Criteria for exact qudit universality

Gavin K. Brennen*, Dianne P. O'Leary, Stephen S. Bullock

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

59 Citations (Scopus)


We describe criteria for implementation of quantum computation in qudits. A qudit is a d-dimensional system whose Hilbert space is spanned by states 0, 1,..., d-1. An important earlier work [A. Muthukrishnan and C.R. Stroud, Jr., Phys. Rev. A 62, 052309 (2000)] describes how to exactly simulate an arbitrary unitary on multiple qudits using a 2d-1 parameter family of single qudit and two qudit gates. That technique is based on the spectral decomposition of unitaries. Here we generalize this argument to show that exact universality follows given a discrete set of single qudit Hamiltonians and one two-qudit Hamiltonian. The technique is related to the QR-matrix decomposition of numerical linear algebra. We consider a generic physical system in which the single qudit Hamiltonians are a small collection of Hjkx=Ω(kj+jk) and Hjky=Ω(ikj-ijk). A coupling graph results taking nodes 0,..., d-1 and edges jkiffHjkx,y are allowed Hamiltonians. One qudit exact universality follows iff this graph is connected, and complete universality results if the two-qudit Hamiltonian H=Ωd-1,d-1d-1,d-1 is also allowed. We discuss implementation in the eight dimensional ground electronic states of Rb87 and construct an optimal gate sequence using Raman laser pulses.

Original languageEnglish
Article number052318
Pages (from-to)1-7
Number of pages7
JournalPhysical Review A - Atomic, Molecular, and Optical Physics
Issue number5
Publication statusPublished - May 2005
Externally publishedYes


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