How to efficiently select an arbitrary Clifford group element

Abstract

We give an algorithm which produces a unique element of the Clifford group ${\mathcal{C}}_n$ on $n$ qubits from an integer $0\le i<|{\mathcal{C}}_n|$ (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 $Sp(2n,{\mathbb{F}}_2)$ 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 random elements of ${\mathcal{C}}_n$ 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 $O(n^3)$.

7 pages plus 4 1/2 pages of python code

References in corpus (3)

• Randomized Benchmarking of Quantum Gates
• Robust randomized benchmarking of quantum processes
• Instantaneous non-local computation of low T-depth quantum circuits

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