Characterization of universal two-qubit Hamiltonians
Andrew M. Childs, Debbie Leung, Laura Mančinska, Māris Ozols · Quantum Information and Computation · 2011
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, 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 (with the third condition relevant only when the global phase of the unitary matters). A 2-non-universal 2-qubit Hamiltonian can still be n-universal for some n \geq 3. We give some partial results on 3-universality.