only a subset of EM parameters and struggle with the high-dimensional, non-linear dependence of dispersive Hamiltonian parameters on device geometry.

Methodology
The authors propose two complementary deep neural network (DNN) approaches to address this inverse-design problem at different levels of the design stack. The workflow follows a data-driven template: sampling candidate geometries, evaluating them via finite-element simulation to generate a labeled dataset, training a DNN to invert the forward map (geometry $\to$ response), and validating predictions via end-to-end re-simulation.

1. Cavity Geometry Inverse Design:

   • Goal: Generate SRF cavity geometries that produce target cavity observables.
   • Inputs: Two types of DNNs were trained. The first takes the TM$_{010}$-like passband width (frequency difference between specific modes) as input. The second takes a five-point axial $E_z$ field profile at the prospective qubit location as input.
   • Outputs: Cavity wall geometries parametrized by cubic splines. For multicell (TESLA-like) cavities, the output defines control points for the meridional contour; for cylindrical cavities, it defines control points and length.
   • Training Data: 2,000 multicell and 2,000 cylindrical geometries were sampled and simulated in COMSOL.
   • Architecture: Feed-forward networks with ReLU activations. The frequency-input network uses a polynomial expansion of the input, while the field-input network processes the 5-point vector directly.

2. Transmon Inverse Design:

   • Goal: Generate transmon qubit designs that produce target qubit-cavity parameters.
   • Inputs: Target parameters $(g, \nu_q, \alpha)$, representing the coupling rate, dressed qubit frequency, and anharmonicity. Notably, the cross-Kerr interaction $\chi_{0m}$ is excluded as an input because it is algebraically dependent on the other parameters in the dispersive regime and varies orders of magnitude, making it a poor regression target.
   • Outputs: Three transmon design variables: junction insertion depth ($z_{trans}$), Josephson inductance ($L_J$), and shunt-pad width ($w_{pad}$).
   • Training Data: 981 "clean" data points derived from HFSS eigenmode simulations processed through the pyEPR framework. A physics-based filter removed "bad solves" (e.g., mode mixing or breakdown of the dispersive approximation).
   • Architecture: A fully connected feed-forward network mapping the 3 input parameters to the 3 design variables.

Key Contributions

• Dual-Level Inverse Design: The work presents a complete pipeline for inverse design covering both the macroscopic cavity structure and the microscopic qubit parameters, addressing the full design stack for 3D cQED systems.
• Integration with Characterization Pipelines: The transmon DNN is trained directly on parameters extracted via the Energy Participation Ratio (EPR) method (implemented in pyEPR). This embeds the machine learning model within the standard quantum-device characterization workflow, effectively inverting the geometry-dependent set of self- and cross-Kerr interactions.
• Handling One-to-Many Mappings: By training on specific distributions of viable geometries, the deterministic feed-forward networks learn to select a single viable solution branch for a given target, avoiding the ambiguity inherent in the inverse problem within the sampled design space.

Results

• Cavity Performance:
  • The passband-width DNN recovered geometries matching target passband widths within ~5%. The network focused on dominant features (e.g., iris radii) while varying less critical geometric details.
  • The $E_z$ field DNN achieved even higher fidelity, matching the input field profile point-by-point within ~4%.
  • End-to-end re-simulation in COMSOL confirmed these results.
• Transmon Performance:
  • The transmon DNN achieved a mean relative root mean squared error (RRMSE) of 1.81 ± 0.96% across 15 random train/test shuffles, with a best-case shuffle reaching 0.72%.
  • End-to-end validation involved re-simulating the DNN-predicted designs in HFSS with pyEPR extraction. The recovered parameters $(g, \nu_q, \alpha)$ deviated from the targets by at most ~2% (specifically ~1.6% for $\alpha$, ~1% for $g$, and ~0.5% for $\nu_q$).
  • The model successfully generalized across different operating regimes, from long-storage (low cross-Kerr) to fast-control (high cross-Kerr) applications.

Significance and Claims
The paper claims that these DNN approaches provide a fast alternative to the iterative simulation studies typically required for designing superconducting quantum hardware. By mapping desired device behavior directly to candidate designs, the method bypasses the need for repeated manual parameter sweeps, which can take minutes per iteration and accumulate over many cycles.

The authors position this work within the "AI-for-quantum" direction, emphasizing its utility as quantum devices scale and their parameter spaces expand. The approach is presented as a method to tailor universal gate sets to specific applications through hardware-software co-design. The authors note that while the current work uses deterministic networks that produce a single solution, future extensions could employ generative or invertible architectures to handle cases where multiple viable geometries exist for a single target, thereby offering a broader range of design options for different coupling regimes (e.g., favoring long-lived storage vs. fast control).