Power of Anisotropic Exchange Interactions: Universality and Efficient Codes for Quantum Computing

We study the quantum computational power of a generic class of anisotropic solid state Hamiltonians. A universal set of encoded logic operations are found which do away with difficult-to-implement single-qubit gates in a number of quantum computer proposals, e.g., quantum dots and donor atom spins with anisotropic exchange coupling, quantum Hall systems, and electrons floating on helium.We show how to make the corresponding Hamiltonians universal by encoding one qubit into two physical qubits, and by controlling nearest neighbor interactions.