This noise can be viewed as extra probabilistic gates on top of the perfect Clifford gates. As we try to obtain the error thresholds of the codes, we often need to introduce various forms of noise into the circuits based on the underlying physical implementations. For each code, there exists an error threshold of the circuit components below which the computational error can be made arbitrarily small by scaling up the code. One important example is the circuits used to implement quantum error correction codes. The Gottesman-Knill theorem 13, 14 states that any quantum circuits involving only Clifford gates can be perfectly simulated in polynomial time on a classical computer. In this article, twirling is discussed as a technique for simulating noise and the impact of the noise on the performance of quantum error correction codes 12. More recently, twirling was used as means to boost the performance of NISQ through error mitigations 8, 9, 10, 11 in which it enables a controlled increase of the gate error rates for error extrapolations.

Then it appeared again as an integral part in randomised benchmarking 3, 4 and was also used to reduce the number of experimental runs needed in quantum process tomography 5, 6, both are critical in benchmarking the performance of quantum systems, especially “Noisy Intermediate-Scale Quantum” (NISQ) systems 7.

It was first used for mapping a diverse range of states into a canonical form in entanglement purification 1, 2. Twirling is a technique that has been long established in the quantum information literature.

Clifford group quantum error correction