  Non-Clifford gates are frequently exclusively implemented on fault-tolerant
architectures by first distilling magic states in specialised magic-state
factories. In the rest of the architecture, the computational space, magic
states can then be consumed by a stabilizer circuit to implement non-Clifford
operations. We show that the connectivity between the computational space and
magic state factories forms a fundamental bottleneck on the rate at which
non-Clifford operations can be implemented. We show that the nullity of the
magic state, $\nu(|D\rangle)$ for diagonal gate $D$, characterizes the
non-local resources required to implement $D$ in the computational space. As
part of our proof, we construct local stabilizer circuits that use only
$\nu(|D\rangle)$ ebits to implement $D$ in the computational space that may be
useful to reduce the non-local resources required to inject non-Clifford gates.
Another consequence is that the edge-disjoint path compilation algorithm
[arXiv:2110.11493] produces minimum-depth circuits for implementing
single-qubit diagonal gates.
