Multiqubit Clifford groups are unitary 3-designs

TL;DR

This paper proves that multiqubit Clifford groups in even prime-power dimensions are unitary 3-designs, revealing their minimality and implications for quantum information, phase space physics, and quantum computation.

Contribution

It demonstrates that multiqubit Clifford groups are unitary 3-designs and explores their minimality, connections to phase space physics, and implications for quantum information science.

Findings

Multiqubit Clifford groups are unitary 3-designs in even prime-power dimensions.

Orbit of pure states of Clifford group forms a complex projective 3-design.

No discrete Wigner function is covariant with respect to the Clifford group.

Abstract

Unitary $t$-designs are a ubiquitous tool in many research areas, including randomized benchmarking, quantum process tomography, and scrambling. Despite the intensive efforts of many researchers, little is known about unitary $t$-designs with $t\geq3$ in the literature. We show that the multiqubit Clifford group in any even prime-power dimension is not only a unitary 2-design, but also a 3-design. Moreover, it is a minimal 3-design except for dimension~4. As an immediate consequence, any orbit of pure states of the multiqubit Clifford group forms a complex projective 3-design; in particular, the set of stabilizer states forms a 3-design. In addition, our study is helpful to studying higher moments of the Clifford group, which are useful in many research areas ranging from quantum information science to signal processing. Furthermore, we reveal a surprising connection between unitary 3-designs and the physics of discrete phase spaces and thereby offer a simple explanation of why no discrete Wigner function is covariant with respect to the multiqubit Clifford group, which is of intrinsic interest to studying quantum computation.