  Twirling noise affecting quantum gates is essential in understanding and
controlling errors, but applicable operations to noise are usually restricted
by symmetries inherent in quantum gates. In this Letter, we propose symmetric
Clifford twirling, a Clifford twirling utilizing only symmetric Clifford
operators that commute with certain Pauli subgroups. We fully characterize how
each Pauli noise is converted through the twirling and show that certain Pauli
noise can be scrambled to a noise exponentially close to the global white
noise. We further demonstrate that the effective noise of some highly
structured circuits, such as Trotterized Hamiltonian simulation circuits, is
scrambled to global white noise, and even a single use of CNOT gate can
significantly accelerate the scrambling. These findings enable us to mitigate
errors in non-Clifford operations with minimal sampling overhead in the early
stages of fault-tolerant quantum computing, where executing non-Clifford
operations is expected to be significantly more challenging than Clifford
operations. Furthermore, they offer new insights into various fields of physics
where randomness and symmetry play crucial roles.
