  Mapping a quantum algorithm to any practical large-scale quantum computer
will require a sequence of compilations and optimizations. At the level of
fault-tolerant encoding, one likely requirement of this process is the
translation into a topological circuit, for which braided circuits represent
one candidate model. Given the large overhead associated with encoded circuits,
it is paramount to reduce their size in terms of computation time and qubit
number through circuit compression. While these optimizations have typically
been performed in the language of three-dimensional diagrams, such a
representation does not allow an efficient, general, and scalable approach to
reduction or verification. We propose the use of the ZX-calculus as an
intermediate language for braided circuit compression, demonstrating advantage
by comparing results using this approach with those previously obtained for the
compression of A- and Y-state distillation circuits. We then provide a
benchmark of our method against a small set of Clifford+T circuits, yielding
compression percentages of 77%. Our results suggest that the overheads of
braided, defect-based circuits are comparable to those of their lattice-surgery
counterparts, restoring the potential of this model for surface-code quantum
computation.
