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\(x₀.x₁x₂ldots x_l\) = sum_j^l πfrac{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 \[[\(k₂+2\)\(k₁+2\), k_1k2, 4\]] code.