Abstract
In this thesis, we introduce several ef?cient quantum algorithms for a range of problems and show that they are more ef?cient than their classical counterparts. First, we give an introduction to the basic concepts in quantum computation and a review of related quantum algorithms, such as the quantum adiabatic algorithm, quantum phase estima-tion, quantum Hamiltonian simulation and the quantum algorithm for solving linear equations. We then present a quantum adiabatic-like measurement-driven algorithm to obtain the ground state of a frustration-free Hamiltonian. We show that the energy gap plays an important role in bounding the number of measurements required. We prove that there always exists a frustration-free path. We also give a measurement-driven algorithm for solving the well-known 3-SAT problem. A bound for the energy gap for intermediate Hamiltonians along an evolution path for solving 3-SAT problems is also given. We also introduce quantum algorithms to approximate matrix functions for low-rank and sparse matrices, which can be decomposed into basic matrix operations including addition, mul-tiplication, Kronecker sum, tensor product, Hadamard product and single matrix functions, with bounded errors. We also show that we can estimate the trace and Schatten p-norm of a matrix function ef?ciently. Furthermore, we describe ef?cient quantum algorithms to obtain higher-order singular value decomposition and multi-mode principal component analysis of tensors. We compare these quantum algorithms with their classical counterparts and show that the former one is exponen-tially faster for a certain class of tensors. Finally, we construct a quantum graph data structure based on quantum graph states, then we give quantum algorithms to perform a number of graph operations, such as complementing various classes of edges and comparison operations. We also compare our quantum data struc-ture with classical counterparts and show that the graph state data structure performs better for a set of operations than any possible classical structure.