Upper bounds on fault tolerance thresholds of noisy Clifford-based quantum computers

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.