The Clifford group forms a unitary 3-design

TL;DR

This paper proves that the Clifford group forms a unitary 3-design, providing a better approximation to Haar-random unitaries than previously known, with implications for quantum information processing.

Contribution

The paper establishes that the Clifford group is a 3-design, a significant advancement in understanding its approximation to Haar-random unitaries, and characterizes its limitations as a 4-design.

Findings

Clifford group is a 3-design, better approximating Haar randomness.

Clifford group does not form a 4-design.

Generalized Clifford group for qudits is not a 3-design unless dimension is a power of 2.

Abstract

Unitary $k$-designs are finite ensembles of unitary matrices that approximate the Haar distribution over unitary matrices. Several ensembles are known to be 2-designs, including the uniform distribution over the Clifford group, but no family of ensembles was previously known to form a 3-design. We prove that the Clifford group is a 3-design, showing that it is a better approximation to Haar-random unitaries than previously expected. Our proof strategy works for any distribution of unitaries satisfying a property we call Pauli 2-mixing and proceeds without the use of heavy mathematical machinery. We also show that the Clifford group does not form a 4-design, thus characterizing how well random Clifford elements approximate Haar-random unitaries. Additionally, we show that the generalized Clifford group for qudits is not a 3-design unless the dimension of the qudit is a power of 2.