Quantum computation is a multifaceted field of research aiming for the physical realization of quantum systems and their manipulation. This thesis discusses the combination of two notable approaches in the pursuit of a fully operational quantum computer---circuit quantum electrodynamics and topological quantum computation based on Majorana quasiparticles. In circuit quantum electrodynamics quantum information is stored into small superconducting circuit elements whose interaction with electromagnetic radiation in the range of microwaves allows to process quantum information very efficiently. This approach has proven extremely useful for control and readout of superconducting qubits, i.e., small circuit elements that carry quantum information. Because of remarkably strong light-matter couplings that can be achieved for superconducting qubits in microwave resonators, the circuit quantum electrodynamics architecture is particularly useful to perform highly sensitive quantum measurements.Superconductivity by itself is an intriguing state of matter that shows a great variety of different phenomena. In particular, the discovery of topological phases in superconductors opened new horizons for quantum computation. One notable system that admits topological superconductivity is a semiconductor-superconductor nanowire with special zero modes occurring at its ends. These so-called Majorana zero modes are remarkably robust against decoherence and therefore well-suited for fault-tolerant quantum computation.The first part of this thesis examines the coupling of Majorana zero modes to electromagnetic radiation with microwave frequencies. The light-matter coupling mechanism that is considered here arises for Majorana zero modes located at a voltage-biased superconducting tunneling junction. The emission of microwave radiation in presence of Majorana zero modes gives rise to coherent radiation which is emitted at half of the usual Josephson frequency. On the basis of this fractional Josephson radiation, we propose a microwave readout scheme for Majorana qubits. As usual for typical measurements in circuit quantum electrodynamics, the proposed readout implements a quantum non-demolition measurement for Majorana qubit.In the last part of the thesis we propose a novel scheme for the implementation of measurement-induced entanglement between remote superconducting qubits as required for quantum communication. By detecting a single photon, that passes a Mach-Zehnder interferometric setup, deterministic entanglement with single-shot efficiency is achieved. This scheme essentially relies on the strong coupling between the qubits and the photon.
Quantum computation is a multifaceted field of research aiming for the physical realization of quantum systems and their manipulation. This thesis discusses the combination of two notable approaches in the pursuit of a fully operational quantum computer---circuit quantum electrodynamics and topological quantum computation based on Majorana quasiparticles. In circuit quantum electrodynamics quantum information is stored into small superconducting circuit elements whose interaction with electromagnetic radiation in the range of microwaves allows to process quantum information very efficiently. This approach has proven extremely useful for control and readout of superconducting qubits, i.e., small circuit elements that carry quantum information. Because of remarkably strong light-matter couplings that can be achieved for superconducting qubits in microwave resonators, the circuit quantum electrodynamics architecture is particularly useful to perform highly sensitive quantum measurements.Superconductivity by itself is an intriguing state of matter that shows a great variety of different phenomena. In particular, the discovery of topological phases in superconductors opened new horizons for quantum computation. One notable system that admits topological superconductivity is a semiconductor-superconductor nanowire with special zero modes occurring at its ends. These so-called Majorana zero modes are remarkably robust against decoherence and therefore well-suited for fault-tolerant quantum computation.The first part of this thesis examines the coupling of Majorana zero modes to electromagnetic radiation with microwave frequencies. The light-matter coupling mechanism that is considered here arises for Majorana zero modes located at a voltage-biased superconducting tunneling junction. The emission of microwave radiation in presence of Majorana zero modes gives rise to coherent radiation which is emitted at half of the usual Josephson frequency. On the basis of this fractional Josephson radiation, we propose a microwave readout scheme for Majorana qubits. As usual for typical measurements in circuit quantum electrodynamics, the proposed readout implements a quantum non-demolition measurement for Majorana qubit.In the last part of the thesis we propose a novel scheme for the implementation of measurement-induced entanglement between remote superconducting qubits as required for quantum communication. By detecting a single photon, that passes a Mach-Zehnder interferometric setup, deterministic entanglement with single-shot efficiency is achieved. This scheme essentially relies on the strong coupling between the qubits and the photon.
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.
The technology of quantum computing is believed to solve certain computational problems significantly faster than classical computers, enabling classically inaccessible problems. However, the technology is still in its infancy and it is still too early to conclude which physical system(s) will form the basis of tomorrow's quantum computer. Small-scale quantum processors, built on the superconducting transmon qubit, demonstrated already the anticipated quantum speed-up. Despite this tremendous milestone, scaling up to a full-fledged quantum computer is far from trivial: with these qubits, the flux-control causes crosstalk between qubits and heating, the room-temperature microwave control is hardly scalable and comes with high energetic radiation, and most importantly, the loss of quantum information in time leads to computational errors. Recent advances in various hybrid semiconductor materials enabled novel voltage-controlled transmons, gatemons, which are less susceptible to heating and crosstalk. Even more exciting, gatemons can be designed to host Majorana zero modes in a way that renders the qubit inherently protected against decoherence. In order to induce Majorana zero modes in such nanowire systems strong magnetic fields are required. Problematically, magnetic fields destroy the superconductivity that these microwave circuits rely on. In this thesis we integrate three types of hybrid Josephson junctions in magnetic field compatible microwave devices. Chapter 4 demonstrates the first graphene-based transmon. Due to the mono-atomic thickness of graphene in combination with magnetic field resilient coplanar waveguide resonators, we are able to operate the transmon circuit at an in-plane magnetic field of 1 Tesla. Chapter 5 embeds an InAs-Al semiconducting nanowire Josephson junction in a high quality factor superconducting transmission line resonator, demonstrating on-chip microwave generation. The gate-controllable semiconducting platform lends itself for fast pulse control, providing a perspective for the coherent on-chip manipulation of artificial two-level systems, in particular superconducting qubits such as transmons. Chapter 6 continues the development of InAs-Al nanowire transmons. The offset-charge-sensitive regime, additional plunger gates and magnetic field compatibility prepares the platform for the detection of coherent coupling between Majorana zero modes, a phenomena which unfortunately still remains unobserved. Additionally, we realise the first InSb-Al gatemon. The higher spin-orbit coupling makes InSb to be a preferred material in the search for Majorana signatures. Chapter 7 reports on the dynamics of quasiparticle tunneling events in real-time across the InAs-Al nanowire junction in a transmon architecture. The magnetic field compatibility of our device up to 1 Tesla together with in-situ voltage-controlled quasiparticle trap engineering, allows us to measure the survival of the charge-parity lifetime up to strong magnetic fields. A result which is extremely important in the research field of topological quantum computing, where the qubit space is defined in the charge-parity.
Quantum computers are devices that can solve certain problems faster than ordinary, classical computers. The fundamental units of quantum information are qubits, superpositions of two states, a "zero" state and a "one" state. There are various approaches to construct such two-level systems, among others, using superconducting circuits, trapped ions or photons. A common feature of these physical systems is that their coherence times are relatively short compared to the length of useful computations. Superconducting qubits, for instance, are currently the most advanced solid-state qubits, but they decohere after around 100 microseconds, and any information stored in these qubits is lost. On the other hand, useful quantum computations may require quantum information to survive on time scales that are many orders of magnitude longer, as their runtimes can reach several hours or even days. Topological quantum computing is an approach to construct qubits that survive for the entire duration of such a long computation. Topological quantum computing comes in two flavors. The condensed matter approach is to build error-resilient qubits using exotic quasiparticles in topological materials, most prominently Majorana zero modes in topological superconductors. Even though no such qubit has been built to date, the hope is that their coherence times may be significantly longer than the coherence times of currently available solid-state qubits, but are still expected to be too short for large-scale quantum computing. The quantum information approach is to combine many error-prone qubits to build more robust logical qubits using topological error-correcting codes, e.g., surface codes. Even though the first approach is hardware-based and the second approach is software-based, they are deeply related. With Majorana-based qubits, the main logical operations are Majorana fermion parity measurements. By replacing Majorana-based qubits with surface-code patches and parity measurements with lattice-surgery operations, schemes for quantum computation with Majorana-based qubits or with surface codes can be identical. In this thesis, we explore how to construct a large-scale topological fault-tolerant quantum computer that can perform useful quantum computations. Here, topological refers to the nature of the quantum error-correcting code, while the underlying hardware may be based on non-topological qubits, but could also be composed of Majorana-based qubits. We provide a complete picture of such a large-scale device, breaking down large quantum computations into logical qubits and logical operations, describing how these logical operations are performed on the level of physical qubits and physical gates, and finally discussing how these physical qubits can be pieced together in a Majorana-based system using topological superconducting nanowires.
Majorana qubits offer a promising way to store and manipulate quantum information by encoding it into the state of Majorana zero modes. As the information is stored in a topological property of the system, local noise cannot lead to decoherence. Manipulation of the information is achieved by braiding the zero modes. The measurement, however, is challenging as the information is well hidden and thus inherently hard to access. Here, we discuss a setup for measuring the state of a Majorana qubit by employing standard tools of microwave engineering. The basic physical effect that we employ is the fact that a voltage-biased Josephson junction hosting Majorana fermions allows photons to be emitted and absorbed at half the Josephson frequency. We show that in the dispersive regime, our setup allows us to perform a quantum nondemolition measurement and to reach the quantum limit. An appealing feature of our setup is that the interaction of the Majorana qubit with the measurement device can be turned on and off at will by changing the dc bias of the junction.
One of the key elements in circuit quantum electrodynamics (cQED) is the superconducting microwave resonator, which can enable strong coupling to and between nanoscale quantum systems, and as a result has emerged as an essential tool for both quantum information and quantum sensing. In this thesis, I will discuss new applications of superconducting resonators in both of these fields, which illustrates how quantum information and quantum sensing can be strongly tied together. As the field of quantum information continues to grow, various platforms for realizing a quantum computer have been proposed. One attractive platform is to control and manipulate the spin of an electron to form spin qubits, which have demonstrated a long coherence time, weak coupling to the environment, and high fidelity single- and two-qubit gates. However, scaling beyond two qubits has proven challenging, as has incorporating long-range coupling, although both are crucial for making spin qubits a scalable quantum computing platform. The confinement of electromagnetic fields in superconducting resonators enables strong coupling to small systems, such as the single electrons of a spin qubit. In the first part of this thesis, I will describe how a singlet-triplet spin qubit coupled to a high-impedance superconducting resonator constitutes a promising architecture for mediating long-ranged interactions based on a longitudinal coupling between the resonator and qubit. I will discuss the technical developments needed to successfully implement this novel type of interaction in a hybrid spin qubit-resonator device. This type of interaction has several advantages over the more commonly studied transverse coupling scheme: it does not require careful tuning of the qubit frequency to match that of the resonator, which facilitates scaling to a large number of qubits, and also avoids the loss of coherence, which is inevitable in resonant coupling. This type of interaction therefore opens promising paths for generating high-fidelity long-ranged two-qubit coupling by employing superconducting microwave resonators. By adapting techniques used in quantum information, we have also developed a new technique in quantum sensing, where we employ superconducting resonators as a probe of quantum materials such as mesoscopic systems, two-dimensional materials, and heterostructures, which are otherwise challenging to experimentally access with conventional techniques. In the second part of my thesis, I will discuss how implementing cQED techniques from quantum information allows us to probe the symmetry of the order parameter in unconventional superconductors. Unconventional superconductors are not readily available, and we therefore study an engineered system that comprises of interfacing a ferromagnet and superconductor, in which unconventional pairing can be induced into the ferromagnet. This platform allows us to simultaneously excite spin waves, a collective excitation of spins, in the ferromagnet, which opens new directions for studying interactions between coherently coupled magnons (quanta of spin waves) and unconventional superconducting states. Finally, by applying the same technique to a two-dimensional (2D) superconductor, tungsten ditelluride, we leverage this platform in an attempt to study the order parameter of its intrinsic superconducting phase. The nature of the order parameter in this material remains an open question, as both topological and superconducting phases can exist, possibly simultaneously. This work thus demonstrates a unique use of superconducting resonators beyond quantum information applications.
Topological quantum computing seeks to store and manipulate information in a protected manner using topological phases of matter. Information encoded in the degenerate state space of pairs of non-Abelian anyons or defects is robust to local perturbations, reducing its susceptiblity to environmental errors and potentially providing a scalable approach to quantum computing. However, topological quantum computing faces significant challenges, not least of which is identifying an experimentally accessible platform supporting non-Abelian topological physics. In this thesis, we critically analyze topological quantum computing with Majorana zero modes, non-Abelian defects of a topological superconductor. We identify intrinsic error sources for Majorana-based systems and propose quantum computing architectures that minimize their effects. Additionally, we consider a new approach for realizing and detecting non-Abelian topological defects in fractional Chern insulators. Topological quantum computing is predicated on the idea that braiding non-Abelian anyons adiabatically can implement quantum gates fault tolerantly. However, any braiding experiment will necessarily depart from the strict adiabatic limit. We begin by analyzing the nature of diabatic errors for anyon braiding, paying particular attention to how such errors scale with braiding time. We find that diabatic errors are unfavorably large and worryingly sensitive to details of the time evolution. We present a measurement-based correction protocol for such errors, and illustrate its application in a particular Majorana-based qubit design. We next propose designs for Majorana-based qubits operated entirely by a measurement-based protocol, thereby avoiding the diabatic errors discussed above. Our designs can be scaled into large two dimensional arrays amenable to long-term quantum computing goals, whose core components are testable in near-term devices. These qubits are robust to quasiparticle poisoning, anticipated to be one of the dominant error sources coupling to Majorana zero modes. We demonstrate that our designs support topologically protected Clifford operations and can be augmented to a universal gate set without requiring additional control parameters.While topological protection greatly suppresses errors, residual coupling to noise limits the lifetimes of our proposed Majorana-based qubits. We analyze the dephasing times for our quasiparticle-poisoning-protected qubits by calculating their charge distribution using a particle number-conserving formalism. We find that fluctuations in the electromagnetic environment couple to an exponentially suppressed topological dipole moment. We estimate dephasing times due to 1/f noise, thermal quasiparticle excitations, and phonons for different qubit sizes.The residual errors discussed above will necessarily require error correction for a sufficiently long quantum computation. We develop physically motivated noise models for Majorana-based qubits that can be used to analyze the performance of a quantum error correcting code. We apply this noise model to estimate pseudo-thresholds for a small subsystem code, identifying the relative importance of difference error processes from a fault tolerance perspective. Our results emphasize the necessity of suppressing long-lived quasiparticle excitations that can spread across the code.Finally, we turn our attention to a different platform that could host non-Abelian topological defects: fractional Chern insulators in graphene. We study the edge states of fractional Chern insulators using the field theory of fractional quantum Hall edges supplemented with a symmetry action. We find that lattice symmetries impose a quantized momentum difference for edge electrons in a fractional state of a C=2 Chern band. This momentum difference can be used to selectively contact the different edge states, thereby allowing detection of topological defects in the bulk with a standard four terminal measurement. Our proposal could be implemented in graphene subject to an artificially patterned lattice.