In the early 1980’s, it was suggested that a computer obeying the laws of quantum mechanics would be able to solve problems beyond the capabilities of a classical computer. The novel ways in which such a quantum computer works relies on the quantum properties of the quantum bits (qubits) used to store the information. Any candidate for a quantum computer must thus be able to sustain these properties and offer means to manipulate and read-out the information. One such candidate are quantum mechanical, superconducting circuits, where the logical bit is encoded in the energy eigenstates of the system. In circuit quantum electrodynamics, such a qubit is coupled to a microwave cavity allowing the qubit to be coherently controlled and read-out by probing the cavity. In this thesis, we theoretically investigate the destructive effects of noise which couple to the system as one tries to measure and control the qubits. We study the so called quantum capacitance read-out scheme, where the state of the qubit is mapped onto an equivalent capacitance of the circuit. It is shown that this is quantum limited, in the sense that the state of the qubit can be determined while simultaneously adding a minimum amount of noise to the system. Apart from the added noise, a measurement on a qubit will perturb it, causing the state to collapse to one of the measurement eigenstates. Such a state collapse can be utilized to generate entanglement between qubits by measuring on the cavity. We show that high-fidelity entangled states can be produced in this way and discuss the potential of the measurement to violate a bound given by local hidden variable theories. The possibility to prolong the life-time of the quantum state by active error correction is also investigated and we discuss limits on gate operation times to benefit from such a code, given realistic values for the error probabilities. This work was supported by the European Commission through IST-015708 EuroSQIP integrated pro ject, and by the Swedish Research Council.
In the early 1980’s, it was suggested that a computer obeying the laws of quantum mechanics would be able to solve problems beyond the capabilities of a classical computer. The novel ways in which such a quantum computer works relies on the quantum properties of the quantum bits (qubits) used to store the information. Any candidate for a quantum computer must thus be able to sustain these properties and offer means to manipulate and read-out the information. One such candidate are quantum mechanical, superconducting circuits, where the logical bit is encoded in the energy eigenstates of the system. In circuit quantum electrodynamics, such a qubit is coupled to a microwave cavity allowing the qubit to be coherently controlled and read-out by probing the cavity. In this thesis, we theoretically investigate the destructive effects of noise which couple to the system as one tries to measure and control the qubits. We study the so called quantum capacitance read-out scheme, where the state of the qubit is mapped onto an equivalent capacitance of the circuit. It is shown that this is quantum limited, in the sense that the state of the qubit can be determined while simultaneously adding a minimum amount of noise to the system. Apart from the added noise, a measurement on a qubit will perturb it, causing the state to collapse to one of the measurement eigenstates. Such a state collapse can be utilized to generate entanglement between qubits by measuring on the cavity. We show that high-fidelity entangled states can be produced in this way and discuss the potential of the measurement to violate a bound given by local hidden variable theories. The possibility to prolong the life-time of the quantum state by active error correction is also investigated and we discuss limits on gate operation times to benefit from such a code, given realistic values for the error probabilities. This work was supported by the European Commission through IST-015708 EuroSQIP integrated pro ject, and by the Swedish Research Council.
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.
In this thesis, experimentally relevant aspects and open questions of quantum information processing and measurement in circuit quantum electrodynamics have been investigated theoretically. Circuit quantum electrodynamics is a relatively young field combining superconducting transmission line resonators on-chip with superconducting quantum bits which serve as artificial atoms. Remarkable experiments have underlined the prospects of circuit QED as a possible architecture for quantum information processing as well as a framework within which quantum optics experiments can be performed. In contrast to their optical counterparts, these experiments reach the ultra-strong coupling limit and allow e.g. the generation of Fock states, the study of decoherence and the observation of quantum jumps. We present the physics of superconducting circuits based on the Josephson junction as a non-linear circuit element. We discuss the various types of quantum bits (qubits) before turning to an introduction on cavity QED and conclude the introduction with a brief presentation of the most relevant experiments and a short introduction to the principles and prospects of quantum information processing. Building on the pioneering theoretical and experimental work in which up to three qubits have been integrated on a chip and successfully coupled via a cavity mode, we propose a scheme to overcome the natural limits concerning scalability in these systems. Usually, scalability is severely limited by the resonance width of the qubit transitions and the attainable frequency range for both qubit transition and cavity resonance frequencies. This limits the present setups to at most ten qubits. As a possible solution, we propose and investigate a cross-bar grid layout of cavities with qubits at the intersections. It turns out that this setup, allowing for up to about 1000 qubits, offers significant advantages over e.g. nearest-neighbor coupling schemes. In addition, it can serve as a building block for a truly scalable scheme for fault-tolerant quantum information processing with superconducting circuits by combining 7x7 arrays in a staircase manner. Furthermore, we discuss and simulate different possibilities to improve upon the suggested scheme by e.g. using resonant gates and optimal control theory. Another focus of the present thesis is measurement physics. In order to investigate e.g. the possibilities of using a circuit QED system to dispersively detect single microwave photons or to generate multi-qubit entangled states, we introduce in a pedagogical manner the basics of stochastic master equations which generate quantum trajectories before turning to concrete applications. Quantum trajectories have proven to be a state-of-the-art tool to analyze measurement situations in a very realistic manner by giving access to both the measurement record as well as the internal quantum dynamics conditioned on this record. Using the method of quantum trajectories as generated by stochastic master equations, we discuss a scheme to efficiently generate multi-qubit entangled states in a very flexible way. The method is based on a collective, dispersive qubit readout. By this measurement, a suitable product state of N qubits can be reliably converted into e.g. an N-qubit Greenberger-Horne-Zeilinger or W state. We propose and discuss the scheme and investigate the effects of decoherence and parameter spread as they would be encountered in experiment. Last but not least, we propose and fully analyze a scheme to dispersively detect single itinerant microwave photons. While experiments to generate single microwave photons have already been successfully performed, their single-shot quantum non-demolition detection has not been possible so far. The scheme presented in this thesis closes this gap in the circuit QED toolbox and reaches detection efficiencies of 30% only limited by fundamental quantum mechanics in this case the Quantum Zeno effect. This detection scheme could be used in quantum communication or quantum cryptography applications.
Circuit quantum electrodynamics (cQED), exemplified by a coupled system of a superconducting qubit and a microwave cavity, is a powerful framework for the quantum control and measurement of macroscopic quantum states in solid-state circuits. Quantum circuits based on superconducting qubits have emerged as one of the leading platforms in the pursuit of fault-tolerant quantum computing. Despite significant progress in extending qubit lifetimes, refining experimental protocols, and scaling up quantum processors, errors arising from the control and measurement of superconducting qubits continue to hinder the realization of a universal quantum computer. To investigate the error mechanisms for state-of-the-art qubits, we realize long-lived superconducting transmon qubits that exhibit fluctuating lifetimes, with an average of 0.2 ms and a maximum exceeding 0.4 ms. We then investigate their dominant error mechanism. By introducing novel time-resolved error measurements that are synchronized with the operation of the pulse tube cooler in a dilution refrigerator, we find that mechanical vibrations from the pulse tube induce nonequilibrium dynamics in highly coherent qubits, leading to their correlated bit-flip errors. Our findings not only deepen our understanding of qubit error mechanisms but also provide valuable insights into potential error-mitigation strategies for achieving fault tolerance by decoupling superconducting qubits from their mechanical environments. We also present a novel scheme for operating a cQED system in a regime where the cavity is largely displaced. We find that a qubit drive at the bare cavity frequency can effectively displace the cavity field while minimizing the cavity-drive-induced dephasing and the ac Stark shift. Furthermore, we demonstrate that standard cQED protocols, such as qubit control, readout, and cavity single-photon generation, remain compatible with largely displaced cavity fields, even beyond the critical photon number. These findings open new pathways for exploring cQED dynamics in large-photon-number regimes, including the study of nonlinear phenomena in microwave cavities, the saturation of two-level system baths for extending the coherence of microwave quantum states, and the acceleration of the dispersive readout. Finally, mechanical resonators have emerged as integral components in quantum computing for storing and transducing quantum information. Low-frequency mechanical resonators, with extremely low energy dissipation rates, are particularly interesting for extending the lifetimes of stored quantum information. However, integrating low-frequency mechanical resonators with qubits is challenging because their small energy gaps make coherent energy exchange and ground-state preparation difficult. In this thesis, we report the first steps towards the characterization of a hybrid quantum device, consisting of a Transmon qubit and a circuit optomechanical system, both coupled to a 3D microwave cavity.
In a quantum computer, the information carriers, which are bits in ordinary computers, are implemented as devices that exhibit coherent superpositions of physical states and entanglement. Such components, known as quantum bits or qubits, can be realized with various different types of two-state quantum systems. Quantum computers will be built for computational speed, with hoped for applications especially in cryptography and in other tasks where classical computers remain inefficient. Circuit quantum electrodynamics (cQED) is a quantum-computer architecture which employs superconducting electronic components and microwave photon fields as building blocks. Compared to cavity quantum electrodynamics (CQED), where atoms are trapped in physical cavities, cQED is more attractive in that its qubits are tunable and conveniently integrable with the electronics already in use. This architecture has shown some of the most promising qubit designs, despite their coherence times reaching tens of microseconds, are still below the state of the art with spin qubits, which reach milliseconds. Coherence times are historically the most relevant parameters describing the fitness of a qubit, although these days not necessarily the limiting factor. This thesis presents a comprehensive set of theoretical and experimental methods for measuring the characteristic parameters of superconducting qubits. We especially study transmission-line-shunted plasma oscillation qubits, or transmons, and presents experimental results for a single sample. Transmon capacitively couples a superconducting quantum interference device (SQUID) with a coplanar waveguide (CPW) resonator, often with added frequency tunability utilizing an external magnet. The number of superconducting charge carriers tunnelled through a junction in the SQUID are used as qubit degrees of freedom. Readout of the qubit state is carried out by measuring transmission through the CPW. A cryogenic setup is employed with measurement and driving pulses delivered from microwave sources. Steady-state spectroscopy is employed to determine the resonance frequencies of the qubit and the resonator, qubit-resonator coupling constants, and the energy parameters of the qubit. Pulse-modulated measurements are employed to determine the coherence times of the qubit. The related analysis- and simulation programs and scripts are collected togithub.com/patomaki.
Among the potential physical implementations of quantum bits (qubits) solid state devices appear most promising for large scale applications and integration in electronic circuits. Devices based on low-capacitance Josephson junctions exploit the coherence of the superconducting state, combined with the possibility to control individual charges by Coulomb blockade effects. The logical states in these systems differ by one Cooper-pair charge. Single- and two-bit operations can be performed by applying a sequence of gate voltages. The phase coherence time is sufficiently long to allow a series of these steps. If, in addition, the strength of the Josephson coupling is controlled (e.g. by currents in SQUID loops) the system is ideal for quantum computation. Recent experiments by Nakamura et al. show the feasibility of this approach. In addition to the manipulations of qubits the resulting quantum state has to be read out. This quantum measurement process can be accomplished by coupling a single-electron transistor (SET) capacitively to the qubit. We investigate the process by analyzing the time-evolution of the density matrix of the coupled system. The measurement process is characterized by three time scales: the dephasing time, which is fast when a dissipative current flows in the SET, a longer ‘measurement time’ needed to read out the signal, and a third, even longer ‘mixing time’, characterizing measurement induced transitions which destroy the information about the initial quantum state. The noise spectrum of the current in the SET displays these time scales and allows their direct observation.
Real-time monitoring of a quantum state provides powerful tools for studying the backaction of quantum measurement and performing quantum feedback. Historically, this monitoring capability has been the exclusive province of atomic and optical physics. This thesis describes the implementation of the first such high-fidelity readout scheme in a solid state circuit, a superconducting quantum bit (qubit) coupled to a microwave cavity in the circuit quantum electrodynamics (circuit QED) architecture. The qubit-state-dependent resonance frequency of the cavity is probed with a microwave drive tone, and the resulting signal amplified using a fast, ultralow-noise superconducting parametric amplifier. This arrangement enables the observation of quantum jumps between the qubit states in real time. The ability to monitor the qubit continuously with high fidelity and resolve quantum jumps can be used to investigate the backaction of the measurement process on the qubit. This thesis examines the quantum Zeno effect--where strong measurement inhibits the evolution of a quantum system-- as well as the transition to non-ideal measurement with increasing measurement strength in the circuit QED architecture, a phenomenon shown to be due to the upconversion of low-frequency dephasing noise. These data allow probes of ``universal'' flux noise in previously inaccessible frequency ranges. The work presented here opens the door for quantum feedback and error correction in solid-state quantum systems using continuous weak measurement.