  Group twirling is crucial in quantum information processing, particularly in
randomized benchmarking and random compiling. While protocols based on Pauli
twirling have been effectively crafted to transform arbitrary noise channels
into Pauli channels for Clifford gates -- thereby facilitating efficient
benchmarking and mitigating worst-case errors -- practical twirling groups for
multi-qubit non-Clifford gates are lacking. In this work, we study the issue of
finding twirling groups for generic quantum gates within a widely used circuit
structure in randomized benchmarking or random compiling. For multi-qubit
controlled phase gates, which are essential in both the quantum Fourier
transform and quantum search algorithms, we identify optimal twirling groups
within the realm of classically replaceable unitary operations. In contrast to
the simplicity of the Pauli twirling group for Clifford gates, the optimal
groups for such gates are much larger, highlighting the overhead of tailoring
noise channels in the presence of global non-Clifford gates.
