How do we actually go about sampling from a unitary 2-design? Because the size of the 2-design grows quickly with the number of qubits, it seems challenging to sample.

Some of the references I've found mostly focus on utilizing Hamiltonian dynamics to create the state or discuss how to create a random unitary on a classical computer.


1 Answer 1


Unitary 2-designs without efficient sampling access are arguably not very useful. Indeed, if you go by the textbook definition, then a unitary design is nothing but a set of unitaries and there's generally no efficient way of sampling from this set due to its size.

That is one reason why the Clifford group is used that often. It is a unitary 2-design which has a group structure, its group operations are efficiently computable, and there's an efficient sampling algorithm (with runtime $O(n^2)$).

However, for moderately sized systems, Clifford unitaries may yet be too noisy to be useful, due to their linear circuit depth requirements. A more flexible alternative are random quantum cirucits (RQCs). These are usually efficient to sample from, and they form approximate unitary designs where the approximation error can be controlled by the circuit depth. This is usually enough for most (all) applications. Thus, RQCs are a good and practically simple option when randomisation is needed. However, it is not a priori clear whether the required circuit depth for RQCs is indeed lower than for Clifford unitaries. In general, this depends on the application.


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