Geometric and holonomic quantum gates perform transformations that only dependon the geometry of a loop covered by the parameters controlling the gate. Thesegates require adiabatic time evolution, which is achieved in the limit when the looptakes infinite time to complete. However, it is of interest to also know thetransformation properties of the gates for finite run times. It has been shown [Phys.Rev. A 73, 022327 (2006)] that some holonomic gates for a trapped ion system showrevival structures, i.e., for some finite run time the gate performs the sametransformation as it does in the adiabatic limit. The purpose of this thesis is to investigate if similar revival structures are shown alsofor geometric and holonomic quantum gates for spin systems. To study geometricquantum gates an NMR setup for spin-1/2 particles is used, while an NQR setup forspin-3/2 particles is used to study holonomic quantum gates. Furthermore, for thegeometric quantum gates the impact of some open system effects are examined byusing the quantum jump approach. The non-adiabatic time evolution operators of thesystems are calculated and compared to the corresponding adiabatic time evolutionoperators by computing their operator fidelity. The operator fidelity ranges between0 and 1, where 1 means that the gates are identical up to an unimportant phasefactor. All gates show an oscillating dependency on the run time, and some Abeliangates even show true revivals, i.e., the operator fidelity reaches 1.

No description available.

Doctor of Philosophy / Electrical and Computer Engineering / Sanjoy Das / Quantum circuit synthesis is one of the major areas of current research in the field of quantum computing. Analogous to its Boolean counterpart, the task involves constructing arbitrary quantum gates using only those available within a small set of universal gates that can be realized physically. However, unlike the latter, there are an infinite number of single qubit quantum gates, all of which constitute the special unitary group SU(2). Realizing any given single qubit gate using a given universal gate family is a complex task. Although gates can be synthesized to arbitrary degree of precision as long as the set of finite strings of the gate family is a dense subset of SU(2), it is desirable to accomplish the highest level of precision using only the minimum number of universal gates within the string approximation. Almost all algorithms that have been proposed for this purpose are based on the Solovay-Kitaev algorithm. The crux of the Solovay-Kitaev algorithm is the use of a procedure to decompose a given quantum gate into a pair of group commutators with the pair being synthesized separately. The Solovay-Kitaev algorithm involves group commutator decomposition in a recursive manner, with a direct approximation of a gate into a string of universal gates being performed only at the last level, i.e. in the leaf nodes of the search tree representing the execution of the Solovay-Kitaev algorithm. The main contribution of this research is in integrating conventional optimization procedures within the Solovay-Kitaev algorithm. Two specific directions of research have been studied. Firstly, optimization is incorporated within the group commutator decomposition, so that a more optimal pair of group commutators are obtained. As the degree of precision of the synthesized gate is explicitly minimized by means of this optimization procedure, the enhanced algorithm allows for more accurate quantum gates to be synthesized than what the original Solovay-Kitaev algorithm achieves. Simulation results with random gates indicate that the obtained accuracy is an order of magnitude better than before. Two versions of the new algorithm are examined, with the optimization in the first version being invoked only at the bottom level of Solovay-Kitaev algorithm and when carried out across all levels of the search tree in the next. Extensive simulations show that the second version yields better results despite equivalent computation times. Theoretical analysis of the proposed algorithm is able to provide a more formal, quantitative explanation underlying the experimentally observed phenomena. The other direction of investigation of this research involves formulating the group commutator decomposition in the form of bi-criteria optimization. This phase of research relaxed the equality constraint in the previous approach and with relaxation, a bi-criteria optimization is proposed. This optimization algorithm is new and has been devised primarily when the objective needs to be relaxed in different stages. This bi-criteria approach is able to provide comparably accurate synthesis as the previous approach.

This thesis introduces the emerging field of quantum computing, emphasizing its capability to surpass traditional computing by solving complex problems that are beyond the reach of classical computers. Unlike classical systems that operate with bits and logic gates, quantum computing utilizes qubits and quantum gates, exploiting the vast computational space offered by quantum mechanics. A focal point of this study is topological quantum computing, a novel approach designed to overcome the inherent vulnerability of quantum systems to errors, such as decoherence and operational inaccuracies. At the heart of this method lies the use of non-Abelian anyons, with a particular focus on Fibonacci anyons, whose unique topological characteristics and braiding operations present a viable path to fault-tolerant quantum computation. This thesis aims to elucidate how the braiding of Fibonacci anyons can be employed to construct the necessary quantum gates for topological quantum computing. By offering a foundational exploration of quantum computing principles, especially topological quantum computing, and detailing the process for creating quantum gates through braiding of Fibonacci anyons, the work sets the stage for further research and development in this transformative computing paradigm.

Nessa dissertação, abordamos o problema de dois qubits interagindo com campos externos e entre si controladamente, de acordo com um Hamiltoniano considerado realista para implementação da porta lógica quântica XOR. Introduzimos acoplamentos entre as observáveis do sistema de dois qubits e um banho de osciladores harmônicos a fim de tratarmos o problema da dissipação e da decoerência. Primeiramente nós consideramos o limite no qual a decoerência é mais rápida que qualquer processo gerado pelo Hamiltoniano do sistema. Prosseguimos então, através do método numérico conhecido como Integrador Unitário, com o estudo da matriz densidade do sistema durante a operação da porta lógica quântica sem incluir, inicialmente, o acoplamento com o banho de osciladores harmônicos. Finalmente, implementamos o método numérico conhecido como Propagador quase adiabático para estudar a decoerência e a dissipação durante a operação da porta lógica quântica XOR, a fim de analisarmos os aspectos perturbativos do sistema quântico de dois qubits. / In this dissertation, we approach the problem of two qubits interading with themselves and with externa1 fields in a controlled way, according to a Hamiltonian considered realistic to implement the XOR quantum gate. We introduce couplings between the observables of the two-qubits system and of a bath of harmonic oscillators, to treat the problems of dissipation and decoherence. Preliminarly, we consider the limit in which decoherence is faster than any process dictated by the Hamiltonian evolution of the system. Then, through a unitary-integrator numerical method, we proceed with the study of the evolution of the density matrix of the system during the operation of the logical quantum gate, initially, without the coupling with the bath of harmonic oscillators. Finally, we use the quasiadiabatic path integral method to study the dissipation and decoherence during the logical operation, through the inclusion of the bath.

This thesis deals with a series of quantum computer implementation issues from the Kane 31P in 28Si architecture to Shor’s integer factoring algorithm and beyond. The discussion begins with simulations of the adiabatic Kane CNOT and readout gates, followed by linear nearest neighbor implementations of 5-qubit quantum error correction with and without fast measurement. A linear nearest neighbor circuit implementing Shor’s algorithm is presented, then modified to remove the need for exponentially small rotation gates. Finally, a method of constructing optimal approximations of arbitrary single-qubit fault-tolerant gates is described and applied to the specific case of the remaining rotation gates required by Shor’s algorithm.

Nessa dissertação, abordamos o problema de dois qubits interagindo com campos externos e entre si controladamente, de acordo com um Hamiltoniano considerado realista para implementação da porta lógica quântica XOR. Introduzimos acoplamentos entre as observáveis do sistema de dois qubits e um banho de osciladores harmônicos a fim de tratarmos o problema da dissipação e da decoerência. Primeiramente nós consideramos o limite no qual a decoerência é mais rápida que qualquer processo gerado pelo Hamiltoniano do sistema. Prosseguimos então, através do método numérico conhecido como Integrador Unitário, com o estudo da matriz densidade do sistema durante a operação da porta lógica quântica sem incluir, inicialmente, o acoplamento com o banho de osciladores harmônicos. Finalmente, implementamos o método numérico conhecido como Propagador quase adiabático para estudar a decoerência e a dissipação durante a operação da porta lógica quântica XOR, a fim de analisarmos os aspectos perturbativos do sistema quântico de dois qubits. / In this dissertation, we approach the problem of two qubits interading with themselves and with externa1 fields in a controlled way, according to a Hamiltonian considered realistic to implement the XOR quantum gate. We introduce couplings between the observables of the two-qubits system and of a bath of harmonic oscillators, to treat the problems of dissipation and decoherence. Preliminarly, we consider the limit in which decoherence is faster than any process dictated by the Hamiltonian evolution of the system. Then, through a unitary-integrator numerical method, we proceed with the study of the evolution of the density matrix of the system during the operation of the logical quantum gate, initially, without the coupling with the bath of harmonic oscillators. Finally, we use the quasiadiabatic path integral method to study the dissipation and decoherence during the logical operation, through the inclusion of the bath.

This thesis is concerned with studying the fascinating quantum properties of real-world nanostructures embedded in a noisy condensed matter environment. The interaction with light is used for controlling and manipulating the quantum state of the systems considered here. In some instances, laser pulses also provide a way of actively probing and controlling environmental interactions. The first two research chapters assess two different ways of performing all-optical spin qubit gates in self-assembled quantum dots. The principal conclusion is that an `adiabatic' control technique holds the promise of achieving a high fidelity when all primary sources of decoherence are taken into account. In the next chapter, it is shown that an optically driven quantum dot exciton interacting with the phonons of the surrounding lattice acts as a heat pump. Further, a model is developed which predicts the temperature-dependent damping of Rabi oscillations caused by bulk phonons, finding an excellent agreement with experimental data. A different system is studied in the following chapter: two electron spin qubits with no direct interaction, yet both exchange-coupled to an optically active mediator spin. The results of this study show that these general assumptions are sufficient for generating controlled electron spin entanglement over a wide range of parameters, even in the presence of noise. Finally, the Radical Pair model of the avian compass is investigated in the light of recent experimental results, leading to the surprising prediction that the electron spin coherence time in this molecular system seems to approach the millisecond timescale.

In this project a qutrit tripod system is studied to implement quantum gates using non-Abelian geometric phases, allowing for holonomic quantum computation which in turn results in more robust computations. First, a general foundation of the theory is presented. This includes the relevant theory of matrices in Hilbert space, as well as theory of the quantum mechanics used in the report. The method is then described in depth, showing how the pulse area is fixed. Using properties of the Hamiltonian as well as the time-evolution operator of the tripod system the computational subspace can be derived. These findings are combined to show how the computational subspace evolves in time, resulting in the unitary matrix used to form quantum gates. Using educated guesses to find the necessary parameters or utilizing iterative methods to find the parameters are the two main approaches used for constructing the considered gates. Three of the suggested quantum gates are successfully implemented through educated guesses, namely X, T and Z using an angle parametrization of the phase and amplitude of the pulses. The last desired gate is the Hadamard-gate, but the implementation of said gate required numerical approximation. The reasons as to why this is the case, are later discussed.

The progess in NMRQIP can be outlined in to four parts.1) Implementation of theoretical protocols on small number of qubits. 2) Demonstration of QIP on various NMR systems. 3) Designing and implementing the algorithms for mixed initial states. 4) Developing the techniques for coherent and decoherent control on higher number(up to 15) of qubits. This thesis contains some efforts in the direction of first three points. Quantum-state discrimination has important applications in the context of quantum communication and quantum cryptography. One of the characteristic features of quantum mechanics is that it is impossible to devise a measurement that can distinguish nonorthogonal states perfectly. However, one can distinguish them with a finite probability by an appropriate measurement strategy. In Chapter 2, we describe the implementation of a theoretical protocol of programmable quantum-state discriminator, on a two-qubit NMR System. The projective measurement is simulated by adding two experiments. This device does the unambiguous discrimination of a pair of states of the data qubit that are symmetrically located about a fixed state. The device is used to discriminate both linearly polarized states and eillipitically polarized states. The maximum probability of successful discrimination is achieved by suitably preparing the ancilla quubit. The last step of any QIP protocol is the readout. In NMR-QIP the readout is done by using density matrix tomography. It was first proposed by Ernst and co-workers that a two-dimensional method can be used to correlate input and output states. This method uses an extra (aniclla) qubit, whose transitions indicate the quantum states of the remaining qubits. The 2D spectrum of ancilla qubit represent the input and output states along F1 and F2 dimensions respectively. However the 2D method requires several t1 increments to achieve the required spectral width and resolution in the indirect dimension, hence leads to large experimental time. In chapter 3, the conventional 2D NMRQIP method is speeded-up by using Hadamard spectroscopy. The Hadamard method is used to implement various two-, three-qubit gates and qutrit gates. We also use Hadamard spectroscopy for information storage under spatial encoding and to implement a parallel search algorithm. Various slices of water sample can be spatially encoded by using a multi-frequency pulse under the field gradient. Thus the information of each slice is projected to the frequency space. Each slice represents a classical bit, where excitation and no excitation corresponds to the binary values 0 and 1 respectively. However one has to do the experiment for each binary information, by synthesizing a suitable multi-frequency pulse. In this work we show that by recording the data obtained by various Hadamard encoded multi-frequency pulses, one can suitably decode it to obtain any birnary information, without doing further experiments. Geometric phases depend only on the geometry of the path executed in the projective Hilbert space, and are therefore resilient to certain types of errors. This leads to the possibility of an intrinsically fault-tolerant quantum computation. In liquid state NMRQIP. Controlled phase shift gates are achieved by using qubit selective pulses and J evolutions, and also by using geometir phases. In order to achieve higher number of qubits in NMR, one explores dipolar couplings which are larger in magnitude, yielding strongly coupled spectra. In such systems since the Hamiltonian consists of terms, it is difficult to apply qubit selective pulses. However such systems have been used for NMRQIP by considering 2n eigen states as basis states of an n-qubit system. In chapter 4, it is shown that non-adiabatic geometric phases can be used to implement controlled phase shift gates in strongly dipolar coupled systems. A detailed theoretical explanation of non-adiabatic geometric phases in NMR is given, by using single transition operators. Using such controlled phase shift gates, the implementation of Deutsch-Jozsa and parity algorithms are demonstrated. Search algorithms play an important role in the filed of information processing. Grovers quantum search algorithm achieves polynomial speed-up over the classical search algorithm. Bruschweiler proposed a Liouville space search algorithm which achieve polymonial speed-up. This algorithm requires a weakly coupled system with a mixed initial state. In chapter 5 we modified the Bruschweiler’s algorithm, so that it can be implemented on a weakly as well as strongly coupled system. The experiments are performed on a strongly dipolar coupled four-qubit system. The experiments from four spin-1/2 nuclei of a molecule oriented in a liquid crystal matrix. Chapter 6 describes the implementation of controlled phase shift gates on a quadrupolar spin-7/2 nucleus, using non-adiabatic geometric phases. The eight energy levels of spin-7/2 nucleus, form a three qubit system. A general procedure is given, for implementing a controlled phase shift gate on a system consisting of any number of energy levels. Finally Collin’s version of three-qubit DJ algorithm using multi-frequency pulses, is implemented in the spin-7/2 system.