In this thesis, we generate a lower bound on the security of quantum protocols for secure function evaluation. Central to our proof is the concept of gentle measurements of quantum states, which do not greatly disturb a quantum state if a certain outcome is obtained with high probability. We show how a cheating party can leverage gentle measurements to learn more information than should be allowable. To quantify our lower bound, we reduce a specific cryptographic task known as die-rolling to secure function evaluation and use the concept of gentle measurements to relate their security notions. Our lower bound is then obtained using a known security bound for die-rolling known as Kitaev's bound. Due to the generality of secure function evaluation, we are able to apply this lower bound to obtain lower bounds on the security of quantum protocols for many quantum tasks. In particular, we provide lower bounds for oblivious transfer, XOR oblivious transfer, the equality function, the inner product function, Yao's millionaires' problem, and the secret phrase problem. Note that many of these lower bounds are the first of their kind, which is a testament to the utility of our lower bound. As a consequence, these bounds prove that unconditional security for quantum protocols is impossible for these applications, and since these are constant lower bounds, this rules out any form of boosting toward perfect security. Our work lends itself to future research on designing optimal protocols for the above listed tasks, and potentially others, by providing constant lower bounds to approximate or improve. / Master of Science / Quantifying the cryptographic security of quantum applications is the focus of much research in the quantum cryptography discipline. Quantum protocols might have better security than their classical counterparts, and this advantage might make the adoption of quantum cryptographic protocols a viable option. In this thesis, we introduce a method for generating constant lower bounds on the security of a variety of quantum applications. This is accomplished through finding a lower bound on the security of a protocol that is general, and by virtue of its generality, can be scoped to quantum applications such that the lower bound can be applied, and constant lower bounds generated for these applications. The significance of the work in this thesis is that many of the constant lower bounds presented are the first of their kind for these quantum applications, thus proving the impossibility of them having unconditional security. This also proves that one cannot asymptotically boost towards perfect security in these quantum tasks by any means. These constant lower bounds also provide a foundation for future work in the study of these quantum applications, specifically in the search for upper and lower bounds on their cryptographic security, as well as in the search for protocols that approximate these bounds.

The 20th century saw the first revolution of quantum mechanics, setting the rules for our understanding of light, matter, and their interaction. The 21st century is focused on using these quantum mechanical laws to develop technologies which allows us to solve challenging practical problems. One of the directions is the use quantum devices which promise to surpass the best computers and best known classical algorithms for solving certain tasks. Crucial to the design of realistic devices and technologies is to account for the open nature of quantum systems and to cope with their interactions with the environment. In the first part of this dissertation, we show how to tackle classical optimization problems of interest in the physical sciences within one of these quantum computing paradigms, known as quantum annealing (QA). We present the largest implementation of QA on a biophysical problem (six different experiments with up to 81 superconducting quantum bits). Although the cases presented here can be solved on a classical computer, we present the first implementation of lattice protein folding on a quantum device under the Miyazawa-Jernigan model. This is the first step towards studying optimization problems in biophysics and statistical mechanics using quantum devices. In the second part of this dissertation, we focus on the problem of excitonic energy transfer. We provide an intuitive platform for engineering exciton transfer dynamics and we show that careful consideration of the properties of the environment leads to opportunities to engineer the transfer of an exciton. Since excitons in nanostructures are proposed for use in quantum information processing and artificial photosynthetic designs, our approach paves the way for engineering a wide range of desired exciton dy- namics. Finally, we develop the theory for a two-dimensional electronic spectroscopic technique based on fluorescence (2DFS) and challenge previous theoretical results claiming its equivalence to the two-dimensional photon echo (2DPE) technique which is based on polarization. Experimental realization of this technique confirms our the- oretical predictions. The new technique is more sensitive than 2DPE as a tool for conformational determination of excitonically coupled chromophores and o↵ers the possibility of applying two-dimensional electronic spectroscopy to single-molecules.

<p dir="ltr">The two-dimensional electron gas (2DEG) in shallow InAs quantum wells, combined with epitaxial aluminum, is commonly used to study topological superconductivity. Key features include strong spin-orbit coupling, a high effective g-factor, and the ability to manage proximity-induced superconductivity. My thesis discusses two aspects of this unique material. In the first section, I report on the transport characteristics of shallow InGaAs/InAs/InGaAs quantum wells and evaluate the effect of modulation doping on these shallow InAs quantum well structures. We systematically investigate the magnetotransport properties in relation to doping density and spacer thickness. Optimized samples show peak mobilities exceeding 100,000 cm²/Vs at n_2DEG < 10¹²cm^-2 in gated Hall bar, marking the highest mobility observed in this type of heterostructure. Our findings suggest that the doping layer moves the electron wave function away from the surface, minimizing surface scattering and enhancing mobility. This mobility improvement does not compromise Rashba spin-orbit coupling or induced superconductivity. In the second section, motivated by a theoretical study by Peng et al., we explore tunneling spectroscopy measurements on DC current biased planar Josephson junctions made on an undoped hybrid epitaxial Al-InAs 2DEG heterostructure. We observe four tunneling conductance peaks in the spectroscopy that can be adjusted by DC current bias. Our analysis indicates that these results come from strong coupling between the tunneling probe and the superconducting leads, rather than from Floquet engineering. We also touch on potential improvements to the device's design and materials. This work lays the groundwork for further investigation of Floquet physics in planar Josephson junctions. This thesis ends with a discussion of other unusual physics that could be explored in these novel shallow InAs quantum wells coupled with epitaxial aluminum.</p>

In thinking about quantum causality one would like to approach rigorous QFT from outside the perspective of QFT, which one expects to recover only in a specific physical domain of quantum gravity. This thesis considers issues in causality using Category Theory, and their application to field theoretic observables. It appears that an abstract categorical Machian principle of duality for a ribbon graph calculus has the potential to incorporate the recent calculation of particle rest masses by Brannen, as well as the Bilson-Thompson characterisation of the particles of the Standard Model. This thesis shows how Veneziano n point functions may be recovered in such a framework, using cohomological techniques inspired by twistor theory and recent MHV techniques. This distinct approach fits into a rich framework of higher operads, leaving room for a generalisation to other physical amplitudes. The utility of operads raises the question of a categorical description for the underlying physical logic. We need to consider quantum analogues of a topos. Grothendieck's concept of a topos is a genuine extension of the notion of a space that incorporates a logic internal to itself. Conventional quantum logic has yet to be put into a form of equal utility, although its logic has been formulated in category theoretic terms. Axioms for a quantum topos are given in this thesis, in terms of braided monoidal categories. The associated logic is analysed and, in particular, elements of linear vector space logic are shown to be recovered. The usefulness of doing so for ordinary quantum computation was made apparent recently by Coecke et al. Vector spaces underly every notion of algebra, and a new perspective on it is therefore useful. The concept of state vector is also readdressed in the language of tricategories.

A fundamental component of theoretical computer science is the application of logic. Logic provides the formalisms by which we can model and reason about computational questions, and novel computational features provide new directions for the development of logic. From this perspective, the unusual features of quantum computation present both challenges and opportunities for computer science. Our existing logical techniques must be extended and adapted to appropriately model quantum phenomena, stimulating many new theoretical developments. At the same time, tools developed with quantum applications in mind often prove effective in other areas of logic and computer science. In this thesis we explore logical aspects of this fruitful source of ideas, with category theory as our unifying framework. Inspired by the success of diagrammatic techniques in quantum foundations, we begin by demonstrating the effectiveness of string diagrams for practical calculations in category theory. We proceed by example, developing graphical formulations of the definitions and proofs of many topics in elementary category theory, such as adjunctions, monads, distributive laws, representable functors and limits and colimits. We contend that these tools are particularly suitable for calculations in the field of coalgebra, and continue to demonstrate the use of string diagrams in the remainder of the thesis. Our coalgebraic studies commence in chapter 3, in which we present an elementary formulation of a representation result for the unitary transformations, following work developed in a fibrational setting in [Abramsky, 2010]. That paper raises the question of what a suitable "fibred coalgebraic logic" would be. This question is the starting point for our work in chapter 5, in which we introduce a parameterized, duality based frame- work for coalgebraic logic. We show sufficient conditions under which dual adjunctions and equivalences can be lifted to fibrations of (co)algebras. We also prove that the semantics of these logics satisfy certain "institution conditions" providing harmony between syntactic and semantic transformations. We conclude by studying the impact of parameterization on another logical aspect of coalgebras, in which certain fibrations of predicates can be seen as generalized invariants. Our focus is on the lifting of coalgebra structure along a fibration from the base category to an associated total category of predicates. We show that given a suitable parameterized generalization of the usual liftings of signature functors, this induces a "fibration of fibrations" capturing the relationship between the two different axes of variation.

Este projeto se destina ao estudo de sistemas quânticos não relativísticos de dois, quatro e oito níveis de energia que descrevem partículas com spin s=1/2 sujeitas à ação de campos externos e interagentes entre si. São apresentadas soluções exatas para as equações que regem esses sistemas. Tais sistemas possuem uma vasta aplicação em diversas áreas da física, dentre as quais é possível destacar a computação quântica. Possíveis aplicações dos resultados são a construção de portas lógicas quânticas universais. Estas portas lógicas quânticas representam um elemento essencial no desenvolvimento dos chamados computadores quânticos. A análise e a implementação destes computadores quânticos exige a manipulação de sistemas de vários níveis, sujeitos a campos externos dependentes do tempo. Neste trabalho é apresentada a solução para o assim chamado Problema de Rabi, um particular problema de dois níveis. Um exemplo de solução para o sistema de quatro níveis, aqui relativo a um problema de dois spins também é discutido. Foram obtidas soluções exatas para sistemas de oito níveis cuja possível aplicação é a Correção Quântica de Erros. / This project aims to study the non-relativistic quantum systems of two, four and eight energy levels that describe particles with spin s=1/2 in external .elds and interacting with each other. We find exact analitical solutions for these systems. Such systems have extensive applications in various areas of physics, among which its possible to highlight quantum computing. Possible applications of the results are the construction of quantum universal logic gates.These quantum logic gates are an essential element in the development of so-called quantum computers. The analysis and implementation of quantum computers requires handling systems of various levels, subject to time-dependent external fields. This work presents a solution to the so-called Rabi problem, a particular problem at two levels. An example of a solution to the system of four levels, related to two spins problem is also investigated. We obtained exact solutions for systems of eight levels with possible application to the Quantum Error Correction.

Ocorre atualmente um movimento de aumento da importância que a manutenção da segurança da informação vem adquirindo em redes de informação de crítica. Ao longo das últimas décadas a utilização de ferramentas criptográficas, especialmente aquelas baseadas em problemas de díficil solução computacional, foram suficientes para garantir a segurança dos sistemas de comunicação. Contudo, o desenvolvimento da nova técnica de processamento de informação conhecida como computação quântica e os resultados téoricos e experimentais apresentados por esta mostram que é possível inviabilizar alguns dos sistemas de criptografia atuais amplamente utilizados. A existência de tal vulnerabilidade representa um fator crítico em redes em que falhas de segurança da informação podem estar associadas a riscos de segurança física. Uma alternativa para os métodos criptográficos atuais consiste na utilização de sistemas quânticos na obtenção de um método criptográfico, o que se conhece como criptografia quântica. Este novo paradigma tem seu fundamento resistente mesmo na presença de capacidade tecnológica ilimitada, incluindo o cenário com disponibilidade de computação quântica. Este trabalho tem como objetivo levantar os impactos que o desenvolvimento da computação quântica têm sobre a segurança dos atuais sistemas criptográficos, apresentar e desenvolver alternativas de protocolos de criptografia quântica disponíveis, e realizar um estudo de caso por meio da avaliação da utilização de criptografia quântica no contexto da Aeronautical Telecommunication Network (ATN). Isto é feito por meio do desenvolvimento de um ambiente de simulacão que permite avaliar o comportamento de um protocolo de criptografia quântica em um cenário em um ambiente com requisitos de missão crítica, como é o caso da ATN. / The importance of security maintenance in critical information networks has been rising in recent times. Over the past decades, the utilization of cryptography tools, mainly those based on computationally intractable problems, was enough to ensure the security of communications systems. The development of the new information processing technique known as quantum computation and the theoretical and experimental results showed by this approach demonstrated that could be possible to cripple the current widely used cryptography techniques. This vulnerability represents a critical issue for networks where a security fault could be associated to a safety fault. An alternative for the current cryptography methods consists in the utilization of quantum systems to obtain a new cryptographic method. The new paradigm presented by this approach has solid principles even in the presence of unlimited computational capacity, including the scenario with availability of quantum computation. The aim of this work is the assessment of impacts that the development of quantum computation has over the current cryptographic methods security, the presentation and development of alternatives based on quantum cryptography protocols, and the development of a case study using the case of Aeronautical Telecommunication Network (ATN). This aim is reached by means of the development of a simulation environment that allows the evaluation of a quantum cryptography protocol behavior in an environment with mission critical requirements, like the ATN case.

Ocorre atualmente um movimento de aumento da importância que a manutenção da segurança da informação vem adquirindo em redes de informação de crítica. Ao longo das últimas décadas a utilização de ferramentas criptográficas, especialmente aquelas baseadas em problemas de díficil solução computacional, foram suficientes para garantir a segurança dos sistemas de comunicação. Contudo, o desenvolvimento da nova técnica de processamento de informação conhecida como computação quântica e os resultados téoricos e experimentais apresentados por esta mostram que é possível inviabilizar alguns dos sistemas de criptografia atuais amplamente utilizados. A existência de tal vulnerabilidade representa um fator crítico em redes em que falhas de segurança da informação podem estar associadas a riscos de segurança física. Uma alternativa para os métodos criptográficos atuais consiste na utilização de sistemas quânticos na obtenção de um método criptográfico, o que se conhece como criptografia quântica. Este novo paradigma tem seu fundamento resistente mesmo na presença de capacidade tecnológica ilimitada, incluindo o cenário com disponibilidade de computação quântica. Este trabalho tem como objetivo levantar os impactos que o desenvolvimento da computação quântica têm sobre a segurança dos atuais sistemas criptográficos, apresentar e desenvolver alternativas de protocolos de criptografia quântica disponíveis, e realizar um estudo de caso por meio da avaliação da utilização de criptografia quântica no contexto da Aeronautical Telecommunication Network (ATN). Isto é feito por meio do desenvolvimento de um ambiente de simulacão que permite avaliar o comportamento de um protocolo de criptografia quântica em um cenário em um ambiente com requisitos de missão crítica, como é o caso da ATN. / The importance of security maintenance in critical information networks has been rising in recent times. Over the past decades, the utilization of cryptography tools, mainly those based on computationally intractable problems, was enough to ensure the security of communications systems. The development of the new information processing technique known as quantum computation and the theoretical and experimental results showed by this approach demonstrated that could be possible to cripple the current widely used cryptography techniques. This vulnerability represents a critical issue for networks where a security fault could be associated to a safety fault. An alternative for the current cryptography methods consists in the utilization of quantum systems to obtain a new cryptographic method. The new paradigm presented by this approach has solid principles even in the presence of unlimited computational capacity, including the scenario with availability of quantum computation. The aim of this work is the assessment of impacts that the development of quantum computation has over the current cryptographic methods security, the presentation and development of alternatives based on quantum cryptography protocols, and the development of a case study using the case of Aeronautical Telecommunication Network (ATN). This aim is reached by means of the development of a simulation environment that allows the evaluation of a quantum cryptography protocol behavior in an environment with mission critical requirements, like the ATN case.

This project is a comprehensive investigation into the application of the exchange interaction, particularly with the realization of the SWAP^1/n quantum operator, in quantum information processing. We study the generation, characterization and application of entanglement in such systems. Given the non-commutativity of neighbouring SWAP^1/n gates, the mathematical study of combinations of these gates is an interesting avenue of research that we have explored, though due to the exponential scaling of the complexity of the problem with the number of qubits in the system, numerical techniques, though good for few-qubit systems, are found to be inefficient for this research problem when we look at systems with higher number of qubits. Since the group of SWAP^1/n operators is found to be isomorphic to the symmetric group Sn, we employ group-theoretic methods to find the relevant invariant subspaces and associated vector-states. Some interesting patterns of states are found including onedimensional invariant subspaces spanned by W-states and the Hamming-weight preserving symmetry of the vectors spanning the various invariant subspaces. We also devise new ways of characterizing entanglement and approach the separability problem by looking at permutation symmetries of subsystems of quantum states. This idea is found to form a bridge with the entanglement characterization tool of Peres-Horodecki's Partial Positive Transpose (PPT), for mixed quantum states. We also look at quantum information taskoriented 'distance' measures of entanglement, besides devising a new entanglement witness in the 'engle'. In terms of applications, we define five different formalisms for quantum computing: the circuit-based model, the encoded qubit model, the cluster-state model, functional quantum computation and the qudit-based model. Later in the thesis, we explore the idea of quantum computing based on decoherence-free subspaces. We also investigate ways of applying the SWAP^1/n in entanglement swapping for quantum repeaters, quantum communication protocols and quantum memory.

Orientador: Alexandre César Rodrigues da Silva / Abstract: Research in the field of quantum circuits has increased as technology advances in the development of quantum computers. IBM offers access to quantum computers via the cloud service called IBM Q. However, these architectures have some restrictions regarding the types of quantum gates that can be realized. This work proposes a methodology for the mapping of quantum and reversible circuits to the architectures made available by the IBM Q project. The methodology consists in finding CNOT mappings using a set of defined qubits movements to satisfy the architectures constraints by adding as few gates as possible. In order to reduce the number of CNOT gates needing mapping, the permutation of the circuit can be changed. One alternative to find this permutation is trough exhaustive search. However, is not feasible as the number of qubit increases. To solve this problem, the permutation problem was formulated as an Integer Linear Programming problem. The mapping of quantum circuits realized with non-implementable gates and reversible Toffoli circuits to the IBM quantum architectures were proposed in this work as well. This was done by adapting the developed CNOT mappings along with the Integer Linear Programming formulation. The proposed methodology was evaluated by mapping quantum and reversible circuits to an IBM quantum architectures with 5 and 16 qubits. The results were compared with two algorithms that map quantum circuits to IBM architectures. The cost metric used in the evalua... (Complete abstract click electronic access below) / Resumo: Pesquisa no campo de circuitos quânticos tem alavancado conforme a tecnologia avança no desenvolvimento de computadores quânticos. Atualmente, a IBM oferece acesso a computadores quânticos através do serviço em nuvem chamado IBM Q. No entanto, essas arquiteturas têm algumas restrições com relação aos tipos de portas quânticas e qubits em que uma porta CNOT pode ser implementada. Neste trabalho foi proposta uma metodologia para o mapeamento de circuitos quânticos e reversíveis para as arquiteturas disponibilizadas pelo projeto IBM Q. A metodologia consiste em mapear as portas CNOT utilizando uma série de movimentos de qubits, mantendo a permutação do circuito inalterada. A fim de reduzir o número de portas CNOT não implementáveis, a permutação do circuito pode ser alterada. Uma alternativa para encontrar essa permutação é a busca exaustiva. No entanto, é inviável conforme o número de qubits aumenta. Para resolver este problema, o problema de permutação foi formulado como um problema de Programação Linear Inteira. Como a metodologia é facilmente adaptável, o mapeamento de circuitos quânticos utilizando portas quânticas não implementáveis e circuitos reversíveis Toffoli também foram propostas neste trabalho. A avaliação da metodologia proposta foi feita com a realização do mapeamento de circuitos quânticos e reversíveis para arquiteturas quânticas com 5 e 16 qubits. Os resultados foram comparados com dois algoritmos que mapeiam circuitos quânticos para arquiteturas IBM. A métric... (Resumo completo, clicar acesso eletrônico abaixo) / Doutor