Criteria for exact qudit universality

We describe criteria for implementation of quantum computation in qudits. A qudit is a -dimensional system whose Hilbert space is spanned by states ∣0⟩, ∣1⟩, …, . An important earlier work [] describes how to exactly simulate an arbitrary unitary on multiple qudits using a 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 -matrix decomposition of numerical linear algebra. We consider a generic physical system in which the single qudit Hamiltonians are a small collection of and . A coupling graph results taking nodes 0, …, and edges are allowed Hamiltonians. One qudit exact universality follows iff this graph is connected, and complete universality results if the two-qudit Hamiltonian is also allowed. We discuss implementation in the eight dimensional ground electronic states of and construct an optimal gate sequence using Raman laser pulses.

Figure 1

A single $d=8$ qudit encoded in the ground-state hyperfine levels of ${}^{87}\mathrm{Rb}$. A pair of lasers can couple states in different hyperfine manifolds according to the selection rule $\Delta M_F=0,\pm 1$. Projective measurements of population in state ∣7⟩ are made by observing resonant fluorescence on a cycling transition to the excited state. Any pair of states can be coupled by swapping neighbors together pairwise and similarly any state can be measured by swapping to ∣7⟩.

Figure 2

This is the coupling graph for the coupled hyperfine states of ${}^{87}\mathrm{Rb}$ (see Fig. 1.) As it is connected, the collection of atom-laser couplings allows for universal one-qudit computation.