How to efficiently select an arbitrary Clifford group element

Abstract

We give an algorithm which produces a unique element of the Clifford group on n qubits ( Cn) from an integer 0 ≤ i < C n (the number of elements in the group). The algorithm involves O(n3) operations and 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 random elements of C n which are 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(n3).