  In a recent work arXiv:2201.07655v2 we showed that there is a constant
$\lambda >0$ such that it is possible to efficiently classically simulate a
quantum system in which (i) qudits are placed on the nodes of a graph, (ii)
each qudit undergoes at most $D$ diagonal gates, (iii) each qudit is
destructively measured in the computational basis or bases unbiased to it, and
(iv) each qudit is initialised within $\lambda^{-D}$ of a diagonal state
according to a particular distance measure. In this work we explicitly compute
$\lambda$ for any two qubit diagonal gate, thereby extending the computation of
arXiv:2201.07655v2 beyond CZ gates. For any finite degree graph this allows us
to describe a two parameter family of pure entangled quantum states (or three
parameter family of thermal states) which have a non-trivial classically
efficiently simulatable "phase" for the permitted measurements, even though
other values of the parameters may enable ideal cluster state quantum
computation. The main the technical tool involves considering separability in
terms of "cylindrical" sets of operators. We also consider whether a different
choice of set can strengthen the algorithm, and prove that they are optimal
among a broad class of sets, but also show numerically that outside this class
there are choices that can increase the size of the classically efficient
regime.
