Characterization of universal two-qubit hamiltonian
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 onn 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 ofnon-universal 2-qubit Hamiltonians has been elusive. Our main result is a completecharacterization of 2-non-universal 2-qubit Hamiltonians. In particular, there are threeways that a 2-qubit Hamiltonian H can fail to be universal: (1) H shares an eigenvectorwith the gate that swaps two qubits, (2) H acts on the two qubits independently (inany of a certain family of bases), or (3) H has zero trace (with the third conditionrelevant only when the global phase of the unitary matters). A 2-non-universal 2-qubitHamiltonian can still be n-universal for some n ≥ 3. We give some partial results on3-universality.