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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?

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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:

https://github.com/qiskit-community/qiskit-community-tutorials/blob/master/terra/qis_adv/Clifford_Group.ipynb

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