Some properties of the ferromagnetic XXZ spin chain and their applications to quantum computation

One of the key requirements of quantum computation is the ability to encode qubits and construct unitary gates that are protected from the effects of environmental noise. The ferromagnetic XXZ chain is one of the best studied quantum spin models. In this thesis we extend the study of the low lying spectrum of the one dimensional ferromagnetic XXZ chain with special boundary conditions by proving the existence of isolated interface states called 'kinks'. We then capitalize on the isolated nature of these kink states to use them to encode a qubit and construct an implementation scheme for quantum gates.;In chapter 3 we investigate the low-lying excited states of the spin J ferromagnetic XXZ chain with Ising anisotropy Delta and kink boundary conditions. Since the third component of the total magnetization, M, is conserved, it is meaningful to study the spectrum for each fixed value of M. In this chapter we prove that for J ≥ 3/2 and sufficiently large Delta, the lowest excited eigenvalues of the XXZ chain determined by a fixed value of M are separated by a gap from the rest of the spectrum, uniformly in the length of the chain.;The existence of isolated eigenvalues in the low energy spectrum is very good from a quantum computing point of view since they provide natural protection from the noise effects. In chapter 4 we demonstrate an implementation scheme for constructing quantum gates using unitary evolutions of the one-dimensional spin-J ferromagnetic XXZ chain. We present numerical results based on simulations of the chain using the time-dependent DMRG method and techniques from optimal control theory. Using only a few control parameters, we find that it is possible to implement one- and two-qubit gates on a system of spin-3/2 XXZ chains, such as Not, Hadamard, Pi-8, Phase, and C-Not, with fidelity levels exceeding 99%.;Our methods could easily be adapted to a variety of other systems.