  The ZX-calculus is a graphical language which allows for reasoning about
suitably represented tensor networks - namely ZX-diagrams - in terms of rewrite
rules. Here, we focus on problems which amount to exactly computing a scalar
encoded as a closed tensor network. In general, such problems are #P-hard.
However, there are families of such problems which are known to be in P when
the dimension is below a certain value. By expressing problem instances from
these families as ZX-diagrams, we see that the easy instances belong to the
stabilizer fragment of the ZX-calculus. Building on previous work on efficient
simplification of qubit stabilizer diagrams, we present simplifying rewrites
for the case of qutrits, which are of independent interest in the field of
quantum circuit optimisation. Finally, we look at the specific examples of
evaluating the Jones polynomial and of counting graph-colourings. Our
exposition further champions the ZX-calculus as a suitable and unifying
language for studying the complexity of a broad range of classical and quantum
problems.
