Suppose we can apply a given 2-qubit Hamiltonian H to any (ordered) pair of
qubits. We say H is n-universal if it can be used to approximate any unitary
operation on n qubits. While it is well known that almost any 2-qubit
Hamiltonian is 2-universal (Deutsch, Barenco, Ekert 1995; Lloyd 1995), an
explicit characterization of the set of non-universal 2-qubit Hamiltonians has
been elusive. Our main result is a complete characterization of 2-non-universal
2-qubit Hamiltonians. In particular, there are three ways that a 2-qubit
Hamiltonian H can fail to be universal: (1) H shares an eigenvector with the
gate that swaps two qubits, (2) H acts on the two qubits independently (in any
of a certain family of bases), or (3) H has zero trace. A 2-non-universal
2-qubit Hamiltonian can still be n-universal for some n >= 3. We give some
partial results on 3-universality. Finally, we also show how our
characterization of 2-universal Hamiltonians implies the well-known result that
almost any 2-qubit unitary is universal.
