  Recent advances in classical simulation of Clifford+T circuits make use of
the ZX calculus to iteratively decompose and simplify magic states into
stabiliser terms. We improve on this method by studying stabiliser
decompositions of ZX diagrams involving the triangle operation. We show that
this technique greatly speeds up the simulation of quantum circuits involving
multi-controlled gates which can be naturally represented using triangles. We
implement our approach in the QuiZX library and demonstrate a significant
simulation speed-up (up to multiple orders of magnitude) for random circuits
and a variation of previously used benchmarking circuits. Furthermore, we use
our software to contract diagrams representing the gradient variance of
parametrised quantum circuits, which yields a tool for the automatic numerical
detection of the barren plateau phenomenon in ans\"atze used for quantum
machine learning. Compared to traditional statistical approaches, our method
yields exact values for gradient variances and only requires contracting a
single diagram. The performance of this tool is competitive with tensor network
approaches, as demonstrated with benchmarks against the quimb library.
