01458nas a2200145 4500008004100000245005700041210005600098260001500154520102700169100002301196700001801219700002101237700001701258856003701275 2010 eng d00aCharacterization of universal two-qubit Hamiltonians0 aCharacterization of universal twoqubit Hamiltonians c2010/04/093 a 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.
1 aChilds, Andrew, M.1 aLeung, Debbie1 aMancinska, Laura1 aOzols, Maris uhttp://arxiv.org/abs/1004.1645v2