Characterization of non-universal two-qubit Hamiltonians

Hamiltonian;universal;quantum;universality;Combinatorics and Optimization

Mancinska, Laura

University of Waterloo

关键词: Hamiltonian;    universal;    quantum;    universality;    Combinatorics and Optimization;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/4873/1/Non-universal%20Hamiltonians.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

It is known that almost all 2-qubit gates are universal for quantum computing (Lloyd 1995; Deutsch, Barenco, Eckert 1995). However, an explicit characterization of non-universal 2-qubit gates is not known. We consider a closely related problem of characterizing the set of non-universal 2-qubit Hamiltonians. We call a 2-qubit Hamiltonian n-universal if, when applied on different pairs of qubits, it can be used to approximate any unitary operation on n qubits. It follows directly from the results of Lloyd and Deutsch, Barenco, Eckert, that almost any 2-qubit Hamiltonian is 2-universal. Our main result is a complete characterization of 2-non-universal 2-qubit Hamiltonians. There are three cases when a 2-qubit Hamiltonian H is not 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);(3) H has zero trace.The last condition rules out the Hamiltonians that generate SU(4)---it can be omitted if the global phase is not important.A Hamiltonian that is not 2-universal can still be 3-universal. We give a (possibly incomplete) list of 2-qubit Hamiltonians that are not 3-universal. If this list happens to be complete, it actually gives a classification of n-universal 2-qubit Hamiltonians for all n >= 3.