Quantum information science explores ways in which quantum physical laws can be harnessed to control the acquisition, transmission, protection, and processingof information. This field has seen explosive growth in the past several years from progress on both theoretical and experimental fronts. Essential to this endeavor are methods for controlling quantum information.
In this thesis, I present three new approaches for controlling quantum information. First, I present a new protocol for continuously protecting unknown quantumstates from noise. This protocol combines and expands ideas from the theories of quantum error correction and quantum feedback control. The result can outperform either approach by itself. I generalize this protocol to all known quantumstabilizer codes, and study its application to the three-qubit repetition code in detail via Monte Carlo simulations.
Next, I present several new protocols for controlling quantum information that are fault-tolerant. These protocols require only local quantum processing due tothe topological properties of the quantum error correcting codes upon which they are built. I show that each protocol's fault-dependence behavior exhibits an order-disorder phase transition when mapped onto an associated statistical-mechanical model. I review the critical error rates of these protocols found by numerical studyof the associated models, and I present new analytic bounds for them using a self-avoiding random walk argument. Moreover, I discuss fault-tolerant procedures for encoding, error-correction, computing, and decoding quantum information using these protocols, and calculate the accuracy threshold of fault-tolerant quantum memory for protocols using them.
I end by presenting a new class of quantum algorithms that solve combinatorial optimization problems solely by measurement. I compute the running times ofthese algorithms by establishing an explicit dynamical model for the measurement process. This model, the digitized version of von Neumann's measurement model,is recognized as Kitaev's phase estimation algorithm. I show that the running times of these algorithms are closely related to the running times of adiabatic quantum algorithms. Finally, I present a two-measurement algorithm that achieves a quadratic speedup for Grover's unstructured search problem.
