How to efficiently select an arbitrary Clifford group element

We give an algorithm which produces a unique element of the Clifford group C_n on n qubits from an integer 0\le i < |C_n| (the number of elements in the group). The algorithm involves O(n^3) operations. It is a variant of the subgroup algorithm by Diaconis and Shahshahani which is commonly applied to compact Lie groups. We provide an adaption for the symplectic group Sp(2n,F_2) which provides, in addition to a canonical mapping from the integers to group elements g, a factorization of g into a sequence of at most 4n symplectic transvections. The algorithm can be used to efficiently select randomelements of C_n which is often useful in quantum information theory and quantum computation. We also give an algorithm for the inverse map, indexing a group element in time O(n^3).

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