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arXiv · 1406.2170

How to efficiently select an arbitrary Clifford group element

Robert Koenig·John A. Smolin
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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 $O(n^3)$.

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Robert Koenig, John A. Smolin. 2014-06-09. How to efficiently select an arbitrary Clifford group element. https://doi.org/10.1063/1.4903507

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