Building a quantum computer is a formidable challenge. In this thesis, we focus on two projects, which tackle very different aspects of quantum computation, and yet still share a common goal in hopefully getting us closer to implementing a quantum computer on a large scale. The first project involves a numerical error threshold calculation of a quantum error correcting code called a surface code. These are local check codes, which means that only nearest neighbour interaction is required to determine where errors occurred. This is an important advantage over other approaches, as in many physical systems, doing operations on arbitrarily spaced qubits is often very difficult. An error threshold is a measure of how well a given error correcting scheme performs. It gives the experimentalists an idea of which approaches to error correction hold greater promise. We simulate both toric and planar variations of a surface code, and numerically calculate a threshold value of approximately $6.0 \times 10^{-3}$, which is comparable to similar calculations done by others \cite{Raussendorf2006,Raussendorf2007,Wang2009}. The second project deals with coupling superconducting flux qubits together. It expands the scheme presented in \cite{Plourde2004} to a three qubit, two coupler scenario. We study L-shaped and line-shaped coupler geometries, and show how the coupling strength changes in terms of the dimensions of the couplers. We explore two cases, the first where the interaction energy between two nearest neighbour qubits is high, while the coupling to the third qubit is as negligible as possible, as well as a case where all the coupling energies are as small as possible. Although only an initial step, a similar scheme can in principle be extended further to implement a lattice required for computation on a surface code.

Building a quantum computer is a formidable challenge. In this thesis, we focus on two projects, which tackle very different aspects of quantum computation, and yet still share a common goal in hopefully getting us closer to implementing a quantum computer on a large scale. The first project involves a numerical error threshold calculation of a quantum error correcting code called a surface code. These are local check codes, which means that only nearest neighbour interaction is required to determine where errors occurred. This is an important advantage over other approaches, as in many physical systems, doing operations on arbitrarily spaced qubits is often very difficult. An error threshold is a measure of how well a given error correcting scheme performs. It gives the experimentalists an idea of which approaches to error correction hold greater promise. We simulate both toric and planar variations of a surface code, and numerically calculate a threshold value of approximately $6.0 \times 10^{-3}$, which is comparable to similar calculations done by others \cite{Raussendorf2006,Raussendorf2007,Wang2009}. The second project deals with coupling superconducting flux qubits together. It expands the scheme presented in \cite{Plourde2004} to a three qubit, two coupler scenario. We study L-shaped and line-shaped coupler geometries, and show how the coupling strength changes in terms of the dimensions of the couplers. We explore two cases, the first where the interaction energy between two nearest neighbour qubits is high, while the coupling to the third qubit is as negligible as possible, as well as a case where all the coupling energies are as small as possible. Although only an initial step, a similar scheme can in principle be extended further to implement a lattice required for computation on a surface code.

Josephson junctions are dissipationless elements used notably in superconducting nanocircuits. While being indispensable for the making of superconducting quantum bits, they are plagued by intrinsic noise mechanisms that reduce the coherence time of the quantum devices. An important source of such fluctuations may come from the non-cristallinity and disorder of the oxide layer sandwiched between the two superconducting leads. In this work, roughness in a Josephson junction is modeled as a set of pinholes with a universal bimodal distribution of transmission eigenvalues that sum incoherently in the noise power. Each of these channels is treated as a ballistic quantum point contact with a thin barrier that determines the transmission eigenvalue. The noise spectrum is calculated using the quasiclassical Green's function method to analyze high and low transmission limits at non-zero temperature for all interesting frequencies. As suggested by experiments, low transmission channels generate shot noise while fast switching between subgap states creates strong non-poissonian low-frequency noise. However, when analyzed for three different universal models of disorder, the principal contribution to noise is found to come from the partially opened channels. Finally, fluctuations of the noise from sample to sample is seen to be dominated by the contribution of opened channels which may reduce the reproducibility of results between different experiments.

Josephson junctions are dissipationless elements used notably in superconducting nanocircuits. While being indispensable for the making of superconducting quantum bits, they are plagued by intrinsic noise mechanisms that reduce the coherence time of the quantum devices. An important source of such fluctuations may come from the non-cristallinity and disorder of the oxide layer sandwiched between the two superconducting leads. In this work, roughness in a Josephson junction is modeled as a set of pinholes with a universal bimodal distribution of transmission eigenvalues that sum incoherently in the noise power. Each of these channels is treated as a ballistic quantum point contact with a thin barrier that determines the transmission eigenvalue. The noise spectrum is calculated using the quasiclassical Green's function method to analyze high and low transmission limits at non-zero temperature for all interesting frequencies. As suggested by experiments, low transmission channels generate shot noise while fast switching between subgap states creates strong non-poissonian low-frequency noise. However, when analyzed for three different universal models of disorder, the principal contribution to noise is found to come from the partially opened channels. Finally, fluctuations of the noise from sample to sample is seen to be dominated by the contribution of opened channels which may reduce the reproducibility of results between different experiments.

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Quantum information and quantum computation are linked by a common mathematical and physical framework of quantum mechanics. The manipulation of the predicted dynamics and its optimization is known as quantum control. Many techniques, originating in the study of nuclear magnetic resonance, have found common usage in methods for processing quantum information and steering physical systems into desired states. This thesis expands on these techniques, with careful attention to the regime where competing effects in the dynamics are present, and no semi-classical picture exists where one effect dominates over the others. That is, the transition between the diabatic and adiabatic error regimes is examined, with the use of such techniques as time-dependent diagonalization, interaction frames, average-Hamiltonian expansion, and numerical optimization with multiple time-dependences. The results are applied specifically to superconducting systems, but are general and improve on existing methods with regard to selectivity and crosstalk problems, filtering of modulation of resonance between qubits, leakage to non-compuational states, multi-photon virtual transitions, and the strong driving limit.

Quantum information and quantum computation are linked by a common mathematical and physical framework of quantum mechanics. The manipulation of the predicted dynamics and its optimization is known as quantum control. Many techniques, originating in the study of nuclear magnetic resonance, have found common usage in methods for processing quantum information and steering physical systems into desired states. This thesis expands on these techniques, with careful attention to the regime where competing effects in the dynamics are present, and no semi-classical picture exists where one effect dominates over the others. That is, the transition between the diabatic and adiabatic error regimes is examined, with the use of such techniques as time-dependent diagonalization, interaction frames, average-Hamiltonian expansion, and numerical optimization with multiple time-dependences. The results are applied specifically to superconducting systems, but are general and improve on existing methods with regard to selectivity and crosstalk problems, filtering of modulation of resonance between qubits, leakage to non-compuational states, multi-photon virtual transitions, and the strong driving limit.

Abstract The topic of this thesis is superconducting electric circuits. Technical advances have made possible the experimental study of Josephson junction based circuit elements which sustain quantum mechanical properties long enough to be denoted as quantum devices. The quantum state can be controlled with electronic variables and measured using standard electrical setups. The research is motivated by the possibility to examine quantum phenomena in circumstances that can be customized, prospects of new quantum devices, and the development of quantum information processing. This thesis presents theoretical studies on the nonlinear and stochastic driving of a superconducting quantum two-level system (qubit). We first investigate the energy level shifts a single-Cooper-pair transistor under large amplitude driving realized via the inherently nonlinear Josephson energy by using an external magnetic flux. The effective driving field substantially deviates from a circular polarization and linear coupling. The energy level shifts are compared to the cases of a vanishing and a weak driving field, measured as the Stark shift and the generalized Bloch-Siegert shift, respectively. We describe criteria for the natural basis of the analytical and the numerical calculations. In addition to that, we develop a formalism based on the Floquet method for the weak probe measurement of the strongly driven qubit. In the latter part of the thesis research, we study utilization of a stochastic driving field whose time evolution is not regular but follows probabilistic laws. We concentrate on the motional averaging phenomenon and show that it can be measured with an unparalleled accuracy by employing a flux-modulated transmon qubit. As the stochastically modulated qubit is simultaneously measured with a moderate driving field, we develop a theoretical description accounting the possible interference effects between the modulation and the drive. The comparison with experimental results shows good agreement. Motional averaging phenomenon can be applied to estimate the properties of fluctuation processes occurring in qubits, e.g., the quasiparticle tunneling or the photon shot noise. Resting on the motional averaging, we anticipate that the qubit dephasing times can be improved if one can accelerate the dynamics of two-level fluctuators. We apply a semiclassical formalism where the qubit is treated with quantum mechanical concepts whereas the driving fields are classical. In the solution procedure, the numerical results support the main analytical understanding. As the theoretical results are extensively compared to reflection measurements, we construct an explicit connection between the dynamics of the studied quantum devices and the measured reflection coefficient.

A computação quântica adiabática tem sua pedra angular no teorema adiabático, cuja eficiência está relacionada tradicionalmente à proporção da variação temporal do Hamiltoniano que descreve o sistema e o gap mínimo entre o estado fundamental e o primeiro excitado. Normalmente, esse gap tende a diminuir quando aumenta o número de recursos (bit quântico: qubit) de um processador quântico, exigindo dessa maneira variações lentas do Hamiltoniano para assim garantir uma dinâmica adiabática. Entre os candidatos para a sua implementação física, estão os qubits baseados em circuitos supercondutores os quais têm um grande potencial, por causa de seu alto controle e escalabilidade promissora. No entanto, quando esses qubits são implementados, eles têm uma fonte intrínseca de ruído devido a erros de fabricação, que não podem ser desprezados. Por isso, nesta tese nós estudamos como os efeitos causados pelas flutuações dos parâmetros físicos do qubit afetam o comportamento da fidelidade da computação, realizando com esse propósito a simulação da dinâmica de cadeias de spin pequenas desordenadas. A partir do análise exaustivo desse estúdio foi possível propor uma estratégia que permite aumentar a fidelidade considerando um sistema ruidoso. Por outro lado, motivados pelo interesse de obter critérios suficientes e necessários para satisfazer uma computação quântica adiabática e pelo fato que ainda não existe uma condição de adiabaticidade geral apesar de existir inúmeras propostas, nós apresentamos um novo critério que manifesta suficiência para sistemas mais gerais e finalmente apresentamos evidências de que tal condição seria um quantificador consistente. / Adiabatic quantum computation has its cornerstone in the adiabatic theorem, whose efficiency is traditionally related to the ratio of the Hamiltonian temporal variation that describes the system and the minimum gap between the ground state and the first excited state. Usually, this gap tends to decrease when the number of quantum resources (quantum bit: qubit) of a quantum processor increases, thus it requires slow variations of the Hamiltonian to ensure an adiabatic dynamic. Among the candidates for its physical implementation are the qubits superconducting circuit-based which have great potential because of their high control and promising scalability. However, when these qubits are implemented, they have an intrinsic source of noise due to manufacturing errors that can not be despised. Therefore, in this thesis we study how the effects caused by the fluctuations of the physical parameters of the qubit affect the behavior of the fidelity of the computation, accomplishing with this purpose the simulation of the dynamics of small disordered spin chains. From the exhaustive analysis of this studio, it was possible to propose a strategy that allows to increase the fidelity considering a noisy system. On the other hand, motivated by the interest of obtaining sufficient and necessary criteria to satisfy an adiabatic quantum computation and the fact that there is still no general adiabaticity condition despite there being numerous proposals, we present a new criterion that manifests sufficiency for more general systems and we finally presented evidence that such a condition would be a consistent quantifier.

A highly successful architecture for the exchange of single quanta between coupled quantum systems is circuit quantum electrodynamics (QED), in which the electrical interaction between a qubit and a high-quality microwave resonator offers the possibility to reliably control, store, and read out quantum bits of information on a chip. This architecture has also been implemented with mechanical resonators, showing that a vibrational mode can in principle be manipulated via a coupled qubit. The work presented in this thesis consists of realising an acoustic version of circuit QED that we call circuit quantum acoustodynamics (QAD), in which a superconducting qubit is piezoelectrically coupled to an acoustic cavity based on surface acoustic waves (SAWs). Designing and building this novel platform involved the following main accomplishments: a systematic characterisation of SAW resonators at low temperatures; successfully developing a recipe for the fabrication of Josephson junction on quartz and diamond; measuring the coherence time of superconducting 3D transmon qubits on these substrates and demonstrating the dispersive coupling between a SAW cavity and a qubit on a planar geometry. This thesis presents evidence of the coherent interaction between a SAW cavity and a superconducting qubit in several ways. First of all, a frequency shift of the mechanical mode as a function of qubit frequency is observed. We also measure the acoustic Stark shift of the qubit due to the population of the SAW cavity. The extracted coupling is in agreement with theoretical expectations. A time delayed acoustic Stark shift serves to further demonstrate that the Stark shifts that we observe are indeed due to the acoustic field of the SAW mode. The dispersive coupling between these two quantum systems offers the possibility to perform qubit spectroscopy using the SAW resonator as readout component, indicating that these acoustic resonators can, in principle, be adopted as an alternative qubit readout scheme in quantum information processors. We finally present preliminary measurements of the direct coupling between a SAW resonator and a transmon on diamond, suggesting that strong coupling can in principle be obtained.