Geometric methods in quantum computation

Recent advances in the physical sciences and engineering have created great hopes for new computational paradigms and substrates. One such new approach is the quantum computer, which holds the promise of enhanced computational power. Analogous to the way a classical computer is built from electrical circuits containing wires and logic gates, a quantum computer is built from quantum circuits containing quantum wires and elementary quantum gates to transport and manipulate quantum information. Therefore, design of quantum gates and quantum circuits is a prerequisite for any real application of quantum computation.; In this dissertation we apply geometric control methods from differential geometry and Lie group representation theory to analyze the properties of quantum gates and to design optimal quantum circuits. Using the Cartan decomposition and the Weyl group, we show that the geometric structure of nonlocal two-qubit gates is a 3-Torus. After further reducing the symmetry, the geometric representation of nonlocal gates is seen to be conveniently visualized as a tetrahedron. Each point in this tetrahedron except on the base corresponds to a different equivalent class of nonlocal gates. This geometric representation is one of the cornerstones for the discussion on quantum computation in this dissertation.; We investigate the properties of those two-qubit operations that can generate maximal entanglement. It is an astonishing finding that if we randomly choose a two-qubit operation, the probability that we obtain a perfect entangler is exactly one half. We prove that given a two-body interaction Hamiltonian, it is always possible to explicitly construct a quantum circuit for exact simulation of any arbitrary nonlocal two-qubit gate by turning on the two-body interaction for at most three times, together with at most four local gates. We also provide an analytic approach to construct a universal quantum circuit from any entangling gate supplemented with local gates. Closed form solutions have been derived for each step in this explicit construction procedure. Moreover, the minimum upper bound is found to construct a universal quantum circuit from any Controlled-Unitary gate. A near optimal explicit construction of universal quantum circuits from a given Controlled-Unitary is provided. For the Controlled-NOT and Double-CNOT gate, we then develop simple analytic ways to construct universal quantum circuits with exactly three applications, which is the least possible for these gates. We further discover a new quantum gate (named B gate) that achieves the desired universality with minimal number of gates. Optimal implementation of single-qubit quantum gates is also investigated. Finally, as a real physical application, a constructive way to implement any arbitrary two-qubit operation on a spin electronics system is discussed.