The Clifford group fails gracefully to be a unitary 4design
Abstract
A unitary tdesign is a set of unitaries that is "evenly distributed" in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group. In various fields  e.g. quantum information theory  one frequently encounters constructions that rely on matrices drawn uniformly at random from the unitary group. Often, it suffices to sample these matrices from a unitary tdesign, for sufficiently high t. This results in more explicit, derandomized constructions. The most prominent unitary tdesign considered in quantum information is the multiqubit Clifford group. It is known to be a unitary 3design, but, unfortunately, not a 4design. Here, we give a simple, explicit characterization of the way in which the Clifford group fails to constitute a 4design. Our results show that for various applications in quantum information theory and in the theory of convex signal recovery, Clifford orbits perform almost as well as those of true 4designs. Technically, it turns out that in a precise sense, the 4th tensor power of the Clifford group affords only one more invariant subspace than the 4th tensor power of the unitary group. That additional subspace is a stabilizer code  a structure extensively studied in the field of quantum error correction codes. The action of the Clifford group on this stabilizer code can be decomposed explicitly into previously known irreps of the discrete symplectic group. We give various constructions of exact complex projective 4designs or approximate 4designs of arbitrarily high precision from Clifford orbits. Building on results from coding theory, we give strong evidence suggesting that these orbits actually constitute complex projective 5designs.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.08172
 Bibcode:
 2016arXiv160908172Z
 Keywords:

 Quantum Physics
 EPrint:
 49 pages, many representations. Not. A. Single. Figure. See also related work by Helsen, Wallman, and Wehner