Multiqubit Clifford groups are unitary 3designs
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 ≥3 in the literature. We show that the multiqubit Clifford group in any even primepower dimension is not only a unitary 2design, but also a 3design. Moreover, it is a minimal 3design except for dimension 4. As an immediate consequence, any orbit of pure states of the multiqubit Clifford group forms a complex projective 3design; in particular, the set of stabilizer states forms a 3design. In addition, our study is helpful in 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 3designs 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 in studying quantum computation.
 Publication:

Physical Review A
 Pub Date:
 December 2017
 DOI:
 10.1103/PhysRevA.96.062336
 arXiv:
 arXiv:1510.02619
 Bibcode:
 2017PhRvA..96f2336Z
 Keywords:

 Quantum Physics;
 Mathematical Physics
 EPrint:
 7 pages, published in Phys. Rev. A