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MACROSCOPIC QED FORMALISM FOR EVALUATING TRANSMON EXCHANGE COUPLING RATE IN PRACTICAL DEVICES

Ghazi Khan

Published Jul 23, 2026• 0 citations
Publisher

Abstract

Superconducting qubits are one of the leading platforms for creating practical quantum computers. However, these architectures face significant engineering challenges as they scale up and move towards devices capable of error correction. To aid these efforts, three-dimensional (3D) full-wave computational electromagnetics tools are increasingly being utilized to characterize these devices; however, standard approaches relying on eigenmode solvers are often computationally prohibitive and lack the scalability required to analyse devices beyond a few qubits. To overcome these limitations, this thesis introduces an efficient, alternative modeling framework for superconducting circuits, applied specifically to the evaluation of the qubit-qubit exchange coupling rate in transmon architectures. This coupling rate is a key design parameter that governs the entanglement speed for multi-qubit gates and influences error channels such as qubit crosstalk. Leveraging a field-based formalism within the broader context of macroscopic quantum electrodynamics, we establish that the exchange coupling rate is directly related to the electromagnetic dyadic Green’s function. We then map the imaginary part of the Green’s function to the system’s classical impedance response. Since the impedance response can be evaluated quickly and accurately in standard full-wave solvers, this new formulation provides a scalable way to characterize large-scale superconducting devices. We validate this framework by simulating four practical multi-qubit circuits, benchmarking the extracted coupling rates against both 3D numerical diagonalization methods and experimental data. We also demonstrate how the exchange coupling rate impacts qubit crosstalk, and characterize a multi-coupler device to find operational sweet spots, where the crosstalk is minimized while maintaining strong enough qubit-qubit coupling to enable fast multi-qubit gates. We then conclude this thesis and outline some plans for future work.

DOI
10.25394/pgs.33072392

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Superconducting qubits are one of the leading platforms for creating practical quantum computers. However, these architectures face significant engineering challenges as they scale up and move towards devices capable of error correction. To aid these efforts, three-dimensional (3D) full-wave computational electromagnetics tools are increasingly being utilized to characterize these devices; however, standard approaches relying on eigenmode solvers are often computationally prohibitive and lack the scalability required to analyse devices beyond a few qubits. To overcome these limitations, this thesis introduces an efficient, alternative modeling framework for superconducting circuits, applied specifically to the evaluation of the qubit-qubit exchange coupling rate in transmon architectures. This coupling rate is a key design parameter that governs the entanglement speed for multi-qubit gates and influences error channels such as qubit crosstalk. Leveraging a field-based formalism within the broader context of macroscopic quantum electrodynamics, we establish that the exchange coupling rate is directly related to the electromagnetic dyadic Green’s function. We then map the imaginary part of the Green’s function to the system’s classical impedance response. Since the impedance response can be evaluated quickly and accurately in standard full-wave solvers, this new formulation provides a scalable way to characterize large-scale superconducting devices. We validate this framework by simulating four practical multi-qubit circuits, benchmarking the extracted coupling rates against both 3D numerical diagonalization methods and experimental data. We also demonstrate how the exchange coupling rate impacts qubit crosstalk, and characterize a multi-coupler device to find operational sweet spots, where the crosstalk is minimized while maintaining strong enough qubit-qubit coupling to enable fast multi-qubit gates. We then conclude this thesis and outline some plans for future work.
Superconducting qubits are one of the most mature platforms for achieving practical quantum computers, but significant performance improvements are still needed. To improve the engineering of these systems, three-dimensional (3D) full-wave computational electromagnetics analyses are increasingly being turned to. Unfortunately, existing analysis approaches often rely on full-wave simulations using eigenmode solvers that are typically cumbersome, not robust, and computationally prohibitive if devices with more than a few qubits are to be analyzed. To improve the characterization of superconducting circuits while circumventing these drawbacks, this work begins the development of an alternative modeling framework that we illustrate in the context of evaluating the qubit-qubit exchange coupling rate between transmon qubits. This quantity is a key design parameter that determines the entanglement rate for fast multiqubit gate performance and also affects decoherence sources like qubit crosstalk. Our modeling framework uses a field-based formalism in the context of macroscopic quantum electrodynamics, which we use to show that the qubit-qubit exchange coupling rate can be related to the electromagnetic dyadic Green's function linking the qubits together. We further show how the needed quantity involving the dyadic Green's function can be related to the impedance response of the system that can be easily and efficiently computed with classical computational electromagnetics tools. We demonstrate the validity and efficacy of this approach by simulating four practical multiqubit superconducting circuits and evaluating their qubit-qubit exchange coupling rates. We validate our results against a 3D numerical diagonalization method and against experimental data where available. We also demonstrate the impact of the qubit-qubit exchange coupling rate on qubit crosstalk by simulating a multicoupler device and identifying operating points where the qubit crosstalk becomes zero.
Superconducting qubits are one of the most mature approaches to realize quantum computers. Unfortunately, existing design approaches use full-wave computational electro-magnetics methods to model these devices in inefficient ways that become impractical if even a few qubits are considered. This work uses a field-based formalism to derive a simple formula for the exchange coupling rate between multiple qubits, which is a key design parameter for creating large-scale quantum architectures. This approach is relevant for the common situation of transmon qubits dispersively coupled via an electromagnetic resonator or environment. Then the accuracy of this approach is validated by comparing it to existing modeling approaches on a realistic superconducting circuit topology.
Superconducting qubits have emerged as a leading platform for realizing quantum computers. Accurate modeling of these devices is essential for predicting performance, improving design, and optimizing control. Many modeling approaches currently rely on lumped circuit approximations or other simplified treatments that can be limited in resolving the interplay between the qubit dynamics and the electromagnetic circuitry, leading to significant experimental deviations from numerical predictions at times. To address many of these limitations, methods that self-consistently solve the Schrödinger equation for qubit dynamics with the classical Maxwell's equations have been developed and shown to accurately predict a wide range of effects related to superconducting qubit control and readout. Despite these successes, these methods have not been able to consider multi-qubit effects that give rise to qubit-qubit entanglement. Here, we address this by rigorously deriving how multi-qubit coupling effects between transmon qubits can be embedded into Maxwell-Schrödinger methods. To support this, we build on earlier first-principles derivations of Maxwell-Schrödinger methods for the specific case of two transmon qubits coupled together through a common electromagnetic system in the dispersive regime. To aid in validating aspects of the Maxwell-Schrödinger framework, we also provide a new interpretation of Maxwell-Schrödinger methods as an efficient simulation strategy to capture the class of non-Markovian open quantum system dynamics. Our results demonstrate that these effects can give rise to strong classical crosstalk that can significantly alter multi-qubit dynamics, which we demonstrate for the cross-resonance gate. These classical crosstalk effects have been noted in cross-resonance experiments, but previous quantum theory and device analysis could not explain their origin.
Devices built using superconducting circuits are a popular approach being pursued to develop quantum information processing hardware. These systems involve the quantized interactions of electromagnetic fields with large coherent collections of charges in superconductors, and so are typically referred to as macroscopic circuit quantum electrodynamics (QED) devices or simply circuit QED devices. Although significant progress has been made recently, substantial engineering efforts are required to improve performance and scale these devices so they can address problems of practical interest. Performing these engineering efforts is difficult due to the absence of rigorous numerical modeling approaches. This work begins to address this by providing a new mathematical description of a commonly used circuit QED system, a transmon qubit coupled to microwave transmission lines. Expressed in terms of three-dimensional vector fields, our model is better suited to developing rigorous numerical solvers than approximate lumped element circuit descriptions used in the literature. We present details on the quantization of our model, and derive equations of motion for the coupled field-transmon system. These results can be used in developing full-wave solvers in the future. To make this work more accessible, we assume a limited amount of training in quantum physics and provide many background details.
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 10×1010\times1010×10 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%.
Currently, circuit quantum electrodynamics architectures have emerged as one of the most popular approaches to implement practical quantum information processing hardware. Although significant progress has been made, many technical issues remain that limit the performance of fabricated devices. One approach to accelerate progress in the engineering design of these devices is to develop improved numerical modeling methods. Current modeling methods generally rely on approximate lumped element circuit models to describe the complex network of microwave transmission lines required to operate a circuit quantum electrodynamics device. This reduction in complexity in the theoretical model is valuable for gaining insight into the operation of a device, but does limit the opportunity for using these models to optimize the performance of practical devices. To develop rigorous numerical modeling methods, it is necessary to move toward theoretical descriptions of circuit quantum electrodynamics devices that retain the full details of the three-dimensional vector electromagnetic fields that exist in these systems. In this work, we present details on such a theoretical model for one of the most commonly used circuit quantum electrodynamics systems, a transmon qubit coupled to microwave transmission lines. We discuss the quantization of our new model and show that by adopting relevant approximations our model can be reduced to the same lumped element descriptions commonly used in the literature. We also discuss the derivation of quantum equations of motion for the coupled field-transmon system, which can be used as the starting point for developing full-wave numerical solvers for these circuit quantum electrodynamics systems in the future.