Characterization of universal two-qubit hamiltonian

Authors:
Andrew M. Childs;Debbie Leung;Laura Mancinska;Maris Ozols
Affiliations:
Department of Combinatorics & Optimization and Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada;Department of Combinatorics & Optimization and Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada;Department of Combinatorics & Optimization and Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada;Department of Combinatorics & Optimization and Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada
Venue:
Quantum Information & Computation
Year:
2011

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Abstract

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.