Neste trabalho apresentamos um novo algoritmo quântico eficiente para o Problema do Subgrupo Oculto (PSO) sobre uma classe especial de grupos metacíclicos, Z_p times Z_q^s, com q | (p-1) e p/q= poli(log p), onde p, q são números primos ímpares distintos e s um inteiro positivo qualquer. Em um contexto mais geral, sem impor uma relação entre p e q obtemos um algoritmo quântico com complexidade de tempo 2^{O(sqrt{log p})}. Em qualquer caso, esses resultados são melhores que qualquer algoritmo clássico para o mesmo fim, cuja complexidade é Omega(sqrt{p}). Apresentamos também, algoritmos quânticos para o PSO sobre grupos não abelianos de ordem 2^{n+1} que possuem subgrupos cíclicos de índice 2 e para certos produtos semidiretos de grupos Z_N^m times Z_p, com m, N inteiros positivos e N fatorado de forma especial. / We present a new polynomial-time quantum algorithm that solves the hidden subgroup problem (HSP) for a special class of metacyclic groups, namely Z_{p} times _{q^s}, with q mid (p-1) and p/q= up{poly}(log p), where p, q are any odd prime numbers and s is any positive integer. This solution generalizes previous algorithms presented in the literature. In a more general setting, without imposing a relation between p and q, we obtain a quantum algorithm with time and query complexity 2^{O(sqrt{log p})}. In any case, those results improve the classical algorithm, which needs {Omega}(sqrt{p}) queries. We also present quantum algorithms for the HSP over non-abelian groups of order 2^{n+1} which have a cyclic subgroup of index 2 and for some semidirect product _N^m times _p, where N has a special prime factorization.

Nesta dissertação estudamos, no âmbito teórico, algumas propostas recentes de processamento de informação quântica passiva, isto é, descartando protocolos de correção de erros. Recorrendo à criação de subespaços livres de decoerência através de um sistema físico de quatro spins acoplados ao resto do universo por um agente degenerado, mostramos ser possível construir um conjunto universal de portas lógicas (C-NOT, T e Hadamard) neste mesmo subespaço, alcançando, por conseguinte, a realização de qualquer operação computacional, insensivelmente ao resto do universo. Partimos de um hamiltoniano geral com interações individuais de cada spin com campos externos, além de acoplamentos controlados entre pares de spins. Experimentalmente, hamiltonianos deste tipo são comuns no contexto de junções Josephson, motivo pelo qual tratamos esta implementação em um capítulo especial. Introduzindo perturbativamente ao hamiltoniano operadores espúrios ao subespaço livre de decoerência, incluímos sensibilidade do sistema frente ao ambiente, criando a possibilidade da incursão de erros através de mecanismos de dissipação. Tais mecanismos foram investigados em termos da intensidade do parâmetro de acoplamento entre o sistema e o ambiente, revelando uma clara evidência teórica do Efeito Zenão Quântico, através da excelente concordância entre resultados de operações realizadas em subespaços livres de decoerência e operações realizadas em sistemas fortemente acoplados ao resto do universo. Neste sentido, selecionamos a fidelidade como medida de distância entre um estado em evolução a partir de um certo estado inicial do subespaço livre de decoerência (e submetido a dissipação), e um estado em evolução regida pela mesma operação quântica e a partir das mesmas condições iniciais no caso ideal, livre de decoerência. Essa abordagem explícita permitiu-nos obter a razão necessária entre os parâmetros associados a perturbação (que remove o estado do subespaço original) e acoplamento (entendido como a freqüência entre as medidas promovidas pelo resto do universo), para alcançar a eficiência desejada na realização de uma certa porta lógica. Tecnicamente, o trabalho envolveu vários resultados matemáticos novos e operacionalmente úteis, levando a simplificações importantes durante os cálculos envolvidos. / In this dissertation we studied theoretical aspects of some recent proposals of passive quantum information processing, that is, discarding error correction protocols. Falling back upon the creation of decoherence-free subspaces through a physical system of four spins coupled to the rest of the universe by a degenerate agent, we showed to be possible to build a universal set of logical quantum gates (C-NOT, T and Hadamard) in this same subspace, reaching, consequently, the accomplishment of any computational operation, callously to the rest of the universe. We started from a general Hamiltonian with individual interactions of each spin with external fields, besides controlled couplings between spin pairs. Experimentally, Hamiltonians like this are common in the context of Josephson junctions and, therefore, we treated this implementation in a special chapter. Perturbatively introducing spurious operators to the hamiltonian in the decoherence-free subspace, we included sensibility of the system to the environment, creating the possibility of the incursion of errors through dissipation mechanisms. Such mechanisms were investigated in terms of the intensity of the coupling parameter between the system and the environment, revealing an obvious theoretical evidence of the Quantum Zeno Effect, through the excellent agreement between the results of operations accomplished in decoherence-free subspace and operations accomplished in systems strongly coupled to the rest of the universe. In this sense, we selected the fidelity as the distance measure between a state in evolution starting from a certain initial state of the decoherence-free subspace (and submitted to the dissipation), and a state in evolution governed by the same quantum operation and starting from the same initial conditions in the ideal decoherence-free case. This explicit approach allowed us to obtain the necessary quotient between the associated disturbance parameter (that removes the state from the original subspace) and coupling parameter (understood as the frequency between the measurements promoted by the rest of the universe), to reach the efficiency desired in the accomplishment of a logic gate. Technically, the work involved several new operationally useful mathematical results, leading to important simplifications during the involved calculations.

M.Sc. / In this thesis we study the techniques used to calculate the Hamilton operators related to linear optical quantum computing. We also discuss the basic building blocks of linear optical quantum computing (LOQC) by looking at the logic gates and the physical instruments of which they are made.

Inorganic macrocycles have stimulated interest in recent years for their magnetic properties, their associated host-guest chemistry and their aesthetically appealing structures. These characteristics have led to suggestions that they could be exploited for the purposes of ion recognition, catalysis, as single molecule magnets, MRI agents, antibacterial agents and as part of larger architectures in a molecular machine. This thesis explores the properties of a group of chromium(III) macrocycles, with functionality tailored towards different pursuits. Firstly the magnetic properties of a newly synthesised family of ring dimers are investigated. The nature of magnetic exchange within each ring leads to a net electronic spin which, it has been proposed, could represent a quantum binary digit within a quantum information processing system. By linking together pairs of rings, the degree of inter ring communication can be determined. Such interactions are important for the correlation of spin as initiation of quantum entanglement, a pre-requisite for quantum computing. The rings can also act as fluoro-metallocrowns, hosting the molecule which templated their formation. A range of rings with different guests are synthesised and their solid and solution state structures are explored. On templating about bulky dialkyl amines hybrid organic-inorganic rotaxanes are formed where the guest is fixed. In contrast when using small amines and alkali metals, exchange of guests is possible. The dynamics of all of these systems are investigated with proton NMR, quite remarkable for such highly paramagnetic complexes.

Residual dipolar couplings can be used to increase the number of qubits for quantum information processing. We have used molecules containing 3, 5 and 8 spins oriented in a liquid crystal matrix, and exploited the residual dipolar coupling to demonstrate quantum information processing in them. Transition assignment is performed using HET-Z-COSY experiment and qubit addressability is achieved by transition selective pulses. It is expected that using this protocol higher qubits can be achieved. For the implementations reported in this work, evolution under the internal Hamiltonian was not explored. It is however interesting to investigate how effectively the evolution under internal Hamiltonian can be manipulated to implement quantum algorithms in these systems. Recently an approach has been reported in this direction, where a new method of preparing pseudopure states in oriented systems by exciting selected multiple quantum using evolution under effective dipolar Hamiltonian, has been demonstrated [24].

We experimentally demonstrate time-frequency entangled photons with comb-like spectra via both bulk optical crystals and on-chip microring resonators and explore their characterization in both time and frequency domain using quantum state manipulation techniques. Our characterization of these quantum frequency combs involves the use of unbalanced Mach-Zehnder interferometers and electro-optic modulators for manipulation in time- and frequency-domain, respectively. By creating indistinguishable superposition states using these techniques, we are able to interfere states from various time- and frequency-bins, consequently proving time- and frequency-bin en-tanglement. Furthermore, our time-domain manipulations reveal pair-wise continuous time-energy entanglement that spans multiple frequency bins, while our utilization of electro-optic modulators to verify high-dimensional frequency-bin entanglement constitutes the proof of this phenomenon for a spontaneous four-wave mixing pro-cess. By doing so, we show the potential of these quantum frequency combs for high-dimensional quantum computing with frequency-encoded quantum states, as well as fully secure quantum communications via quantum key distribution by per-forming a nonlocal dispersion cancellation experiment. To show the potential of our entangled photons source for encoding quantum information in the frequency domain, we carry out a frequency-domain Hong-Ou-Mandel interference experiment by implementing a frequency beam splitter. Lastly, we use the high-dimensionality of our time-frequency entangled source in both time and frequency domain to implement deterministic high-dimensional controlled quantum gates, with the quantum information encoded in both the time and frequency degrees of freedom of a single photon. This novel demonstration of deterministic high-dimensional quantum gates paves the way for scalable optical quantum computation, as quantum circuits can be implemented with fewer resources and high success probability using this scheme.<div><br></div><div> </div>

In this thesis, an evolutionary system for searching quantum operators in the form of unitary matrices is implemented. The aim is to propose several representations of candidate solutions and settings of the evolutionary algorithm. Two evolutionary algorithms were applied: the genetic algorithm and evolutionary strategy. Furthermore, a method of generating a unitary matrix is presented which is used for the first time for this task. This method is in some aspects better than the previous ones. Finally, a comparison of all used techniques is shown in experiments.

Kryptografie je velmi důležitým aspektem našeho každodenního života, protože poskytuje teoretický základ informační bezpečnosti. Kvantové výpočty a informace se také stávají velmi důležitou oblastí vědy kvůli mnoha aplikačním oblastem včetně kryptologie a konkrétněji v kryptografii veřejných klíčů. Obtížnost čísel do hlavních faktorů je základem některých důležitých veřejných kryptosystémů, jejichž klíčem je kryptosystém RSA . Shorův kvantový faktoringový al-goritmus využívá zejména kvantový interferenční účinek kvantového výpočtu k faktorovým semi-prime číslům v polynomiálním čase na kvantovém počítači. Ačkoli kapacita současných kvantových počítačů vykonávat Shorův algoritmus je velmi omezená, existuje mnoho rozsáhlých základních vědeckých výzkumů o různých technikách optimalizace algoritmu, pokud jde o faktory, jako je počet qubitů, hloubka obvodu a počet bran. v této práci jsou diskutovány, analyzovány a porovnávány různé varianty Shorova factoringového algoritmu a kvantových obvodů. Některé varianty Shorova algoritmu jsou také simulované a skutečně prováděné na simulátorech a kvantových počítačích na platformě IBM QuantumExperience. Výsledky simulace jsou porovnávány z hlediska jejich složitosti a míry úspěšnosti. Organizace práce je následující: Kapitola 1 pojednává o některých klíčových historických výsledcích kvantové kryptografie, uvádí problém diskutovaný v této práci a představuje cíle, kterých má být dosaženo. Kapitola 2 shrnuje matematické základy kvantového výpočtu a kryptografie veřejných klíčů a popisuje notaci použitou v celé práci. To také vysvětluje, jak lze k rozbití kryptosystému RSA použít realizovatelný algoritmus pro vyhledávání objednávek nebo factoring. Kapitola 3 představuje stavební kameny Shorova algoritmu, včetně kvantové Fourierovy transformace, kvantového odhadu fází, modulární exponentiace a Shorova algoritmu. Zde jsou také uvedeny a porovnány různé varianty optimalizace kvantových obvodů. Kapitola 4 představuje výsledky simulací různých verzí Shorova algoritmu. V kapitole 5 pojednejte o dosažení cílů disertační práce, shrňte výsledky výzkumu a nastíňte budoucí směry výzkumu.

<div>With the transistor dimensions scaling down to a few atoms, quantum phenomena - like quantum tunneling and entanglement - will dictate the operation and performance of the next generation of electronic devices, post-CMOS era. While quantum tunneling limits the scaling of the conventional transistor, Tunneling Field Effect Transistor (TFET) employs band-to-band tunneling for the device operation. This mechanism can reduce the sub-threshold swing (S.S.) beyond the Boltzmann's limit, which is fundamentally limited to 60 mV/dec in a conventional Si-based metal-oxide-semiconductor field-effect transistor (MOSFET). A smaller S.S. ensures TFET operation at a lower supply voltage and, therefore, at lesser power compared to the conventional Si-based MOSFET.</div><div><br></div><div>However, the low transmission probability of the band-to-band tunneling mechanism limits the ON-current of a TFET. This can be improved by reducing the body thickness of the devices i.e., using 2-Dimensional (2D) materials or by utilizing heterojunction designs. In this thesis, two promising methods are proposed to increase the ON-current; one for the 2D material TFETs, and another for the III-V heterojunction TFETs.</div><div><br></div><div>Maximizing the ON-current in a 2D material TFET by determining an optimum channel thickness, using compact models, is presented. A compact model is derived from rigorous atomistic quantum transport simulations. A new doping profile is proposed for the III-V triple heterojunction TFET to achieve a high ON-current. The optimized ON-current is 325 uA/um at a supply voltage of 0.3 V. The device design is optimized by atomistic quantum transport simulations for a body thickness of 12 nm, which is experimentally feasible.</div><div> </div><div>However, increasing the device's body thickness increases the atomistic quantum transport simulation time. The simulation of a device with a body thickness of over 12 nm is computationally intensive. Therefore, approximate methods like the mode-space approach are employed to reduce the simulation time. In this thesis, the development of the mode-space approximation in modeling the triple heterojunction TFET is also documented.</div><div><br></div><div>In addition to the TFETs, quantum computing is an emerging field that utilizes quantum phenomena to facilitate information processing. An extra chapter is devoted to the electronic structure calculations of the Si:P delta-doped layer, using the empirical tight-binding method. The calculations agree with angle-resolved photoemission spectroscopy (ARPES) measurements. The Si:P delta-doped layer is extensively used as contacts in the Phosphorus donor-based quantum computing systems. Understanding its electronic structure paves the way towards the scaling of Phosphorus donor-based quantum computing devices in the future.</div>

Most quantum algorithms that are efficient as opposed to their equivalent classical algorithms are solving variants of the Hidden Subgroup Problem (HSP), therefore HSP is a central problem in the field of quantum computing. In this thesis, we offer some interesting results about the subgroup and coset structure of certain groups, including the dihedral group. We describe classical algorithms to solve the HSP over various abelian groups and the dihedral group. We also discuss some existing quantum algorithms to solve the HSP and give our own novel algorithms and ideas to approach the HSP for the dihedral groups.