Contributions to information-based complexity and to quantum computing

Abstract

Multivariate continuous problems are widely encountered in physics, chemistry, finance and in computational sciences. Unfortunately, interesting real world multivariate continuous problems can almost never be solved analytically. As a result, they are typically solved numerically and therefore approximately. In this thesis we deal with the approximate solution of multivariate problems. The complexity of such problems in the classical setting has been extensively studied in the literature. On the other hand the quantum computational model presents a promising alternative for dealing with multivariate problems. The idea of using quantum mechanics to simulate quantum physics was initially proposed by Feynman in 1982. Its potential was demonstrated by Shor's integer factorization algorithm, which exponentially improves the cost of the best classical algorithm known. In the first part of this thesis we study the tractability of multivariate problems in the worst and average case settings using the real number model with oracles. We derive necessary and sufficient conditions for weak tractability for linear multivariate tensor product problems in those settings. More specifically, we initially study necessary and sufficient conditions for weak tractability on linear multivariate tensor product problems in the worst case setting under the absolute error criterion. The complexity of such problems depends on the rate of decay of the squares of the singular values of the solution operator for the univariate problem. We show a condition on the singular values that is sufficient for weak tractability. The same condition is known to be necessary for weak tractability. Then, we study linear multivariate tensor product problems in the average case setting under the absolute error criterion. The complexity of such problems depends on the rate of decay of the eigenvalues of the covariance operator of the induced measure of the one dimensional problem. We derive a necessary and sufficient condition on the eigenvalues for such problems to be weakly tractable but not polynomially tractable. In the second part of this thesis we study quantum algorithms for certain eigenvalue problems and the implementation and design of quantum circuits for a modification of the quantum NAND evaluation algorithm on k-ary trees, where k is a constant. Finally, we consider the algorithm by Ambainis et.al. that evaluates balanced binary NAND formulas. We design a quantum circuit that implements a modification of the algorithm for k-ary trees, where k is a constant. Furthermore, we design another quantum circuit that consists exclusively of Clifford and T gates. This circuit approximates the previous one with error e using the Solovay-Kitaev algorithm. (Abstract shortened by UMI.)