  Qubits are neither fermions nor bosons. A Fock space description of qubits
leads to a mapping from qubits to parafermions: particles with a hybrid
boson-fermion quantum statistics. We study this mapping in detail, and use it
to provide a classification of the algebras of operators acting on qubits.
These algebras in turn classify the universality of different classes of
physically relevant qubit-qubit interaction Hamiltonians. The mapping is
further used to elucidate the connections between qubits, bosons, and fermions.
These connections allow us to share universality results between the different
particle types. Finally, we use the mapping to study the quantum computational
power of certain anisotropic exchange Hamiltonians. In particular, we prove
that the XY model with nearest-neighbor interactions only is not
computationally universal. We also generalize previous results about universal
quantum computation with encoded qubits to codes with higher rates.
