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arXiv:2106.11352 · IEEE Antennas and Propagation Magazine

The Transmon Qubit for Electromagnetics Engineers: An introduction

Thomas E. Roth, Ruichao Ma, Weng Cho Chew

Published Jun 7, 2022• 74 citations
arXivHTMLPDF

Abstract

One of the most popular approaches being pursued to achieve a quantum advantage with practical hardware is the use of superconducting circuit devices. Although significant progress has been made over the previous two decades, substantial engineering efforts are required to scale these devices so they can be used to solve many problems of interest. Unfortunately, much of this exciting field is described using technical jargon and concepts from physics that are unfamiliar to a classically trained electromagnetics engineer. As a result, this work is often difficult for engineers to become engaged in.

DOI
10.1109/map.2022.3176593

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One of the most popular approaches being pursued to achieve a quantum advantage with practical hardware is the use of superconducting circuit devices. Although significant progress has been made over the previous two decades, substantial engineering efforts are required to scale these devices so they can be used to solve many problems of interest. Unfortunately, much of this exciting field is described using technical jargon and concepts from physics that are unfamiliar to a classically trained electromagnetics engineer. As a result, this work is often difficult for engineers to become engaged in.
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.
In this research, we explore the operation of the transmon qubit, a key superconducting architecture in quantum computing. To do so, we address fundamental concepts such as the quantum anharmonic oscillator, Cooper pairs, quantum tunneling, Josephson junctions (JJ), SQUID devices, and the Cooper pair box. The purpose is to provide a rigorous yet accessible introduction, specifically aimed at engineering professionals, that explains the operational principles of the transmon without requiring advanced knowledge of theoretical physics.
As superconducting circuits emerge as a leading platform for scalable quantum information processing, building comprehensive bridges from the foundational principles of macroscopic quantum phenomena to the architecture of modern quantum devices is increasingly essential for introducing new researchers to the field. This tutorial provides a self-contained, pedagogical introduction to superconducting quantum circuits at the undergraduate level. Beginning with an overview of superconductivity and the Josephson effect, the tutorial systematically develops the quantization of microwave circuits into the framework of circuit quantum electrodynamics (cQED). The transmon qubit is then introduced as a state-of-the-art application, with a detailed derivation of its Hamiltonian and its interaction with control and readout circuitry. The theoretical formalism is consolidated through a numerical simulation of vacuum Rabi oscillations in a driven transmon-resonator system, a canonical experiment that demonstrates the coherent energy exchange characteristic of the strong coupling regime. This work serves as a foundational guide and first point of contact, equipping students and researchers with the conceptual and mathematical tools necessary to understand and engineer superconducting quantum hardware.
Over the past two decades, the performance of superconducting quantum circuits has tremendously improved. The progress of superconducting qubits enabled a new industry branch to emerge from global technology enterprises to quantum computing startups. Here, an overview of superconducting quantum circuit microwave control is presented. Furthermore, we discuss one of the persistent engineering challenges in the field—how to control the electromagnetic environment of increasingly complex superconducting circuits such that they are simultaneously protected and efficiently controllable.
Superconducting qubit circuits are one of the prime candidates for building the quantum computer. In this guide I will provide a complete microwave description of the superconducting qubit circuits consisting of the most popular superconducting qubit the "Transmon". Quantum interaction rate parameters related to simple impedance response formulas will allow microwave engineers to design and analyze superconducting qubit circuits without resorting to heavy machinery of quantum mechanical transformations. All formulas listed below are valid in the dispersive regime of low anharmonicity qubits. This work is a review of [1].