Picturing Counting Reductions with the ZH-Calculus

Counting the solutions to Boolean formulae defines the problem #SAT, which is complete for the complexity class #P. We use the ZH-calculus, a universal and complete graphical language for linear maps which naturally encodes counting problems in terms of diagrams, to give graphical reductions from #SAT to several related counting problems. Some of these graphical reductions, like to #2SAT, are substantially simpler than known reductions via the matrix permanent. Additionally, our approach allows us to consider the case of counting solutions modulo an integer on equal footing. Finally, since the ZH-calculus was originally introduced to reason about quantum computing, we show that the problem of evaluating ZH-diagrams in the fragment corresponding to the Clifford+T gateset, is in FP^#P. Our results show that graphical calculi represent an intuitive and useful framework for reasoning about counting problems.