Multiqubit Clifford groups are unitary 3-designs

QuICS Special Seminar

Huangjun Zhu (ITP, Cologne)
Thursday, May 19, 2016 - 2:00pm
CSS 3100A

We show that the multiqubit (including qubit) Clifford group in any even prime power dimension is not only a unitary 2-design, but also a unitary 3-design. Moreover, it is a minimal unitary 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. By contrast, the Clifford group in any odd prime power dimension is only a unitary 2-design. In addition, we show that no operator basis is covariant with respect to any group that is a unitary 3-design, thereby providing a simple explanation of why no discrete Wigner function is covariant with respect to the multiqubit Clifford group. This result is of interest to understanding the power of quantum computation. Finally, I will mention briefly the decomposition of the fourth tensor power of the Clifford group and construction of Clifford covariant 4-designs as well as potential applications in compressed sensing and quantum process tomography etc.