Apresentamos neste trabalho a primeira verificação experimental de emaranhamento entre os feixes, sinal, complementar e bombeio refletido, produzidos pelo Oscilador Paramétrico Ótico (OPO) acima do limiar. Utilizando o critério de soma de variâncias de van Loock e Furusawa, obtivemos resultados que apontam a existência de emaranhamento tripartite em nosso sistema, evidenciado através da aplicação do critério de Positividade sob Transposição Parcial de Peres-Simon (PPT). A observação desse efeito, previsto em 2006, encontrou dificuldades relacionadas a existência de um ruído clássico espúrio, inserido pelo cristal não-linear. Apresentamos, um modelo teórico para esse ruído que leva em consideração pequenas flutuações na permissividade dielétrica do cristal, associadas às vibrações da rede. Em seguida, mostramos como conseguimos controlar e reduzir o excesso de ruído de modo a obter condições que permitiram à observação do emaranhamento tripartite. / We present in this work the first experimental verification of entanglement between the, signal, idler and reflected pump beams, generated by an Optical Parametric Oscillator (OPO) above the threshold. Using the criterion of sum of variances, proposed by van Loock and Furusawa, we obtained results that indicate the existence of tripartite entanglement in our system. That is confirmed by applying the criterion of positivity under partial transposition by Peres-Simon (PPT). The observation of this effect, predicted in 2006, faced difficulties related to the existence of a spurious classical noise, introduced by the nonlinear crystal. We have also presented a theoretical model for this that takes into account small fluctuations in the dielectric permittivity of the crystal, associated with vibrations of its lattice. We show how to control and reduce the excess noise in order to achieve conditions for the observation of tripartite entanglement.

This thesis studies continuous variable mixed states with the aim of better understanding the fundamental behaviour of quantum correlations in such states, as well as searching for applications of these correlations. I first investigate the interesting phenomenon of discord increase under local loss and explain the behaviour by considering the non-orthogonality of quantum states. I then explore the counter-intuitive result where entanglement can be created by a passive optical beamsplitter, even if the input states are classical, as long as the input states are part of a larger globally nonclassical system. This result emphasises the importance of global correlations in a quantum state, and I propose an application of this protocol in the form of quantum dense coding. Finally, I develop a quantum digital signature protocol that can be described entirely using the continuous variable formalism. Quantum digital signatures provide a method to ensure the integrity and provenance of a message using quantum states. They follow a similar method to quantum key distribution (QKD), but require less post-processing, which means they can sometimes be implemented over channels that are inappropriate for QKD. The method I propose uses homodyne measurement to verify the signature, unlike previous protocols that use single photon detection. The single photon detection of previous methods is designed to give unambiguous results about the signature, but this comes at the cost of getting no information much of the time. Using homodyne detection has the advantage of giving results all the time, but this means that measurement results always have some ambiguity. I show that, even with this ambiguity, the signature protocol based on homodyne measurement outperforms previous protocols, with the advantage enhanced when technical considerations are included. Therefore this represents an interesting new direction in the search for a practical quantum digital signature scheme.

Currently, there is interest in the development of molecular-scale devices for use in quantum information processing (QIP). With this application in mind, physical studies on antiferromagnetically coupled molecular wheels [Cr7MF3(Etglu)(O2CtBu)15(phpy)], where M is a divalent metal cation (M = Mn2+, Zn2+, Ni2+) have been pursued. The heterometallic wheels contain an octagon of metal centres, which are bridged by fluoride ions, pivalate groups and a chiral N-ethyl-D-glutamine molecule which is penta-deprotonated and bound to the metal sites through all available O-donors. They are deep purple in colour and they have been named purple-Cr7M. There is antiferromagnetic coupling between adjacent metal centres, J » -8 cm-1, resulting in a non-zero net spin ground state. The spin-Hamiltonian parameters of this family have been determined.At the heterometal site of purple-Cr7M wheels there is a terminal ligand which can be substituted for a variety of N-donor organic ligands. A series of bidentate N-donor linkers has been used to link Cr7Ni wheels (each wheel Seff = 1/2) to create prototype two-qubit systems. Multi-frequency EPR spectroscopy and SQUID magnetometry has been used to extract the spin-Hamiltonian parameters of this family. It has been shown that the single wheels can be linked together electronically as well as chemically. It has been found that for the unsaturated linkers, there is a weaker interaction between Cr7Ni wheels when longer linkers are used. The strength of interaction is smaller for the saturated linkers than for the unsaturated linkers.The formation of 'green'-Cr7M wheels is different, being templated around a cation. Two new types of wheels have been studied: [tBuCONHC6H12NH2C6H12NHCOtBu][Cr7M2+F8(O2CtBu)16] and [Cs?Cr7MF8(O2CtBu)16]·0.5MeCN (where, M = Mn2+, Zn2+, Ni2+), where the former is templated around a long dialkylammonium group and the latter around a caesium cation. The effect of the templating cation on spectroscopic properties has been determined.Physical studies on a family of antiferromagnetically-coupled homometallic clusters have been pursued. They consist of cyclic arrays of homometallic Cr3+ ions in either a octametallic wheel or hexametallic horseshoes. The horseshoes have the general formula: [CrxFx+5L2x-2]n3- (where L = carboxylate). Cr3+ centres are bridged by pivalate groups and fluorides, while Cr3+ centres at the ends of the chain have terminal fluorides completing their coordination sphere. These terminal fluoride groups are labile enough to be substituted, e.g. [EtNH2][Cr6F7(O2CtBu)10(acac)2] is the product of a substitution reaction with acetylacetone.

<div>This dissertation concerns the development of quantum computing algorithms for solving electronic structure problems. Three projects are contained: comparison of quantum computing methods for the water molecule, the design and implementation of Fully Controlled Variational Quantum Eigensolver(FCVQE) method, and quantum computing for atomic and molecular resonances. </div><div> </div><div>Chapter 1 gives a general introduction to quantum computing and electronic structure calculations. It includes basic concepts in quantum computing, such as quantum bits (qubits), quantum gates, and an important quantum algorithm, Phase Estimation Algorithm(PEA). It also shows the procedure of molecular Hamiltonian derivation for quantum computers.</div><div><br></div><div> </div><div>Chapter 2 discusses several published quantum algorithms and original quantum algorithms to solve molecules' electronic structures, including the Trotter-PEA method, the first- and second-order Direct-PEA methods, Direct Measurement method, and pairwise Variational Quantum Eigensolver(VQE) method. These quantum algorithms are implemented into quantum circuits simulated by classical computers to solve the ground state energy and excited state energies of the water molecule. Detailed analysis is also given for each method's error and complexity. </div><div><br></div><div> </div><div>Chapter 3 proposes an original design for VQE, which is called Fully Controlled Variational Quantum Eigensolver(FCVQE). Based on Givens Rotation matrices, this design constructs ansatz preparation circuits exploring all possible states in the given space. This method is applied to solving the ground state energy curves for different molecules, including NaH, H₂O, and N₂. The results from simulators turn out to be accurate compared with exact solutions. Gate complexity is discussed at the end of the chapter.</div><div><br></div><div> </div><div>Chapter 4 attempts to apply quantum simulation to atomic and molecular resonances. The original design implements the molecule's resonance Hamiltonian into the quantum circuit, and the resonance properties can be obtained from the final measurement results. It is shown that the resonance energy and width of a model system can be calculated by executing the circuit using Qiskit simulators and IBM real quantum computers as well. A proof of concept is also shown for the resonance properties of a real molecule, H₂^-. In the future, when there are more available qubits, longer coherence time, and less noise in quantum computers, this method can be used for larger molecular systems with better accuracy.</div>

This dissertation contains four projects: transforming electronic structure Hamiltonian to approximating Ising-type Hamiltonian to enable electronic structure calculations by quantum annealing, quantum-assisted restricted Boltzmann machine for electronic structure calculations, hybrid quantum classical neural network for calculating ground state energies of molecules and qubit coupled cluster single and double excitations variational quantum eigensolver for electronic structure. In chapter 1 we present a general introduction of quantum computer, including a brief introduction of two quantum computing model: gate model and quantum annealing model. We also give a general review about electronic structure calculations on quantum computer. In chapter 2, we show an approximating mapping between the electronic structure Hamiltonian and the Ising Hamiltonian. The whole mapping is enabled by first enlarging the qubits space to transform the electronic structure Hamiltonian to a diagonal Hamiltonian. Then introduce ancilla qubits to transform the diagonal Hamiltonian to an Ising-type Hamiltonian. We also design an algorithm to use the transformed Hamiltonian to obtain the approximating ground energy of the original Hamiltonian. The numerical simulation results of the transformed Hamiltonian for H₂, He₂, HeH⁺, and LiH molecules match the exact numerical calculations of the original Hamiltonian. This demonstrates that one can map the molecular Hamiltonian to an Ising-type Hamiltonian which could easily be implemented on currently available quantum hardware. In chapter 3, we report a hybrid quantum algorithm employing a restricted Boltzmann machine to obtain accurate molecular potential energy surfaces. By exploiting a quantum algorithm to help optimize the underlying objective function, we obtained an efficient procedure for the calculation of the electronic ground state energy for a small molecule system. Our approach achieves high accuracy for the ground state energy for H₂, LiH, H₂O at a specific location on its potential energy surface with a finite basis set. With the future availability of larger-scale quantum computers, quantum machine learning techniques are set to become powerful tools to obtain accurate values for electronic structures. In chapter 4, we present a hybrid quantum classical neural network that can be trained to perform electronic structure calculation and generate potential energy curves of simple molecules. The method is based on the combination of parameterized quantum circuit and measurements. With unsupervised training, the neural network can generate electronic potential energy curves based on training at certain bond lengths. To demonstrate the power of the proposed new method, we present results of using the quantum-classical hybrid neural network to calculate ground state potential energy curves of simple molecules such as H₂, LiH and BeH₂. The results are very accurate and the approach could potentially be used to generate complex molecular potential energy surfaces. In chapter 5, we introduce a new variational quantum eigensolver (VQE) ansatz based on the particle preserving exchange gate to achieve qubit excitations. The proposed VQE ansatz has gate complexity up-bounded to O(<i>n</i>⁴) where <i>n</i> is the number of qubits of the Hamiltonian. Numerical results of simple molecular systems such as BeH₂, H₂O, N₂, H₄ and H₆ using the proposed VQE ansatz gives very accurate results within errors about 10^-3 Hartree.

Entanglement is a key resource in many quantum information schemes andin the last years the research on multi-qubit entanglement has drawn lots ofattention. In this thesis the experimental generation and characterization ofmulti-qubit entanglement is presented. The qubits are implemented in the polarizationdegree of freedom of photons and we have prepared genuine entangledstates of two, four and six photons. We emphasize that one type of states thatwe produce are invariant entangled states, remaining unchanged under simultaneousidentical unitary transformations of all their individual constituents.Such states can be applied to e.g. decoherence-free encoding, quantum communicationwithout sharing a common reference frame, quantum telecloning,secret sharing and remote state preparation schemes.In the experimental implementation we use a single source of entangledphoton pairs and extract the first, second and third order parametric downconversion.The multi-order processes are not entirely spontaneous, as we getthe right states utilizing bosonic emission enhancement due to indistinguishability.Despite the achievement of six-photon entangled states, is the setupcompletely free from interferometric overlaps making it robust and contributingto high fidelities of the generated states. The analysis results of our experimentalstates are in very good agreement with theory and also show very highvisibilities in their correlations.

In this thesis, I report on how to use Single Photon Entanglement for generating certified quantum random numbers. Single Photon Entanglement is a particular type of entanglement which involves non-contextual correlations between two degrees of freedom of a single photon. In particular, here I consider momentum and polarization. The presence of the entanglement was validated using different attenuated coherent and incoherent sources of light by evaluating the Bell inequality, a well-known entanglement witness. Different non-idealities in the calculation of the inequality are discussed addressing them both theoretically and experimentally. Then, I discuss how to use the Single Photon Entanglement for generating certified quantum random numbers using a semi-device independent protocol. The protocol is based on a partial characterization of the experimental setup and the violation of the Bell's inequality. An analysis of the non-idealities of the devices employed in the experimental setup is also presented In the last part of the thesis, the integrated photonic version of the previously introduced experiments is discussed: first, it is presented how to generate single photon entangled states exploiting different degrees of freedom with respect to the bulk experiment. Second, I discuss how to perform an integrated test of the Bell's inequality.