I've recently started doing some reading on the subject of qudit codes. In particular, i'm interested in the frequently used clock and shift operators.
Can these operators be physically realized? Or, even better, can they be expressed in terms of sums and products of spin operators?
I have found one reference that shows that any 3x3 matrix can be expressed as a (complicated) set of sums and products of the spin-1 operators:
But i'm not sure if this holds for higher spins.
EDIT: I realize my initial question was slightly misleading. Sorry. I am looking at higher spin generalizations of the Toric code in terms of Qudits, whose hamiltionian is written in terms of these clock and shift operators. So my sense of "physically realizable" would be expressing these operators in terms of something I could find in a condensed matter system.