# What is the Clifford gates selection probability distribution used in the generation of randomized benchmarking circuits?

I've read that in standard randomized benchmarking implementations the random quantum circuits are generated through random gate selection from a uniformly distributed Clifford set of either 1 or 2-qubit gates i.e. {H, X, Y, Z, S, T} and {iSWAP, CZ}, however the circuits generated using randomized_benchmarking_seq do not seem to follow this for the following reasons:

1. Random circuits do NOT contain T, iSWAP and CZ gates
2. Random circuits do contain CX gates
3. There does not appear to be a uniform distribution in gate selection, for example gates H and S gates appear more frequently than X, Y, Z and CX gates

Is this observation correct? and if so what is the actual distribution for randomized_benchmarking_seq? and how can uniformly distributed random circuits be generated?

Random Clifford elements are generated using the following function:

https://qiskit.org/documentation/stubs/qiskit.quantum_info.random_clifford.html

Then they are synthesized into basic gates (CX, S, Sdg, H, Paulis) using the following function:

https://qiskit.org/documentation/stubs/qiskit.quantum_info.Clifford.to_circuit.html#qiskit.quantum_info.Clifford.to_circuit

This is based on the following paper:

S. Bravyi and D. Maslov, Hadamard-free circuits expose the structure of the Clifford group. https://arxiv.org/abs/2003.09412

For a simpler explanation on the structure of the 1-qubit and 2-qubit Clifford groups see the following tutorial: