The chapter provides a brief description of theory of superconductors and Josephson effect and of the design and operation of basic superconducting qubits (charge, flux, phase and phase slip) using up-to-date experimental results. The Lagrangian approach to the description of electric circuits (classical and quantum) is introduced.
This chapter explores the foundational concepts underlying superconducting qubits, pivotal in modern quantum computing. It begins with the harmonic oscillator, fundamental to both classical and quantum mechanics. Section 2.1 introduces its classical and quantum descriptions, emphasizing energy quantization and assessing its potential as a qubit. Section 2.2 focuses on the quantum LC oscillator, integral to superconducting circuit theory. Starting with the classical dynamics of an LC circuit, we derive its Hamiltonian and transition to its quantum description using ladder operators. This section bridges classical circuit theory and quantum systems, paving the way to understanding superconducting qubits. Sections 2.3–2.5 examine key qubit types. Section 2.3 introduces phase, charge, and flux qubits, highlighting their operational principles and the physical parameters they encode, such as magnetic flux or electric charge. Section 2.4 delves into the transmon qubit, exploring its Hamiltonians and methods for control and measurement. Finally, Section 2.5 surveys recent advancements in superconducting qubit designs, offering insights into the evolution toward next-generation qubits. This chapter provides a comprehensive framework for understanding superconducting qubits, linking classical systems to quantum technologies, and contextualizing their role in advancing quantum computation.
Thus far, we have examined quantum computing based on single particle states in atoms, ions and semiconductor structures. In this chapter, we will examine quantum states in superconductors and their application as qubits. This chapter is particularly extensive due to the large variety of possible superconducting quantum circuits. We introduce superconductivity and examine the three main types of superconducting qubit: the flux qubit, the charge qubit and the phase qubit. We will also examine the transmon qubit and circuit quantum electrodynamics.
Thus far, we have examined quantum computing based on electron states in atoms, ions and semiconductor structures. In this chapter, we will examine quantum states in superconductors and their application as qubits. This chapter is particularly extensive due to the large variety of possible superconducting quantum circuits. We introduce superconductivity and examine the three main types of superconducting qubit: the flux qubit, the charge qubit and the phase qubit. We will also examine the transmon qubit and circuit quantum electrodynamics.
This chapter explores the evolution of superconducting qubits from traditional superconductors, tracing the scientific and technological milestones that have led to their central role in quantum computing. Beginning with an overview of superconductivity, the Josephson effect, qubits, and their types, the chapter highlights superconducting qubits with their theoretical implementation. It then transitions to the development of modern qubit architectures – charge, flux, and transmon qubits – emphasizing advances in material purity, fabrication, and coherence enhancement. Special focus is given to a short description of fabricating granular aluminum qubits for quantum engineering applications. The chapter concludes with current challenges, including decoherence mitigation, material scalability, and integration with hybrid quantum systems. This work aims to provide readers with a comprehensive understanding of the journey from conventional superconductors to quantum information devices.
Abstract Quantum circuit theory is a powerful tool to describe superconducting circuits. In its language, quantum phase slips (QPSs) are considered to be the exact dual to the Josephson effect. This duality renders the integration of QPS junctions into a unified theoretical framework challenging. As we argue, different existing formalisms may be inconsistent, and the correct inclusion of time-dependent flux driving requires introducing a large number of auxiliary, nonphysical degrees of freedom. We resolve these issues by describing QPS junctions as inductive rather than capacitive elements, and reducing the Hilbert space to account for a compact superconducting phase. Our treatment provides an approach to circuit quantization exclusively in terms of node-flux-node variables, and eliminates spurious degrees of freedom. Finally, the inductive treatment reveals the possibility of a voltage-dependent renormalization of the QPS amplitude, by accounting for spatial variations of the electric field built up across the junction.
We review some basic facts about qubits and qubit processing. After a brief survey of solid-state qubits, we focus on quantized electrical circuits and superconducting qubits based on Josephson junctions. We review the general framework and indicate how the various qubits, such as the superconducting Cooper pair box charge qubit, the persistent current flux qubit, the hybrid charge-phase qubit, and the Andreev-level qubit, can be seen to appear due to different choices of design parameters. Finally, we consider multi-qubit systems and discuss some aspects of decoherence.