What should we add?

The current generation of randomized circuits is fairly inflexible as it only allows to specify a limited number of circuit characteristics: the number of qubits, depth, maximum number of qubits in a quantum gate, and a couple of options allowing for measurements, conditional operations or resets in the random circuit.
Ideally, the generation of random circuits would allow more parameters as an input, e.g. the structure of a circuit, the amount of parameterizable quantum gates, or a finer definition on the permissible types of quantum gates. I would like to use this issue to first discuss which parameters would be reasonable inputs and then track the progress on this.

I have a couple of parameters in mind:

• Density/sparsity
• Qubit connectivity, i.e. all-to-all connectivity, a line, 2D grid, random, clustered, sparse, dense, etc
• Distribution of parameterizable quantum gates to non-parameterizable quantum gates
• Distribution of gates not in a basis gate set
• Distribution of (non-)commuting gates
• Distrbution over n-qubit gates with n in N
• ....
  But please add any you would find useful.