Criteria for exact qudit universality

Abstract

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 of Mathukrishnan and Stroud [1] describes how to exactly simulate an arbitrary unitary on multiple qudits using a 2d-1 parameter family of single qudit and two qudit gates. Their 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 H_{jk}^x=\hbar\Omega (|k><j|+|j><k|) and H_{jk}^y =\hbar\Omega (i|k><j|-i|j><k|). A coupling graph results taking nodes 0,... d-1 and edges j<->k iff H_{jk}^{x,y} are allowed Hamiltonians. One qudit exact universality follows iff this graph is connected, and complete universality results if the two-qudit Hamiltonian H=-\hbar\Omega |d-1,d-1><d-1,d-1| is also allowed. We discuss implementation in the eight dimensional ground electronic states of ^{87}Rb and construct an optimal gate sequence using Raman laser pulses.