  A quantum circuit may be strongly classically simulated with the aid of
ZX-calculus by decomposing its $t$ T-gates into a sum of $2^{\alpha t}$
classically computable stabiliser terms. In this paper, we introduce a general
procedure to find an optimal pattern of vertex cuts in a ZX-diagram to maximise
its T-count reduction at the cost of the fewest cuts. Rather than reducing a
Clifford+T diagram based on a fixed routine of decomposing its T-gates directly
(as is the conventional approach), we focus instead on taking advantage of
certain patterns and structures common to such circuits to, in effect, design
by automatic procedure an arrangement of spider decompositions that is
optimised for the particular circuit. In short, this works by assigning weights
to vertices based on how many T-like gates they are blocking from
fusing/cancelling and then appropriately propagating these weights through any
neighbours which are then blocking weighted vertices from fusing, and so on.
Ultimately, this then provides a set of weightings on relevant nodes, which can
then each be cut, starting from the highest weighted down. While this is a
heuristic approach, we show that, for circuits small enough to verify, this
method achieves the most optimal set of cuts possible $71\%$ of the time.
Furthermore, there is no upper bound for the efficiency achieved by this
method, allowing, in principle, an effective decomposition efficiency
$\alpha\rightarrow0$ for highly structured circuits. Even applied to random
pseudo-structured circuits (produced from CNOTs, phase gates, and Toffolis), we
record the number of stabiliser terms required to reduce all T-gates, via our
method as compared to that of the more conventional T-decomposition approaches
(with $\alpha\approx0.47$), and show consistent improvements of orders of
magnitude, with an effective efficiency $0.1\lesssim\alpha\lesssim0.2$.
