Purifying quantum states: Quantum and classical algorithms

The problems of decoherence and destructive measurement make quantum computers especially unstable to errors. We begin by exploring Kitaev's framework for quantum error correcting codes that exploit the inherent stability of certain topological quantum numbers. Analytic estimates and numerical simulation allow us to quantitatively assess the conditions under which this stability persists. In order to allow quantum computations to be performed on data stored in these and other unconcatenated codes, we present a probabilistic algorithm for executing an encoded Toffoli gate via preparation of a certain three-qubit entangled state as a computational resource. The algorithm works to iteratively purify an initial input state, conditional on favorable measurement outcomes at each stage in the iteration.; Analogous iterative purification methods are identified in the context of classical algorithms for simulating quantum dynamics. Here an ensemble of stochastic trajectories over a classical configuration space is iteratively driven toward a fixed point that yields information about a quantum evolution in an associated Hilbert space. We test this procedure for a restricted subspace of the Heisenberg ferromagnet.; The trajectory ensemble concept is then applied to the problem of mechanism identification in closed-loop quantum control, where a precise definition of mechanism in terms of these trajectories leads to methods for extracting dynamical information directly from the kinds of optimization procedures already realized in control experiments on optical-molecular systems. We demonstrate this on simulated data for controlled population transfer in a seven level molecular system under shaped-pulse laser excitation.