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QUANTUM COMPUTING WITH ELECTRICAL CIRCUITS: HAMILTONIAN CONSTRUCTION FOR BASIC QUBIT-RESONATOR MODELS

Michael R. Geller

Published Jun 1, 2008• 0 citations
Publisher

Abstract

Recent experiments motivated by applications to quantum information processing are probing a new and fascinating regime of electrical engineering—that of quantum electrical circuits—where macroscopic collective variables such as polarization charge and electric current exhibit quantum coherence. Here I discuss the problem of constructing a quantum mechanical Hamiltonian for the low-frequency modes of such a circuit, focusing on the case of a superconducting qubit coupled to a harmonic oscillator or resonator, an architecture that is being pursued by several experimental groups. 1 Quantum gate design In the quantum circuit model of quantum information processing, an arbitrary unitary transformation on N qubits can be decomposed into a sequence of certain universal two-qubit logical operations acting on pairs of qubits, combined with arbitrary single-qubit rotations [1]. The purpose of quantum gate design is to develop experimental protocols or “machine language code ” to implement these elementary operations. For quantum information processing architectures based on superconducting circuits [2, 3], the first step is to construct an effective Hamiltonian for the system. Whereas the fully

DOI
10.1142/9789812709455_0003

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Recent experiments motivated by applications to quantum information processing are probing a new and fascinating regime of electrical engineering—that of quantum electrical circuits—where macroscopic collective variables such as polarization charge and electric current exhibit quantum coherence. Here I discuss the problem of constructing a quantum mechanical Hamiltonian for the low-frequency modes of such a circuit, focusing on the case of a superconducting qubit coupled to a harmonic oscillator or resonator, an architecture that is being pursued by several experimental groups. 1 Quantum gate design In the quantum circuit model of quantum information processing, an arbitrary unitary transformation on N qubits can be decomposed into a sequence of certain universal two-qubit logical operations acting on pairs of qubits, combined with arbitrary single-qubit rotations [1]. The purpose of quantum gate design is to develop experimental protocols or “machine language code ” to implement these elementary operations. For quantum information processing architectures based on superconducting circuits [2, 3], the first step is to construct an effective Hamiltonian for the system. Whereas the fully
Hybrid quantum systems combine the advantages of different physical platforms with the goal of realizing more powerful quantum information processing devices. Mechanical systems, such as bulk acoustic wave resonators, feature many highly coherent harmonic modes in a compact footprint, complementing the strong nonlinearities and fast operation of superconducting quantum circuits. We developed an architecture for mechanical resonator-based quantum computing in which a superconducting qubit is used to perform quantum gates on a collection of mechanical modes. We demonstrate a universal gate set composed of single-qubit gates and controlled arbitrary-phase gates and showcase their use in the quantum Fourier transform and period-finding algorithms. These results show the potential of using mechanical systems to build crucial components for quantum technologies, such as quantum random-access memories.
Hybrid quantum systems combine the unique advantages of different physical platforms with the goal of realizing more powerful and practical quantum information processing devices. Mechanical systems, such as bulk acoustic wave resonators, feature a large number of highly coherent harmonic modes in a compact footprint, which complements the strong nonlinearities and fast operation times of superconducting quantum circuits. Here, we demonstrate an architecture for mechanical resonator-based quantum computing, in which a superconducting qubit is used to perform quantum gates on a collection of mechanical modes. We show the implementation of a universal gate set, composed of single-qubit gates and controlled arbitrary-phase gates, and showcase their use in the quantum Fourier transform and quantum period finding algorithms. These results pave the way toward using mechanical systems to build crucial components for future quantum technologies, such as quantum random-access memories.
We theoretically study single and two-qubit dynamics in the circuit QED architecture. We focus on the current experimental design [Wallraff et al., Nature (London) 431, 162 (2004); Schuster et al., Nature (London) 445, 515 (2007)] in which superconducting charge qubits are capacitively coupled to a single high-QQQ superconducting coplanar resonator. In this system, logical gates are realized by driving the resonator with microwave fields. Advantages of this architecture are that it allows for multiqubit gates between non-nearest qubits and for the realization of gates in parallel, opening the possibility of fault-tolerant quantum computation with superconduting circuits. In this paper, we focus on one- and two-qubit gates that do not require moving away from the charge-degeneracy ``sweet spot.'' This is advantageous as it helps to increase the qubit dephasing time and does not require modification of the original circuit QED. However, these gates can, in some cases, be slower than those that do not use this constraint. Five types of two-qubit gates are discussed, these include gates based on virtual photons, real excitation of the resonator, and a gate based on the geometric phase. We also point out the importance of selection rules when working at the charge degeneracy point.
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.
Quantum information processing systems can be used in a variety of applications. Among proposed realizations of quantum bits nano-electronic circuits appear most promising for integration in electronic circuits and large-scale applications. We discuss Josephson junction circuits in two regimes where they can be used for quantum computing. In one limit, two logic states of the quantum bit differ by a Cooper pair charge, 2e, on a superconducting island. In the other limit different magnetic fluxes penetrate a superconducting loop in two states. Single-qubit and two-qubit logic operations can be performed by means of voltage or current pulses. Weak coupling to the environment allows a series of these elementary steps before the phase coherence is lost. To read out the result of the computation a quantum measurement is needed. For charge qubits this can be accomplished by coupling a single-electron transistor (SET) capacitively to the qubit. Monitoring the current in the transistor one can extract the information about the state of the qubit. We discuss recent experiments and present a suitable set of circuit parameters available with the present-day technology.
A quantum computer will use the properties of quantum physics to solve certain computational problems much faster than otherwise possible. One promising potential implementation is to use superconducting quantum bits in the circuit quantum electrodynamics (cQED) architecture. There, the low energy states of a nonlinear electronic oscillator are isolated and addressed as a qubit. These qubits are capacitively coupled to the modes of a microwave-frequency transmission line resonator which serves as a quantum communication bus. Microwave electrical pulses are applied to the resonator to manipulate or measure the qubit state. State control is calibrated using diagnostic sequences that expose systematic errors. Hybridization of the resonator with the qubit gives it a nonlinear response when driven strongly, useful for amplifying the measurement signal to enhance accuracy. Qubits coupled to the same bus may coherently interact with one another via the exchange of virtual photons. A two-qubit conditional phase gate mediated by this interaction can deterministically entangle its targets, and is used to generate two-qubit Bell states and three-qubit GHZ states. These three-qubit states are of particular interest because they redundantly encode quantum information. They are the basis of the quantum repetition code prototypical of more sophisticated schemes required for quantum computation. Using a three-qubit Toffoli gate, this code is demonstrated to autonomously correct either bit- or phase-flip errors. Despite observing the expected behavior, the overall fidelity is low because of decoherence. A superior implementation of cQED replaces the transmission-line resonator with a three-dimensional box mode, increasing lifetimes by an order of magnitude. In-situ qubit frequency control is enabled with control lines, which are used to fully characterize and control the system Hamiltonian.