How to efficiently select an arbitrary Clifford group element

TL;DR

This paper presents an efficient algorithm for uniquely mapping integers to elements of the Clifford group on n qubits, enabling rapid random selection and decomposition of group elements, which benefits quantum information processing.

Contribution

The authors adapt the subgroup algorithm for the Clifford group and symplectic group, providing a canonical mapping and factorization method with polynomial complexity.

Findings

Algorithm produces unique Clifford group element from integer in O(n^3)

Provides a factorization into symplectic transvections with at most 4n steps

Enables efficient random sampling and inverse mapping of group elements

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 $O(n^3)$.