  Qutrit (or ternary) structures arise naturally in many quantum systems,
particularly in certain non-abelian anyon systems. We present efficient
circuits for ternary reversible and quantum arithmetics. Our main result is the
derivation of circuits for two families of ternary quantum adders, namely
ripple carry adders and carry look-ahead adders. The main difference to the
binary case is the more complicated form of the ternary carry, which leads to
higher resource counts for implementations over a universal ternary gate set.
Our ternary ripple adder circuit has a circuit depth of $O(n)$ and uses only
$1$ ancilla, making it more efficient in both, circuit depth and width than
previous constructions. Our ternary carry lookahead circuit has a circuit depth
of only $O(\log\,n)$, while using with $O(n)$ ancillas. Our approach works on
two levels of abstraction: at the first level, descriptions of arithmetic
circuits are given in terms of gates sequences that use various types of
non-Clifford reflections. At the second level, we break down these reflections
further by deriving them either from the two-qutrit Clifford gates and the
non-Clifford gate $C(X): |i,j\rangle \mapsto |i, j + \delta_{i,2} \mod
3\rangle$ or from the two-qutrit Clifford gates and the non-Clifford gate
$P_9=\mbox{diag}(e^{-2 \pi \, i/9},1,e^{2 \pi \, i/9})$. The two choices of
elementary gate sets correspond to two possible mappings onto two different
prospective quantum computing architectures which we call the metaplectic and
the supermetaplectic basis, respectively. Finally, we develop a method to
factor diagonal unitaries using multi-variate polynomial over the ternary
finite field which allows to characterize classes of gates that can be
implemented exactly over the supermetaplectic basis.
