No / Large-scale data are analysed using the Wigner function. It is shown that the ‘frequency variable’ provides important information, which is lost with other techniques. The method is applied to ‘sentiment analysis’ in data from social networks and also to financial data.

none

The Schrödinger equation ... is considered. The solution of this equation is reduced to the problem of finding the eigenvectors of an infinite matrix. The infinite matrix is truncated to a finite matrix. The approximation due to the truncation is carefully studied. The band structure of the eigenvalues is shown. The eigenvectors of the multiwells potential are presented. The solutions of Schrödinger equation are calculated. The results are very sensitive to the value of the parameter y. Localized solutions, in the case that the energy is slightly greater than the maximum value of the potential, are presented. Wigner and Weyl functions, corresponding to the solutions of Schrödinger equation, are also studied. It is also shown that they are very sensitive to the value of the parameter y. / Garyounis University and Libyan Cultural Affairs

The Schrödinger equation ... is considered. The solution of this equation is reduced to the problem of finding the eigenvectors of an infinite matrix. The infinite matrix is truncated to a finite matrix. The approximation due to the truncation is carefully studied. The band structure of the eigenvalues is shown. The eigenvectors of the multiwells potential are presented. The solutions of Schrödinger equation are calculated. The results are very sensitive to the value of the parameter y. Localized solutions, in the case that the energy is slightly greater than the maximum value of the potential, are presented. Wigner and Weyl functions, corresponding to the solutions of Schrödinger equation, are also studied. It is also shown that they are very sensitive to the value of the parameter y.

none

This thesis deals with the question of what resources are necessary and sufficient for quantum computational speedup. In particular, we study what resources are required to promote fault tolerant stabilizer computation to universal quantum computation. In this context we discover a remarkable connection between the possibility of quantum computational speedup and negativity in the discrete Wigner function, which is a particular distinguished quasi-probability representation for quantum theory. This connection allows us to establish a number of important results related to magic state computation, an important model for fault tolerant quantum computation using stabilizer operations supplemented by the ability to prepare noisy non-stabilizer ancilla states. In particular, we resolve in the negative the open problem of whether every non-stabilizer resource suffices to promote computation with stabilizer operations to universal quantum computation. Moreover, by casting magic state computation as resource theory we are able to quantify how useful ancilla resource states are for quantum computation, which allows us to give bounds on the required resources. In this context we discover that the sum of the negative entries of the discrete Wigner representation of a state is a measure of its usefulness for quantum computation. This gives a precise, quantitative meaning to the negativity of a quasi-probability representation, thereby resolving the 80 year debate as to whether this quantity is a meaningful indicator of quantum behaviour. We believe that the techniques we develop here will be widely applicable in quantum theory, particularly in the context of resource theories.

This thesis deals with the question of what resources are necessary and sufficient for quantum computational speedup. In particular, we study what resources are required to promote fault tolerant stabilizer computation to universal quantum computation. In this context we discover a remarkable connection between the possibility of quantum computational speedup and negativity in the discrete Wigner function, which is a particular distinguished quasi-probability representation for quantum theory. This connection allows us to establish a number of important results related to magic state computation, an important model for fault tolerant quantum computation using stabilizer operations supplemented by the ability to prepare noisy non-stabilizer ancilla states. In particular, we resolve in the negative the open problem of whether every non-stabilizer resource suffices to promote computation with stabilizer operations to universal quantum computation. Moreover, by casting magic state computation as resource theory we are able to quantify how useful ancilla resource states are for quantum computation, which allows us to give bounds on the required resources. In this context we discover that the sum of the negative entries of the discrete Wigner representation of a state is a measure of its usefulness for quantum computation. This gives a precise, quantitative meaning to the negativity of a quasi-probability representation, thereby resolving the 80 year debate as to whether this quantity is a meaningful indicator of quantum behaviour. We believe that the techniques we develop here will be widely applicable in quantum theory, particularly in the context of resource theories.

no / Inverse limits and profinite groups are used in a quantum mechanical context. Two cases are considered: a quantum system with positions in the profinite group Z(p) and momenta in the group Q(p)/Z(p), and a quantum system with positions in the profinite group (Z) over cap and momenta in the group Q/Z. The corresponding Schwatz-Bruhat spaces of wavefunctions and the Heisenberg-Weyl groups are discussed. The sets of subsystems of these systems are studied from the point of view of partial order theory. It is shown that they are directed-complete partial orders. It is also shown that they are topological spaces with T-0-topologies, and this is used to define continuity of various physical quantities. The physical meaning of profinite groups, non-Archimedean metrics, partial orders and T-0-topologies, in a quantum mechanical context, is discussed.

Quantum systems with finite Hilbert space where position x and momentum p take values in Z(d) (integers modulo d) are considered. Symplectic tranformations S(2ξ,Z(p)) in ξ-partite finite quantum systems are studied and constructed explicitly. Examples of applying such simple method is given for the case of bi-partite and tri-partite systems. The quantum correlations between the sub-systems after applying these transformations are discussed and quantified using various methods. An extended phase-space x-p-X-P where X, P ε Z(d) are position increment and momentum increment, is introduced. In this phase space the extended Wigner and Weyl functions are defined and their marginal properties are studied. The fourth order interference in the extended phase space is studied and verified using the extended Wigner function. It is seen that for both pure and mixed states the fourth order interference can be obtained.

Quantum annealing is expected to be a powerful generic algorithm for solving hard combinatorial optimization problems faster than classical computers. Finding the solution to a combinatorial optimization problem is equivalent to finding the ground state of an Ising Hamiltonian. In today's quantum annealers the spins of the Ising Hamiltonian are mapped to superconducting qubits. On the other hand, dissipation processes degrade the success probability of finding the solution. In this thesis we set out to explore a newly proposed architecture for a noise-resilient quantum annealer that instead maps the Ising spins to continuous variable quantum states of light encoded in the field quadratures of a two-photon pumped Kerr- nonlinear resonator based on the proposal by Puri et al. (2017). In this thesis we study the Wigner negativity for this newly proposed architecture and evaluate its performance based on the negativity of the Wigner function. We do this by determining an experimental value to when the presence of losses become too detrimental, such that the Wigner function of the quantum state during the evolution within the anneal becomes positive for all times. Furthermore, we also demonstrate the capabilities of this continuous variable quantum annealer by simulating and finding the best solution of a small instance of the NP-complete subset sum problem and of the number partitioning problem.