Flagging the Clifford hierarchy:~Fault-tolerant logical $\fracπ{2^l}$ 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}}\(\fracπ{2^l}\)$ 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}\(\fracπ{2^l}\)$ or $R_{ZZ}\(\fracπ{2^l}\)$ gates on any $[[k + 2, k, 2]]$ iceberg code and fault-tolerant circuits of size $O(l)$ for preparing $|\fracπ{2^l}\rangle$ resource states in the $[[7,1,3]]$ code, which can be used to perform fault-tolerant $R_{\overline{Z}}\(\fracπ{2^l}\)$ 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 $π\(x_0.x_{1}x_{2}\ldots x_{l}\) = \sum_{j}^{l} π\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 $\fracπ{2}$ iceberg circuit to $4$ through concatenation in two-level iceberg codes. This yields a targeted logical $R_{\overline{Z}}\(\fracπ{2}\)$ gate with fault distance $4$ on any row of logical qubits in an $\[[\(k_2+2\)\(k_1+2\), k_1k_2, 4\]]$ code.