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A unitary t-design is a set of unitaries that is "evenly distributed" in thesense that the average of any t-th order polynomial over the design equals theaverage over the entire unitary group. In various fields -- e.g. quantuminformation theory -- one frequently encounters constructions that rely onmatrices drawn uniformly at random from the unitary group. Often, it sufficesto sample these matrices from a unitary t-design, for sufficiently high t. Thisresults in more explicit, derandomized constructions. The most prominentunitary t-design considered in quantum information is the multi-qubit Cliffordgroup. It is known to be a unitary 3-design, but, unfortunately, not a4-design. Here, we give a simple, explicit characterization of the way in whichthe Clifford group fails to constitute a 4-design. Our results show that forvarious applications in quantum information theory and in the theory of convexsignal recovery, Clifford orbits perform almost as well as those of true4-designs. Technically, it turns out that in a precise sense, the 4th tensorpower of the Clifford group affords only one more invariant subspace than the4th tensor power of the unitary group. That additional subspace is a stabilizercode -- a structure extensively studied in the field of quantum errorcorrection codes. The action of the Clifford group on this stabilizer code canbe decomposed explicitly into previously known irreps of the discretesymplectic group. We give various constructions of exact complex projective4-designs or approximate 4-designs of arbitrarily high precision from Cliffordorbits. Building on results from coding theory, we give strong evidencesuggesting that these orbits actually constitute complex projective 5-designs.

The Clifford group fails gracefully to be a unitary 4-design