Quantum Computation, Complexity, and Many-body Physics

By taking advantage of the laws of physics it is possible to revolutionize the way we communicate (transmit), process or even store information. It is now known that quantum computers, or computers built from quantum mechanical elements, provide new resources to solve certain problems and perform certain tasks more efficiently than today’s conventional computers. However, on the road to a complete understanding of the power of quantum computers there are intermediate steps that need to be addressed. The primary focus of this thesis is the understanding of the possibilities and limitations of the quantum-physical world in the areas of quantum computation and quantum information processing. First I investigate the simulation of quantum systems on a quantum computer (i.e., a quantum simulation) constructed of two-level quantum elements or qubits. For this purpose, I present algebraic mappings that allow one to efficiently obtain physical properties and compute correlation functions of fermionic, anyonic, and bosonic systems with such a computer. By studying the amount of resources required for a quantum simulation, I show that the complexity of preparing a quantum state which contains the desired information is crucial at the time of evaluating the advantages of having a quantum computer over a conventional one. As a small-scale demonstration of the validity of these results, I show the simulation of a fermionic system using a liquid-state nuclear magnetic resonance (NMR) device. Remarkably, the conclusions obtained in the area of quantum simulations can be extended to general quantum computations by means of the notion of generalized entanglement. This is a generalization based on the idea that quantum entanglement (i.e., the existence of non-classical correlations) is a concept that depends on the accesible information, that is, relative to the observer. Then I present a wide class of quantum computations that can be efficiently simulated on a conventional computer and where quantum computers cannot be claimed to be more powerful. The idea is that a quantum algorithm, performed by applying