Flagging the Clifford hierarchy:~Fault-tolerant logical $\frac{\pi}{2^l}$ rotations via measuring circuit gauge operators of non-Cliffords
We provide a recursively defined sequence of flag circuits which will detect logical errors induced by non-fault-tolerant $R_{\overline{Z}}(\frac{\pi}{2^l})$ gates on CSS codes with a fault distance of two. As applications, we give a family of circuits with $O(l)$ gates and ancillae which implement fault-tolerant logical $R_{Z}(\frac{\pi}{2^l})$ or $R_{ZZ}(\frac{\pi}{2^l})$ gates on any $[[k + 2, k, 2]]$ iceberg code and fault-tolerant circuits of size $O(l)$ for preparing $|\frac{\pi}{2^l}\rangle$ resource states in the $[[7,1,3]]$ code, which can be used to perform fault-tolerant $R_{\overline{Z}}(\frac{\pi}{2^l})$ rotations via gate teleportation, allowing for implementations of these gates that bypass the high overheads of gate synthesis when $l$ is small relative to the precision required. We show how the circuits above can be generalized to $\pi( x_0.x_{1}x_{2}\ldots x_{l}) = \sum_{j}^{l} \pi \frac{x_j}{2^j}$ rotations with identical overheads in $l$, which could be useful in quantum simulations where time is digitized in binary. Finally, we illustrate two approaches to increase the fault-distance of our construction. We show how to increase the fault distance of a Cliffordized version of the T gate circuit to $3$ in the Steane code and how to increase the fault-distance of the $\frac{\pi}{2}$ iceberg circuit to $4$ through concatenation in two-level iceberg codes. This yields a targeted logical $R_{\overline{Z}}(\frac{\pi}{2})$ gate with fault distance $4$ on any row of logical qubits in an $[[(k_2+2)(k_1+2), k_1k_2, 4]]$ code.