# RB with Clifford: can I define my own random model?

I'd like to know how much freedom I have when setting up an RB Clifford model.

Is any Clifford circuit sample ok for benchmarking (as long as it characterizes the Clifford group)?

What are the implication of setting up my own way of generating a (uniformly) random sample?

• Can you add more details to this? I am a bit confused about what you are trying to do. The way I understand randomized benchmarking, you build a circuit using the Clifford group such that ideally the gates evaluate to a known result. For example, a circuit of depth d has randomly used Clifford gates for the ideal execution to be an identity. In the presence of noise, it is most definitely not going to be an identity. Commented Aug 1, 2023 at 15:47

## 1 Answer

The core idea of RB is the fact that twirling over the unitary group produces a depolarizing channel, which is what we fit to from the experimental data. However, implementing Haar-random unitaries has some experimental limitations, for example scaling up to larger system sizes (it is known that a Haar-random unitary on $$n$$ qubits requires an exponentially large circuit). Fortunately, the Clifford group is what is known as a unitary 2-design (in fact, a 3-design), which means that it can reproduce the random properties of the Haar measure on the full unitary group up to the 2nd (in fact, 3rd) moment of the distribution. Importantly, this implies that twirling over the Clifford group also produces the same depolarizing channel as if one has twirled over the full unitary group instead. And arbitrary Clifford circuits can be compiled in polynomial circuit complexity, thereby escaping the exponential bottleneck of Haar-random unitaries.

If you were to sample from a distribution other than the uniform Clifford group, then you have to be careful about what this new distribution actually is. If it is a unitary 2-design as well, then you can proceed with RB theory as usual. If not, then the twirl may produce a channel which is not a simple depolarizing channel, and so your fitting procedure would have to be modified appropriately. Such variants are studied using tools from group theory and representation theory, and in fact a number of offshoots of "standard" RB exist which twirl over different groups/distributions. For example, see Real RB, Character benchmarking, and Matchgate benchmarking.

If you still wish to sample from the Clifford group and do standard RB, but are asking specifically how one draws random Clifford elements, then there are many algorithms for doing so. For example, see this paper by Bravyi and Maslov. Regardless of how you compile the circuits, as long as you are generating Clifford circuits uniformly at random then RB will work as expected.

• Thank you so much for being this detailed! Would you mind expand the meaning of "2-design"? Commented Aug 16, 2023 at 15:57
• A unitary 2-design is a distribution of unitaries such that its 2nd moment is equal to that of the unitary group (under the Haar probability measure). There are a number of equivalent mathematical definitions, see for example the Wikipedia page, Olivia Di Matteo's notes, or this tutorial by Pennylane. Commented Aug 22, 2023 at 19:35