Title: Contributions to Information-Based Complexity and to Quantum Computing
Description: 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. First, we study quantum algorithms for the estimation of the ground state energy of the multivariate time-independent Schrodinger equation corresponding to a multiparticle system in a box. The dimension d of the problem depends linearly to the number of particles of the system. We design a quantum algorithm that approximates the lowest eigenvalue with relative error ε for a non-negative potential V, where V as well as its first order partial derivatives are continuous and uniformly bounded by one. The algorithm requires a number of quantum operations that depends polynomially on the inverse of the accuracy and linearly on the number of the particles of the system. We note that the cost of any classical deterministic algorithm grows exponentially in the number of particles.

Thus we have an exponential speedup with respect to the dimension of the problem d, when compared to the classical deterministic case. We extend our results to convex non-negative potentials V, where V as well as its first order partial derivatives are continuous and uniformly bounded by constants C and C' respectively. The algorithm solves the eigenvalue problem for a sequence of convex potentials in order to obtain its final result. More specifically, the quantum algorithm estimates the ground state energy with relative error ε a number of quantum operations that depends polynomially on the inverse of the accuracy, the uniform bound C on the potential and the dimension d of the problem. In addition, we present a modification of the algorithm that produces a quantum state which approximates the ground state eigenvector of the discretized Hamiltonian within Δ. This algorithm requires a number of quantum operations that depends pollynomially on the inverse of ε, the inverse of Δ, the uniform bound C on the potential and the dimension d of the problem.

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 ε using the Solovay-Kitaev algorithm.