A simple method for sampling random Clifford operators

We describe a simple algorithm for sampling n-qubit Clifford operators uniformly at random. The algorithm outputs the Clifford operators in the form of quantum circuits with at most 5n + 2n² elementary gates and a maximum depth of ${\mathcal{O}}\left({n\,{\text{log}}\,n}\right)$ on fully connected topologies. The circuit can be output in a streaming fashion as the algorithm proceeds, and different parts of the circuit can be generated in parallel. The algorithm has an ${\mathcal{O}}\left({{n^2}}\right)$ time complexity, which matches the current state of the art. The main advantage of the proposed algorithm, however, lies in its simplicity and elementary derivation.