  We consider the possibility of adding noise to a quantum circuit to make it
efficiently simulatable classically. In previous works this approach has been
used to derive upper bounds to fault tolerance thresholds - usually by
identifying a privileged resource, such as an entangling gate or a non-Clifford
operation, and then deriving the noise levels required to make it
`unprivileged'. In this work we consider extensions of this approach where
noise is added to Clifford gates too, and then `commuted' around until it
concentrates on attacking the non-Clifford resource. While commuting noise
around is not always straightforward, we find that easy instances can be
identified in popular fault tolerance proposals, thereby enabling sharper upper
bounds to be derived in these cases. For instance we find that if we take
Knill's high threshold proposal together with the ability to prepare any
possible state in the $XY$ plane of the Bloch sphere, then no more than 3.69%
error-per-gate noise is sufficient to make it classical, and 13.71% of Knill's
gamma noise model is sufficient. These bounds have been derived without noise
being added to the decoding parts of the circuits. Introducing such noise in a
toy example suggests that the present approach can be optimised further to
yield tighter bounds.
