I was reading papers on Randomized Benchmarking, such as this and this. (more specifically, equation 30 in the second paper)
It appears to be some kind of averaging but I would like to have a more intuitive and physical picture of what it actually represents in terms of measurements.
(I know very little math, especially in terms of group representation)
I think the way that it is used in deriving bounds on quantum cloning is quite insightful, and hopefully gives you a flavour of the broader context. Where possible I'll skip some of the details in favour of description.
Imagine you want to clone an unknown one-qubit quantum state $|\psi\rangle$, making two copies. We know that this is impossible to do perfectly, but you might want to quantify how close you can get. So, let's theorise that there's some map $\rho=\mathcal{E}(|\psi\rangle\langle\psi|)$ that you're going to implement, producing a two-qubit output. The cloning fidelity in this case is $$ F=\text{Tr}\left((|\psi\rangle\langle\psi|\otimes\mathbb{I}+\mathbb{I}\otimes|\psi\rangle\langle\psi|)\rho\right)/2 $$ Skipping over the details, including the Choi-Jamiolkowski isomorphism, it turns out that you can relate the optimal choice of $\rho$ to the eigenvector with maximum eigenvalue of the operator $$ |\psi\rangle\langle\psi|^\star\otimes(|\psi\rangle\langle\psi|\otimes\mathbb{I}+\mathbb{I}\otimes|\psi\rangle\langle\psi|). $$ Actually, this is straightforward to calculate: the maximum eigenvector is $|\psi\rangle^\star|\psi\rangle|\psi\rangle$, and has eigenvalue 1. i.e. the operation can be done perfectly. However, this formulation presupposes that we know the state we're trying to clone (and obviously, if you know it, you can make arbitrarily many copies). We somehow have to quantify the fact that we don't know what the state is to be cloned.
For this, we need to take a Bayesian approach. If we don't know the state, we assign a probability to each of the possible states. In that case, the expected fidelity of the transformation is $\bar F$, the maximum eigenvalue of the operator
$$
R=\sum_{\psi}p_{\psi}|\psi\rangle\langle\psi|^\star\otimes(|\psi\rangle\langle\psi|\otimes\mathbb{I}+\mathbb{I}\otimes|\psi\rangle\langle\psi|)
$$
You might know something about the possible input state. For example, if you know it's either $|0\rangle$ or $|1\rangle$ with equal probability, you still find the maximum eigenvalue of 1 (which makes sense: you can measure the input state in the Z basis and make as many copies as you want). At the other extreme, if you know nothing about the state at all, you have to average over all single-qubit pure states with equal weights. What is the right way to do this? It's a bit messy as you can perhaps see from looking at the Bloch sphere:
We need to take an average of all points on the surface of the sphere.
One way to achieve this is to say that every $|\psi\rangle=U|0\rangle$ for some $U$, and we average over all possible unitaries, where the correct averaging is determined by the Haar measure. So, we can change our operator $R$ into $$ R=\int U^\star\otimes U\otimes U\left(|0\rangle\langle0|^\star\otimes(|0\rangle\langle0|\otimes\mathbb{I}+\mathbb{I}\otimes|0\rangle\langle0|)\right)U^T\otimes U^\dagger\otimes U^\dagger dU $$ This is basically the twirling operation (except for the slight technicality that there's a complex conjugate on one of the qubits), and certainly captures the intuition of how/why it comes into things.
An average over conjugations is known as a “twirl”. The “twirling” operation originates from invariant theory (where it is sometimes called “transfer homomorphism”). Twirling a quantum channel over $P_1^{⊗n}$, $C_1^{⊗n}$ or $C_n$ takes it to one described by a polynomial number of parameters.
The twirling operation will be useful if it preserves, at least partially, properties of the original channel. Specifically, one would hope that correctable codes of the twirled channel resemble those of the original channel.
See page 31 of "Gaining Information About a Quantum Channel Via Twirling" (Jul 31 2008) by Easwar Magesan.
More in depth references:
"Characterization of addressability by simultaneous randomized benchmarking" (Jan 2 2013), by Jay M. Gambetta, A. D. Corcoles, S. T. Merkel, B. R. Johnson, John A. Smolin, Jerry M. Chow, Colm A. Ryan, Chad Rigetti, S. Poletto, Thomas A. Ohki, Mark B. Ketchen, and M. Steffen.
"Evenly distributed unitaries: on the structure of unitary designs" (May 13 2007), by D. Gross, K. Audenaert, and J. Eisert.
"Exact and Approximate Unitary 2-Designs: Constructions and Applications" (Aug 31 2012), by Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine.
