Die fortschreitende Entwicklung immer leistungsstärkerer Quantencomputer bedroht die Informationssicherheit kryptographischer Anwendungen, die auf dem Faktorisierungsproblem oder dem Problem des diskreten Logarithmus beruhen. Die US-amerikanische Standardisierungsbehörde NIST startete 2017 ein Projekt mit dem Ziel, Kryptographiestandards zu entwickeln, die gegen Angriffe von Quantenrechnern resistent sind. Einer der Kandidaten ist SIKE (Supersingular Isogeny Key Encapsulation), der einzige Vertreter isogeniebasierter Kryptographie im Standardisierungsverfahren. Diese Diplomarbeit enthält eine weitgehend in sich abgeschlossene Beschreibung der SIKE-Protokolle, Sicherheitsbetrachtungen sowie eine einfache Implementierung des Kryptosystems.:1. Einleitung 2. Grundlegende Definitionen 2.1. Elliptische Kurven 2.2. Punktaddition 2.3. Montgomery-Kurven 2.4. Isogenien 2.5. Der Diffie-Hellman-Schlüsselaustausch 2.6. Das Elgamal-Kryptosystem 3. Supersingular Isogeny Key Encapsulation 3.1. Supersingular Isogeny Diffie-Hellman Key Exchange 3.2. Erzeugung der Systemparameter 3.3. Erzeugung der Schlüsselpaare 3.4. Berechnung der gemeinsamen Kurve 3.5. Vom Schlüsselaustausch zum Kryptosystem 3.6. Schlüsseleinschluss (Key Encapsulation) 3.7. Implementierungen 4. Sicherheitsbetrachtungen 4.1. Ciphertext indistinguishability 4.2. Größe der Parameter 4.3. Weitere Aspekte 5. Zusammenfassung A. Implementierung

Heutzutage können ultrakalte Atome in unterschiedlichsten optischen Fallenpotenzialen eingefangen werden, während sich ihre Wechselwirkung durch die Ausnutzung von magnetischen Feshbachresonanzen kontrollieren lässt. Der Einschluss und die resonante Wechselwirkung können zu einer starken Korrelation der Atome führen, welche es erlaubt, mit ihnen physikalische Phänomene zu simulieren, deren Simulation mit heutigen Computern nicht durchführbar wäre. Eine maßgeschneiderte Kontrolle der Korrelationen könnte es schließlich ermöglichen, mit ultrakalten Atomen einen Quantencomputer zu implementieren. Um die Flexibilität und gute Kontrollierbarkeit ultrakalter Atome voll ausnutzen zu können, ist das Ziel dieser Dissertation die präzise theoretische Beschreibung stark korrelierter, eingeschlossener Atome an einer Feshbachresonanz. Das Wechselspiel zwischen dem Einschluss der Atome und einer Feshbachresonanz wird in dieser Arbeit zunächst anhand eines von Grund auf hergeleiteten analytischen Modells einer Feshbachresonanz zwischen Atomen in einer harmonischen Falle untersucht. Basierend auf diesem Modell wird ein Ansatz entwickelt, wechselwirkende Atome an einer Feshbachresonanz in einem optischen Gitter über ein Bose-Hubbard-Modell zu beschreiben. Im Gegensatz zu aufwendigeren numerischen Methoden erlaubt das Bose-Hubbard-Modell mit der Einbeziehung nur weniger Blochbänder die präzise Vorhersage der Eigenenergien und des dynamischen Verhaltens der Atome im optischen Gitter. Weiterhin wird eine Methode zur Lösung der zeitabhängingen Schrödingergleiung für zwei wechselwirkende Atome in einem dynamischen optischen Gitter entwickelt. Schließlich wird ein Ansatz vorgestellt, wie sich mit ultrakalten Atomen in einem dynamischen optischen Gitter ein Quantencomputer implementieren ließe. Als Quantenregister dient der korrelierte Mott-Zustand von repulsiv wechselwirkenden Atomen. Quantenoperationen werden durch periodisches Wackeln des optischen Gitters getrieben. / Today, ultracold atoms can be confined in various optical trapping potentials, while their mutual interaction can be controlled by magnetic Feshbach resonances. The confinement and resonant interaction can lead to a strong correlation of the atoms, which allows for the quantum simulation of physical phenomena whose classical simulation is computationally intractable. A tailored control of these correlations might eventually enable the implementation of a quantum computer with ultracold atoms. In order to take advantage of the flexibility and precise control of ultracold atoms, this thesis aims to provide a precise theoretical description of strongly correlated, confined atoms at a magnetic Feshbach resonance. The interplay between the confinement of the atoms and the Feshbach resonance is investigated by deriving from first principles a model that enables the complete analytic description of harmonically trapped ultracold atoms at a Feshbach resonance. This model is subsequently used to develop a Bose-Hubbard model of atoms in an optical lattice at a Feshbach resonance. In contrast to more elaborate numerical calculations, the model can predict the eigenenergies and the dynamical behavior of atoms in an optical lattice with high accuracy including only a small number of Bloch bands. Furthermore, a method id developed that solves the time-dependent Schrödinger equation for two interacting atoms in a dynamic optical lattice. Finally, a proposal for the implementation of a quantum computer with ultracold atoms in a dynamic optical lattice is presented. It utilizes the correlated Mott-insulator state of repulsively interacting atoms as a quantum register. Quantum operations are driven by a periodic shaking of the optical lattice.

<p>The challenge of building scalable quantum computer lies in striking appropriate balance between designing a reliable system architecture from large number of faulty computational resources and improving the physical quality of system components. The detailed investigation of performance variation with physics of the components and the system architecture requires adequate performance simulation tool. In this thesis we demonstrate a software tool capable of (1) mapping and scheduling the quantum circuit on a realistic quantum hardware architecture with physical resource constraints, (2) evaluating the performance metrics such as the execution time and the success probability of the algorithm execution, and (3) analyzing the constituents of these metrics and visualizing resource utilization to identify system components which crucially define the overall performance.</p><p>Using this versatile tool, we explore vast design space for modular quantum computer architecture based on trapped ions. We find that while success probability is uniformly determined by the fidelity of physical quantum operation, the execution time is a function of system resources invested at various layers of design hierarchy. At physical level, the number of lasers performing quantum gates, impact the latency of the fault-tolerant circuit blocks execution. When these blocks are used to construct meaningful arithmetic circuit such as quantum adders, the number of ancilla qubits for complicated non-clifford gates and entanglement resources to establish long-distance communication channels, become major performance limiting factors. Next, in order to factorize large integers, these adders are assembled into modular exponentiation circuit comprising bulk of Shor's algorithm. At this stage, the overall scaling of resource-constraint performance with the size of problem, describes the effectiveness of chosen design. By matching the resource investment with the pace of advancement in hardware technology, we find optimal designs for different types of quantum adders. Conclusively, we show that 2,048-bit Shor's algorithm can be reliably executed within the resource budget of 1.5 million qubits.</p> / Dissertation

Topological order is a new field of research involving exotic physics. Among other things it has been suggested as a means for realising fault-tolerant quantum computation. Topological degeneracy, i.e. the ground state degeneracy of a topologically ordered state, is one of the quantities that have been used to characterize such states. Topological order has also been suggested as a possible quantum information storage. We study two-dimensional lattice models defined on a closed manifold, specifically on a torus, and find that these systems exhibit topological degeneracy proportional to the genus of the manifold on which they are defined. We also find that the addition of lattice dislocations increases the ground state degeneracy, a behaviour that can be interpreted as artificially increasing the genus of the manifold. We derive the fusion and braiding rules of the model, which are then used to calculate the braiding properties of the dislocations themselves. These turn out to resemble non-abelian anyons, a property that is important for the possibility to achieve universal quantum computation. One can also emulate lattice dislocations synthetically, by adding an external field. This makes them more realistic for potential experimental realisations. / Topologisk ordning är ett nytt område inom fysik som bland annat verkar lovande som verktyg för förverkligandet av kvantdatorer. En av storheterna som karakteriserar topologiska tillstånd är det totala antalet degenererade grundtillstånd, den topologiska degenerationen. Topologisk ordning har också föreslagits som ett möjligt sätt att lagra kvantdata. Vi undersöker tvådimensionella gittermodeller definierade på en sluten mångfald, specifikt en torus, och finner att dessa system påvisar topologisk degeneration som är proportionerlig mot mångfaldens topologiska genus. När dislokationer introduceras i gittret finner vi att grundtillståndets degeneration ökar, något som kan ses som en artificiell ökning av mångfaldens genus. Vi härleder sammanslagningsregler och flätningsregler för modellen och använder sedan dessa för att räkna ut flätegenskaperna hos själva dislokationerna. Dessa visar sig likna icke-abelska anyoner, en egenskap som är viktiga för möjligheten att kunna utföra universella kvantberäkningar. Det går också att emulera dislokationer i gittret genom att lägga på ett yttre fält. Detta gör dem mer realistiska för eventuella experimentella realisationer.

Cette thèse décrit une série de travaux réalisés dans le contexte des qubits de spins, allant de l'utilisation de ces qubits pour stocker de l'information à leur utilisation comme détecteurs ultra-sensibles. Nous utilisons des hétérostructures semi-conductrices d'arséniure de gallium dans lesquelles un électron unique peut être isolé au sein d'un piège électrostatique, une boîte quantique. Le spin de cet électron peut être utilisé pour encoder de l'information, et la boîte quantique contenant ce spin unique est alors vue comme un qubit (quantum bit). Au cours de cette thèse nous démontrons la réalisation expérimentale du transport d'un électron unique le long d'un circuit fermé au sein d'un système composé de quatre boîtes quantiques couplées. En considérant l'interaction spin-orbite, cette expérience ouvre la voie vers des manipulations cohérentes de spins utilisant des effets topologiques. Dans le contexte de l'ordinateur quantique et des qubits de spins, nous étudions les portes logiques à deux qubits. Dans le cadre de deux boîtes quantiques couplées par une barrière tunnel, nous démontrons qu'en contrôlant localement le champ magnétique, la porte logique à deux qubits évoluent de la porte SWAP à la porte C-phase. Nous démontrons ainsi la faisabilité d'une porte C-phase. Finalement nous montrons l'utilisation d'un qubit de spin comme un détecteur de charge ultrasensible. Un singlet-triplet qubit est un système quantique qui peut être réglé de manière à être extrêmement sensible à l'environnement électrostatique. Nous démontrons la faisabilité d'un tel détecteur, et nous montrons qu'il peut être utilisé pour détecter un électron unique. / In this thesis we described a series of experimental works, which have been realized in the context of spin qubits, going from their use as information carriers to their use as very sensitive detectors. We use AlGaAs semiconducting heterostructures in which a single electron can be isolated in an electrostatic trap, the so-called quantum dot. The electron spin can be used in order to encode information, and the quantum dot containing this electron can therefore be seen as a qubit (quantum bit). During this thesis we demonstrate the first experimental realization of a single electron transport along a closed path inside a system composed of four coupled quantum dots. By considering spin-orbit interaction, this experiment opens the way toward coherent topological spin manipulations. In the context of quantum computing and spin qubits, we study the two-qubit gates. By considering two tunnel coupled quantum dots, we demonstrate by controlling the local Zeeman splitting that the natural two-qubit gate for spin qubits evolves from the SWAP gate to the C-phase gate. This work demonstrates the feasibility of the C-phase gate. Finally we use spin qubits as very sensitive detectors. A singlet-triplet qubit is a quantum system which can be tuned in order to be very sensistive to the electrostatic environment. Here we report the use of such a qubit to detect a single electron transported next to the detector.

Quantum computers can theoretically perform certain tasks which classical computers at realistic times could not. Operating a quantum computer requires precise control over the system, for instance achieved by adiabatic evolution, and isolation from the environment to retain coherence. This report combines these two, somewhat contradicting, error preventing techniques. To reduce the run-time a transitionless quantum driving algorithm, or, adiabatic shortcut, is employed. The notion of Noiseless Subsystems (NS), a generalization of decoherence free subspaces, are used for robustness against environmental decoupling, by creating logical qubits which act as a noiseless code. Furthermore, the adiabatic shortcut for the NS code is applied to a refocusing scheme (spin-echo) in order to remove the dynamical phase, sensitive to error propagation, so that only the Berry phase is effectively picked up. The corresponding Hamiltonian is explicitly derived for the only two cases of two-dimensional NS: N=3,4 qubits with total spin of j=1/2,0, respectively. This constitutes geometric quantum computation (GQC) enacting a universal single-qubit gate, which is also explicitly derived. / Kvantdatorer kan teoretiskt utföra vissa uppgifter som klassiska datorer vid realistiska tider inte kan. Att köra en kvantdator kräver exakt kontroll över systemet, till exempel genom adiabatisk utvecking, och isolering från omgiviningen för att behålla koherens. Denna rapport kombinerar dessa två, något motsägelsefulla, tekniker för felhantering. För att minska körtiden används en övergångsfri kvantkörningsalgoritm, också kallad adiabatisk genväg. Konceptet brusfria delsystem, en generalisering av dekoherensfria underrum, används för robusthet mot sammanflätning med omgivningen genom att skapa logiska kvantbitar som fungerar som en brusfri kod. Vidare tillämpas den adiabatiska genvägen för den brusfria koden på ett spinn-eko för att eliminera den dynamiska fasen, som är känslig för felpropagering, så att endast Berrys fas, som är okänslig för felpropagering, effektivt plockas upp. Motsvarande Hamiltonian härleds uttryckligen för de enda två fallen av tvådimensionella brusfria delsystem: 3 eller 4 kvantbitar med respektive totalspinn j = 1/2 och 0. Detta möjliggör beräkning med en geometrisk kvantdator baserad på en universell en-kvantbitsgrind, som också härleds explicit.

The aim of this thesis is to present an extension to the string graphs of Dixon, Duncan and Kissinger that allows the finite representation of certain infinite families of graphs and graph rewrite rules, and to demonstrate that a logic can be built on this to allow the formalisation of inductive proofs in the string diagrams of compact closed and traced symmetric monoidal categories. String diagrams provide an intuitive method for reasoning about monoidal categories. However, this does not negate the ability for those using them to make mistakes in proofs. To this end, there is a project (Quantomatic) to build a proof assistant for string diagrams, at least for those based on categories with a notion of trace. The development of string graphs has provided a combinatorial formalisation of string diagrams, laying the foundations for this project. The prevalence of commutative Frobenius algebras (CFAs) in quantum information theory, a major application area of these diagrams, has led to the use of variable-arity nodes as a shorthand for normalised networks of Frobenius algebra morphisms, so-called "spider notation". This notation greatly eases reasoning with CFAs, but string graphs are inadequate to properly encode this reasoning. This dissertation firstly extends string graphs to allow for variable-arity nodes to be represented at all, and then introduces !-box notation – and structures to encode it – to represent string graph equations containing repeated subgraphs, where the number of repetitions is abitrary. This can be used to represent, for example, the "spider law" of CFAs, allowing two spiders to be merged, as well as the much more complex generalised bialgebra law that can arise from two interacting CFAs. This work then demonstrates how we can reason directly about !-graphs, viewed as (typically infinite) families of string graphs. Of particular note is the presentation of a form of graph-based induction, allowing the formal encoding of proofs that previously could only be represented as a mix of string diagrams and explanatory text.

Microelectronics has shaped the world beyond what was thought possible at the time of its advent. One area of current research in this field is on the solid-state Si:P-based quantum computer (QC). In this machine, each qubit requires an individually addressed fast control-pulse for non-adiabatic drive and measure operations. Additionally, it is increasingly becoming important to be able to interface nanoelectronics with complementary metal-oxide-semiconductor (CMOS) technology. In this work, I have designed and demonstrated full-custom mixed-mode and full-digital fast control-pulse generators fabricated in a silicon-on-sapphire (SOS) CMOS commercial foundry process ?? a radio-frequency (RF) CMOS technology. These circuits are, fundamentally, fast monostable multivibrators. Initially, after the design specifications were decided upon, I characterized NFET and PFET devices and a n+-diffusion resistor from 500 nm and 250 nm commercial SOS-CMOS processes. Measuring their conductance curves at 300 300 K, 4.2 2 K, and sub-K (30 30 mK base to 1000 1000 mK) showed that they function with desirable behaviour although exhibiting some deviations from their 300 300 K characteristics. The mixed-mode first generation control-pulse generator was demonstrated showing that it produced dwell-time adjustable pulses with 100 100 ps rise-times at 300 K, 4.2 2 K, and sub-K with a power dissipation of 12 12 uW at 100 100 MHz. The full-digital second generation control-pulse generator was demonstrated showing accurately adjustable dwell-times settable via a control-word streamed synchronously to a shift-register. The design was based on a ripple-counter with provisions for internal or external clocking. This research has demonstrated that SOS-CMOS technology is highly feasible for the fabrication of control microelectronics for a Si:P-based QC. I have demonstrated full-custom SOS-CMOS mixed-mode and full-digital control circuits at 300 300 K, 4.2 2 K, and sub-K which suitable for qubit control.

The spins of phosphorus doped in silicon are potential candidates for a quantum computing device, with models based on the use of nuclear and/or electron spins suggested. For a quantum computing device, several essential criteria must be demonstrated before any physical implementation, and these include qubit control gates, long decoherence time and scalability. Scalability and compatibility with existing fabrication technologies are strong points in favour of a silicon based system. For spin based schemes, silicon has the potential to provide a host with zero nuclear spin (isotopically purifed 28Si) and also the phosphorus donor provides both nuclear and electron half integer spins (ideal case). In this work, a magnetic resonance method (electron spin resonance) was utilised to investigate these critical issues (controllable quantum gates and decoherence time) for the electron spins of phosphorus donors in silicon. Electron spin resonance (ESR) studies of an ensemble of phosphorus electron spins in silicon were conducted via both continuous wave and pulsed methods. For pulsed ESR operations, two low temperature (4 K and millikelvin) X-band pulsed ESR systems were built. They were designed especially to suit Si:P decoherence time measurements. The design, modelling, construction and evaluation of the probe heads are described. With the aid of computer simulations, the performance of the probe heads was optimised and a rectangular loop gap resonator was found to be the most suitable for wafer type samples. The resonant frequency, quality factor, and coupling coeffcient were calculated via simulation and are in reasonable agreement with experimental results. This demonstrates the effectiveness of such simulations as a tool for optimising the probe head performance. A millikelvin pulsed ESR system was set up through the combination of a dilution refrigerator, superconducting magnet and the in-house construction of a pulsed ESR spectrometer. This novel system allows pulsed ESR experiments on an ensemble system to be realised down to the millikelvin temperature range, hence providing conditions considered most favourable for quantum computing studies. The use of light in combination with the pulsed ESR systems was also explored in an endeavour to overcome the problem of very long spin-lattice relaxation time, T1, allowing the decoherence time to be measured more effciently. With these novel low temperature pulsed ESR units, two-pulse electron spin echo experiments were conducted on phosphorus donors in silicon (both natural silicon (natSi) and 28Si) with the phosphorus concentration in the range of 1015- 1016 P/cm3 and to lower temperatures than previously investigated. Decoherence times measured for both natSi:P and 28Si:P (with similar donor concentrations) were longer than previously reported. Discussions on several effective ways to obtain even longer Si:P decoherence times including variations to sample configurations and experimental conditions are presented. In addition to the pulsed ESR studies, the Si:P controllable quantum gate functions, A gate and J gate, were examined by the continuous wave technique via Stark shift and exchange interaction experiments respectively. Stark shift experiments on bulk samples were carried out to investigate possible manipulation of the spins by the applied electric field. Continuous wave ESR was also used to examine low energy ion implanted Si:P devices, both by single (P+) and dimer (P+2 ) implanted donors. The outcomes from these studies provide materials information useful in formulating a strategy toward the Si:P device fabrication via the top down approach.

Microelectronics has shaped the world beyond what was thought possible at the time of its advent. One area of current research in this field is on the solid-state Si:P-based quantum computer (QC). In this machine, each qubit requires an individually addressed fast control-pulse for non-adiabatic drive and measure operations. Additionally, it is increasingly becoming important to be able to interface nanoelectronics with complementary metal-oxide-semiconductor (CMOS) technology. In this work, I have designed and demonstrated full-custom mixed-mode and full-digital fast control-pulse generators fabricated in a silicon-on-sapphire (SOS) CMOS commercial foundry process ?? a radio-frequency (RF) CMOS technology. These circuits are, fundamentally, fast monostable multivibrators. Initially, after the design specifications were decided upon, I characterized NFET and PFET devices and a n+-diffusion resistor from 500 nm and 250 nm commercial SOS-CMOS processes. Measuring their conductance curves at 300 300 K, 4.2 2 K, and sub-K (30 30 mK base to 1000 1000 mK) showed that they function with desirable behaviour although exhibiting some deviations from their 300 300 K characteristics. The mixed-mode first generation control-pulse generator was demonstrated showing that it produced dwell-time adjustable pulses with 100 100 ps rise-times at 300 K, 4.2 2 K, and sub-K with a power dissipation of 12 12 uW at 100 100 MHz. The full-digital second generation control-pulse generator was demonstrated showing accurately adjustable dwell-times settable via a control-word streamed synchronously to a shift-register. The design was based on a ripple-counter with provisions for internal or external clocking. This research has demonstrated that SOS-CMOS technology is highly feasible for the fabrication of control microelectronics for a Si:P-based QC. I have demonstrated full-custom SOS-CMOS mixed-mode and full-digital control circuits at 300 300 K, 4.2 2 K, and sub-K which suitable for qubit control.