Stabilizer circuits, i.e. quantum circuit that only use the $H$, $\sqrt{Z}$, and $CNOT$ gates, can only implement a finite subset of all possible unitary operations.

I want to iterate over all these operations for a small number of qubits, or sample from the space for a larger number of qubits.

I could just use the stabilizer circuits themselves as stands ins for the operations: iterate over the possible circuits with up to $n$ operations and $k$ qubits, sample from the possible circuits, etc. But many circuits encode the same operation, so focusing on the circuits would create extra work and bias the sampling. Also there's not a clear length cut-off.

What's a straightforward way to generate all the operations in the group? To sample from the group?

up vote 3 down vote accepted

Since stabilizer circuits are exactly those which map stabilizers to stabilizers, you could characterize them via their action on a basis of stabilizers, see for instance Sec 2.3 of http://home.lu.lv/~sd20008/papers/essays/Clifford%20group%20[paper].pdf.

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