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Chapman & Hall/CRC eBooks

Superconducting Quantum Computing Devices

Zhigang Zhang, Goong Chen

Published Sep 19, 2007• 0 citations
Publisher

Abstract

Abstract Superconductivity manifests macroscopic quantum phenomena. A superconducting circuit composed of Josephson junctions, Cooper pair boxes, and rf-/dc-SQUID, properly miniaturized, becomes quantized and demonstrates Rabi oscillations and entanglement. In this paper, we offer an overview of this approach. We begin with an introduction of the history and elementary theory of superconductivity. Then we describe the building blocks of superconducting classical circuits and derive their canonical quantizations. The set-up of qubits and superconducting quantum logic gates are then examined. Finally, ways to make measurements are discussed.

DOI
10.1201/9781584889007-14

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Quantum mechanics and superconductivity are intimately connected on many levels. Quantum mechanics is needed to understand why a metal is a metal. The BCS condensate of Cooper pairs would not exist without quantum mechanics, nor could Cooper pairs tunnel through the barriers of Josephson junctions. The name SQUID is the abbreviation of superconducting quantum interference device. Yet another level of quantumness has been added in the last two decades. It turns out to be possible to design and fabricate superconducting objects that have macroscopic dimensions and properties, but behave as if they are single quantum particles. Amplitude and phase of the superconducting wavefunction act as conjugate quantum variables. Practical devices and circuits make use of Josephson junctions to reach the parameter values that are needed, but there is a clear distinction from the Josephson electronics circuits in the other chapters of this book. The latter may, in this context, well be referred to as being classical.

CRC Press eBooks•Nov 11, 2011•4 citations
Abstract Superconductivity manifests macroscopic quantum phenomena. A superconducting circuit composed of Josephson junctions, Cooper pair boxes, and rf-/dc-SQUID, properly miniaturized, becomes quantized and demonstrates Rabi oscillations and entanglement. In this paper, we offer an overview of this approach. We begin with an introduction of the history and elementary theory of superconductivity. Then we describe the building blocks of superconducting classical circuits and derive their canonical quantizations. The set-up of qubits and superconducting quantum logic gates are then examined. Finally, ways to make measurements are discussed.
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.
The theory of superconducting phase qubits---also known as current-biased Josephson junctions---is presented. In the first part of this thesis, I introduce quantum computation, quantum simulation, and their deep connection with symplectic integration. I then consider the fundamental many-body theory of superconductivity and Josephson junctions and show how the quantum dynamics of a single macroscopic degree of freedom, the gauge invariant phase difference, emerges. A complete study of the Hilbert space structure of such a variable is performed for the current-biased junction. The resulting resonance structure is studied in detail, using various formalisms including the WKB approximation, instanton methods, the complex scaling transformation, basis set stabilization, numerical integration, and dynamical simulation using Lie algebraic wave-packet propagation. The second part of this thesis explores how the current-biased junction can be used as an element of a quantum computer---a quantum bit (qubit). Single qubit operations are studied, followed by the presentation of the theory of coupled qubit devices. My key result is the design and optimization of quantum logic gates with high fidelity (F ~ 0.9999) for capacitively coupled phase qubits with short gate times (~ 10 ns). Finally, I examine an advanced qubit-coupling scheme, a resonant coupling method utilizing a harmonic oscillator as the auxiliary degree of freedom. The models and methods presented here have been developed in direct collaboration with an experimental program. These experiments are the first to show spectroscopic evidence for entanglement between two and three macroscopic degrees of freedom in a superconducting circuit.
The achievements of quantum computation theory, e.g. Shor's factoring algorithm, motivate efforts to realize quantum computers. Among systems proposed for quantum computing, macroscopic superconducting circuits of Josephson junctions appear promising for integration in electronic circuits and large-scale applications. Recently, a superconducting tunnel junction circuit was designed and a sufficiently high quality factor of quantum coherence has been obtained. This indicates that decoherence need not be among the obstacles in building quantum computers with macroscopic Josephson circuits. In this paper we present the setup of some elementary quantum logic with macroscopic Josephson qubits, strengthened by some simulation work, and then study the feasibility of implementing Shor's quantum factoring algorithm on them. It is shown that it would be eventually possible to build a 2-dimensional Josephson qubit array, possibly accompanied by classical computing components, capable of performing useful quantum computations.
Electrons in a superconductor form Cooper pairs and thus condense into a ground state which can be described by a macroscopic quantum wavefunction. The phase of the wavefunction represents a collective movement of millions of Cooper pairs. Using devices based on Josephson tunnel junctions we are able to control and detect the phase and number of Cooper pairs in a superconductor. Josephson devices are therefore excellent systems for demonstrating macroscopic quantum phenomena. We present a short review of the macroscopic quantum phenomena in superconducting Josephson devices, as well as their applications and prospects in quantum computationl.
The concept of a quantum computer was invented in the beginning of the 1980s as a quantum generalization of the reversible classical computer. After discoveries in the middle of the 1990s of quantum algorithms, which wouldsolve some problems considered intractable for classical computers, the field of quantum computing has developed rapidly. A wide range of quantum systems are investigated for their possible ability to implement quantum algorithms in practice. The most promising solid state implementations are superconducting electrical circuits based on the Josephson effect, and the Coulomb blockade effect. Superconducting circuits are macroscopic quantum systems which can be fabricated using standard lithography technologies. Superconducting qubits - basic building blocks of a quantum computer - have been developed at several research laboratories, among others the MC2 here at Chalmers. This thesis presents a theoretical investigation of controllable coupling of superconducting qubits based on the single Cooper-pair box. Two coupling schemes are investigated in detail: current-controlled coupling via large Josephson junctions, and coupling via a superconducting stripline cavity. Both coupling designs are scalable and suitable for the charge regime of the single Cooper-pair box, as well as for the charge-phase regime where the qubit is better protected from environmental noise. Quantum gate protocols which are relevant for these physical couplings are discussed. The investigations show that, for both considered designs, the simplest two-qubit gate is the conditional phase-shift gate. This gate is universal for quantum computing, and it can create maximallyentangled qubit pairs in only one run. The investigation combines methods from electrical engineering, quantum mechanics and quantum computer science.
Quantum mechanics and superconductivity are intimately connected on many levels. Quantum mechanics is needed to understand why a metal is a metal. The BCS condensate of Cooper pairs would not exist without quantum mechanics, nor could Cooper pairs tunnel through the barriers of Josephson junctions. The name SQUID is the abbreviation of superconducting quantum interference device. Yet another level of quantumness has been added in the last two decades. It turns out to be possible to design and fabricate superconducting objects that have macroscopic dimensions and properties, but behave as if they are single quantum particles. Amplitude and phase of the superconducting wavefunction act as conjugate quantum variables. Practical devices and circuits make use of Josephson junctions to reach the parameter values that are needed, but there is a clear distinction from the Josephson electronics circuits in the other chapters of this book. The latter may, in this context, well be referred to as being classical.